University of Southern California Dissertations and Theses

Some Properties Of String Transducers

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Some Properties Of String Transducers

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Content SO M E P R O P E R T IE S O F STRIN G TRA N SD U CERS by R U S SE L L JO S E P H A B B O T T A D is s e r ta tio n P r e s e n te d to th e F A C U L T Y O F T H E G RA D U A TE SCH OO L U N IV ER SITY O F SO U TH ER N C A LIFO RN IA In P a r t i a l F u lfillm e n t of th e R e q u ire m e n ts fo r th e D e g re e D OCTOR O F PH IL O SO PH Y (C o m p u te r S cien ce) S e p te m b e r 1973 INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1.T he sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication th a t the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. 5. PLEASE NOTE: Some pages may have indistinct print. Filmed as received. Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 ABBOTT, Russell Joseph, 1942- SGME PROPERTIES OF STRING TRANSDUCERS. University of Southern California, Ph.D., 1973 Computer Science University Microfilms, A X E R O X Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. UNIVERSITY O F SO U TH ER N CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES. CALI FORNIA 9 0 0 0 7 This dissertation, written by .............RUSSELL JOSEPH .•A?.?.9.T.T under the direction of his.... Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment of requirements of the degree of D O C T O R OF P H IL O S O P H Y £ Dean DISSERTATION COMMITTEE P -yfl * Chairman ACKNOW LEDGEMENT I w ish to e x p r e s s m y g ra te fu l a p p re c ia tio n to m y A d v is o r an d C h a irm a n o f m y C o m m itte e , P r o f e s s o r . P . G ilb e rt fo r h is e n c o u ra g e m e n t, p a tie n c e , s u p p o rt and g u id a n c e . I a ls o w ish to th a n k P r o f e s s o r W . J . C h a n d le r and P r o f e s s o r A . M a rs to n , th e o th e r m e m b e rs of m y d is s e r ta tio n c o m m itte e , fo r th e ir kind w is h e s an d good f e e lin g s . I a m g ra te fu l to th e C h a irm a n of th e C o m p u te r S c ie n c e P r o g r a m a t th e U n iv e rs ity of S o u th e rn C a lifo rn ia , P r o f e s s o r J . M u n u sh ian fo r th e a s s is ta n c e h e p ro v id e d m e . To M r s . K ris P e n d le to n g o es m y w a rm g ra titu d e fo r h e r ty p in g and frie n d s h ip an d to M r s . E s th e r D illo n m y th a n k s fo r a ll h e r a s s is ta n c e . T h is w o rk w a s s u p p o rte d in p a r t by th e O ffice of N av al R e s e a r c h C o n tr a c t N o . N 0 0 0 1 4 -6 7 -A -0 2 6 9 -0 0 2 0 A m m e n d m e n t 2 . A B ST R A C T T h is p a p e r c o m p a re s th e c o m p u ta tio n a l p o w er o f v a rio u s c l a s s e s of t r a n s d u c e r s . A tr a n s d u c e r is an a u to m a to n w h ich tr a n s f o r m s a n in p u t s tr in g in to an o u tp u t s tr in g , w h e re b o th th e in p u t and o u tp u t s tr in g s c o n s is t of sy m b o ls f r o m so m e fin ite a lp h a b e t of s y m b o ls . A t r a n s d u c e r p e r f o r m s its c o m p u ta tio n s on a w o rk in g ta p e of d is c r e te s q u a r e s . E a c h s q u a r e m a y c o n ta in one sy m b o l fro m a fin ite w o rk in g a lp h a b e t. A tr a n s d u c e r M is s a id to b e a m e m b e r o f th e c la s s ^ (L (n )) of L(n) s p a c e b o u n d ed tr a n s d u c e r s if, w h e n e v e r M c o m p le te s a c o m p u ta tio n on an in p u t of le n g th n, th e n M m a k e s u se of no m o re th a n L(n) s q u a r e s of w o rk in g ta p e . I t is show n th a t th e c la s s ty?(log(n)) is c lo s e d u n d e r fin ite c o m p o s itio n . A m u lti- p a s s tr a n s d u c e r is th e c o m p o sitio n o f a tr a n s d u c e r w ith it s e lf so m e n u m b e r o f ti m e s . A m u lti- p a s s tr a n s d u c e r M is c a lle d P(n) p a s s b o u n d ed if, w h e n e v e r M h a lts on an in p u t of le n g th n, th e n M m a k e s no m o re th a n P(n) p a s s e s on th e in p u t. T h e c la s s of tr a n s d u c e r s w h ich a r e b o th P(n) p a s s b o u n d ed a s w e ll a s L(n) s p a c e bou n d ed is d e s ig n a te d ?ft(P(n), L (n )). I t is show n th a t ?7?(L(n)) c ff?(P(n), lo g (n )), w h e re P (n) = lo g (L (n )/lo g (n )). A d d itio n a l r e s u lt s a r e p re s e n te d fo r sin g le p a s s tr a n s d u c e r s . A s p a c e bound L(n) is s a id to b e " s e lf-c o m p u ta b le " if th e re is an L(n) sp a c e bo u n d ed tr a n s d u c e r w h ich m a y b e c o n s tru c te d to " m a rk iii off" its w o rk in g ta p e a r e a . L e t L(n) b e s e lf-c o m p u ta b le , m o n o to n e in c re a s in g an d u n b o u n d ed . It is show n th a t fo r e a c h r e c u r s iv e ly e n u m e ra b le s e t A th e r e is a n M 6 ty?(L(n)) su c h th a t A is th e ra n g e of th e fu n ctio n d e fin e d b y M . A lg e b ra ic s u ffic ie n c y c o n d itio n s fo r c e r ta in ty p e s of c lo s u r e a r e g iv e n . L e t A b e a c la s s o f tr a n s d u c e r s and le t 3~(A) b e th e c la s s of s e ts a c c e p te d b y tr a n s d u c e r s in A . (A s e t is s a id to b e a c c e p te d b y a tr a n s d u c e r if it is th e d o m a in o f th e fu n c tio n d efin e d b y th a t tr a n s d u c e r .) I t is show n th a t: (1) c lo s u r e o f A) u n d e r p r e im a g e b y m e m b e rs of A fo llo w s fro m c lo s u r e of A u n d e r co m p o s itio n w ith m e m b e rs of A; and (2) c lo s u r e of .7(A) u n d e r tr a n s - d u c tio n b y m e m b e rs of A fo llo w s fro m c lo s u r e o f A u n d e r co m p o s itio n w ith in v e r s e s of m e m b e rs of A . A s p a c e b o u n d ed pushdow n a u to m a to n is a s p a c e bo u n d ed a u to m a to n w ith th e a d d itio n a l s to r a g e c a p a b ilitie s o f a n unbounded pushdow n s t o r e . It h a s b e e n a n open q u e s tio n w h e th e r th e L(n) s p a c e b o u n d ed p u shdow n a u to m a ta a r e m o r e p o w e rfu l th a n th e L(n) s p a c e b o u n d ed a u to m a ta w ith o u t th e p u shdow n s to r e [ l ] . It is show n th a t th e L(n) s p a c e b o u n d ed pushdow n tr a n s d u c e r s d e fin e m o re fu n c tio n s th a n th e L(n) s p a c e b o u n d ed tr a n s d u c e r s w ith o u t p u sh dow n s t o r e s . T A B L E O F C O N T EN TS P a g e A C K N O W L E D G E M E N T ............................................................................... ii A B S T R A C T ............................................................................................... . . iii C H A P T E R 1 IN TR O D U C TIO N AND P R E LIM IN A R IES ................................................................... I C H A P T E R 2 M U L T IP L E PA SSES W ITH S P A C E BO U N D ED TR A N SD U C ER S.......................................... 11 2 .1 T r a n s d u c e r s : B a s ic D e fin itio n s . . . . 11 2 .2 C lo s u re of th e C la s s of log(n) T r a n s d u c e r s u n d e r C o m p o s itio n . . . . 19 2 .3 M u lti- P a s s T r a n s d u c e rs : B a s ic D e f in it io n s ...............................................................31 2 .4 S u b stitu tin g P a s s e s fo r S p a c e ......................... 38 2 .5 M u ltip le P a s s e s a s a C o m p lex ity M e a s u r e .......................................................... 46 C H A P T E R 3 SIN G L E PASS T R A N S D U C E R S ...................................54 3 .1 T r a n s d u c e rs and R e c u rs iv e E n u m e r a b i l i t y ........................ 55 3 .2 C lo s u re R e s u l t s ....................................................60 3 .3 T im e , S p ace an d P u sh d o w n S t o r e s .......................................................................67 B IB L IO G R A P H Y ...................................................................... 74 C H A P T E R 1 IN TR O D U C TIO N AND PR E L IM IN A R IE S T h is stu d y o rig in a te d w ith a tte m p ts to e x p lo re new m o d e ls fo r th e p ro c e s s in g of A lg o l-lik e la n g u a g e s . It w a s d is c o v e re d th a t m u lti-h e a d a u to m a ta [ 6 ] , [1 2 ], [1 5 ], [2 3 ], [2 4 ], [2 6 ], [2 7 ], [29] co u ld re c o g n iz e v a rio u s im p o rta n t s u b c la s s e s of th e c o n te x t f r e e la n g u a g e s an d th a t th e s e s u b c la s s e s in c lu d e d th e c o n te x t f r e e e le m e n ts of A lg o l-lik e la n g u a g e s [2 2 ], In a d d itio n , m u lti-h e a d a u to m a ta h a d th e c o m p u tin g p o w er to p r o c e s s c e r ta in f e a tu r e s found in h ig h e r le v e l la n g u a g e s , b u t w h ic h w e r e n o t c o n te x t f r e e in n a tu re , [7 ], [1 6 ], [1 7 ]. M u ltip le P a s s e s w ith S p a c e B ounded T r a n s d u c e rs A n in v e s tig a tio n in to m u lti-h e a d a u to m a ta e n s u e d . It w a s d e te rm in e d th a t th e m u lti-h e a d a u to m a ta w e r e e q u iv a le n t in p o w er to th e log(n) s p a c e b o u n d ed a u to m a ta [1 8 ], and th e r e s e a r c h b r o a d e n e d to in c lu d e s p a c e b o u n d ed a u to m a ta in g e n e r a l [2 8 ], w h en a p p r o p r ia t e . T h e o r ig in a l a r e a of r e s e a r c h , la n g u a g e p ro c e s s in g , s u g g e s te d a n in v e s tig a tio n o f m u ltip le p a s s e s w ith s p a c e b ounded a u to m a ta . C an m u ltip le p a s s e s p ro v id e m o r e c o m p u ta tio n a l p o w er th a n s in g le p a s s e s ? A nd if so , how m u c h m o r e p o w er fo r how m a n y m o r e p a s s e s [10] ? 1 2 T h e th e o ry of c o m p u ta tio n a l c o m p le x ity in v e s tig a te s h i e r a r c h ie s of a u to m a ta fo r w h ic h p a r tic u la r r e s o u r c e s a r e lim ite d in c e r ta in w a y s [ 2 ] , [ 3 ] , [4 ], [2 8 ], W hat s o r ts o f c o m p u ta tio n s a r e p o s s ib le if m e m o ry s p a c e is lim ite d by a fu n c tio n o f th e le n g th of th e in p u t? C o m p u ta tio n a l c o m p le x ity t h e o r i s t s ; a ls o stu d y th e tr a d e - o f f s b e tw e e n d iffe re n t ty p e s of c o m p u ta tio n a l r e s o u r c e s [4 ], [2 5 ], [1 5 ], R e c e n tly , fo r e x a m p le , P a te r s o n show ed th a t an y th in g c o m p u ta b le in t(n) s te p s , w h e re n is th e le n g th o f th e in p u t, could b e c o m p u te d in '/"t(n) s p a c e [2 0 ], In th is stu d y it is show n how to tr a d e p a s s e s fo r s p a c e . T h a t is , w e show th a t a c o m p u ta tio n th a t r e q u ir e s L(n) s p a c e , w h e re n is th e le n g th o f th e in p u t, c a n be p e rfo rm e d in o nly log(n) s p a c e by a m u ltip le p a s s tr a n s d u c e r w h ich ta k e s lo g (L (n )/lo g (n )) p a s s e s . O th e r T r a n s d u c e rs T h e stu d y o f s p a c e b o u n d ed a u to m a ta b ro a d e n e d in a se c o n d d ire c tio n : th e a d d itio n o f a n a u x ilia r y p u sh dow n s to r e [ 1 ] , [ 8 ] , [11], [1 2 ], [1 3 ], [1 5 ], T h e re h a v e b e e n m a n y in v e s tig a tio n s in to th e r e la tio n s h ip b e tw e e n a u to m a ta w ith p u sh dow n s to r e s and a u to m a ta w ith o u t th is ty p e of m e m o ry [4] , [1 4 ], It is a n o p en q u e s tio n w h e th e r th e a d d itio n of a p u sh dow n s to r e to a s p a c e b o u n d ed a u to m a to n ad d s a n y c o m p u ta tio n a l p o w e r [ 1 ] , [4 ], [1 1 ]. In th is stu d y w e g ive a p a r ti a l a n s w e r . O th e r q u e s tio n s a b o u t s p a c e bou n d ed a u to m a ta a r e a ls o of i n t e r e s t . T he r e la tio n s h ip b e tw e e n th e sp a c e b o u n d ed tr a n s d u c e r s 3 and th e r e c u r s iv e ly e n u m e ra b le s e ts [5 ], [1 6 ], [1 9 ], [2 1 ], [2 7 ], is stu d ie d a s w e ll a s d e c is io n p ro b le m s ab o u t sp a c e bounded t r a n s d u c e rs [2 3 ], In a d d itio n , th is stu d y a ls o e x a m in e s c lo s u r e p r o p e r tie s of s p a c e bou n d ed tr a n s d u c e r s [16] an d c lo s u r e p r o p e r tie s th e m s e lv e s a r e s tu d ie d in so m e g e n e r a lity . O v e rv ie w o f R e s u lts T he r e s u lts of th is stu d y a r e p r e s e n te d in C h a p te rs 2 and 3 . C h a p te r 2 d e v e lo p s th e th e o ry o f m u lti- p a s s tr a n s d u c e r s . C h a p te r 3 p r e s e n ts n ew r e s u lt s ab o u t (s in g le p a s s ) s p a c e bou n d ed t r a n s d u c e rs . S e c tio n 2 .2 sh o w s th a t th e c la s s o f fu n c tio n s c o m p u ta b le by log(n) s p a c e b o u n d ed tr a n s d u c e r s is c lo s e d u n d e r c o m p o s itio n . T h is r e s u l t p ro v id e s a c o n s ta n t s p e e d up th e o re m fo r log(n) m u lti- p a s s tr a n s d u c e r s . It is a ls o o f i n t e r e s t in i t s e l f . S e c tio n 2 .3 in tro d u c e s th e n o tio n of th e " c h a r a c t e r is ti c c l a s s . " A c h a r a c te r i s t ic c la s s fo r a c la s s of s in g le o r m u ltip le p a s s tr a n s d u c e r s c h a r a c te r iz e s th e e x te n t to w h ic h tr a n s d u c e r s in th a t c la s s c a n in c r e a s e th e s iz e of an o u tp u t f r o m a n in p u t. S e c tio n 2 .4 c o n ta in s one o f th e m a jo r r e s u lt s : th a t p a s s e s w ith a tr a n s d u c e r c a n b e tr a d e d fo r s p a c e . L e t L(n) b e an y sp a c e b o u n d . F o r an y s in g le p a s s , L(n) s p a c e b o u n d ed tr a n s d u c e r , th e re is a n e q u iv a le n t lo g (L (n )/lo g (n )) p a s s tr a n s d u c e r w h ich o p e r a te s in o n ly log(n) s p a c e . M . B lu m [2] h a s stu d ie d c o m p u ta tio n a l r e s o u r c e s . H e d e v e lo p e d a s e t of a x io m s w h ic h c h a r a c te r iz e th o se 4 w h ich h a v e s e rv e d a c o m p le x ity m e a s u r e s . In S e c tio n 2 .5 it is show n th a t th e n u m b e r of p a s s e s a tr a n s d u c e r m a k e s is su ch a r e s o u rc e and th a t i t s a tis f ie s th e B lu m a x io m s . C h a p te r 3 e x p lo re s v a rio u s o th e r a s p e c ts of sp a c e b ounded t r a n s d u c e r s . S e c tio n 3 .1 show s th a t an y r e c u r s iv e ly e n u m e ra b le 2 s e t is th e ra n g e o f th e fu n c tio n d e fin e d b y so m e log (n) tr a n s d u c e r . In S e c tio n 3 .2 , a u s e fu l a lg e b r a ic p r o p e r ty of fu n c tio n s is il l u s t r a t e d . It r e la te s fu n c tio n a l c o m p o sitio n to th e c lo s u r e of c la s s e s o f s e ts u n d e r fu n c tio n a l p r e - im a g e . F o r an y c la s s o f tr a n s d u c e r s , w e d e fin e th e c la s s of la n g u a g e s a c c e p te d b y th o s e tr a n s d u c e r s in a n a tu r a l w a y . If a c la s s o f tr a n s d u c e r s is c lo s e d u n d e r c o m p o sitio n , th e n th e c la s s of la n g u a g e s a c c e p te d b y th o s e tr a n s d u c e r s is c lo s e d u n d e r p r e - im a g e by an y tr a n s d u c e r in th a t c l a s s . S e c tio n 3 .3 c o m p a re s log(n) s p a c e bou n d ed tr a n s d u c e r s w ith an d w ith o u t th e a d d itio n of a p u sh dow n s t o r e . It is show n th a t th e a d d itio n o f a p u sh dow n s to r e d o es ad d tra n s d u c tio n p o w e r. T h e p ro o f te c h n iq u e it s e lf is a ls o of in t e r e s t s in c e i t show s how a n L(n) sp a c e .b o u n d e d pushdow n s to r e alo n g w ith a n L(n) s]l>ace bounded c o u n te r m a y b e u se d to co u n t fro m 0 to c " ^ n \ B a s ic N o ta tio n s n o t F o und in th e T e x t T he fo llo w in g l i s t c o n ta in s n o ta tio n s fro m s e t th e o ry . T h ey a r e a ll s ta n d a rd a n d b a s ic and a r e n o t e x p la in e d f u r th e r in th e te x t. N o tatio n N x 6 X A c B A U B a n b |A ( A XB I n te r p r e ta tio n s e t o f p o s itiv e in te g e r s x is an e le m e n t o f th e s e t X th e s e t A is a s u b s e t o f th e s e t B th e s e t c o n s is tin g of a ll m e m b e rs of th e s e ts A and B th e s e t c o n s is tin g o f a ll e le m e n ts w h ich b elo n g to b o th th e s e ts A and B th e n u ll s e t o r th e s e t w ith no m e m b e r s th e c a r d in a lity o f th e s e t A th e s e t o f p a ir s o f e le m e n ts , th e f i r s t of w h ich b e lo n g s to A and th e seco n d to B N o ta tio n s fro m th e T e x t T he n o ta tio n s in th e l i s t b elow a r e d efin e d in th e te x t. T h ey a r e re p ro d u c e d h e r e fo r th e c o n v e n ie n c e o f th e r e a d e r . 3(M ) 2 . 1 .4 N o ta tio n D e fin itio n I n te r p r e ta tio n ID 2 . 1 .2 in s ta n ta n e o u s d e s c rip tio n h r - N r 2 .1 .3 y ie ld s M(w) = v 2 . 1 .4 th e r e s u lt o f ap p ly in g M to w is v {(w, v) { M(w) = v) 6 N o ta tio n 3(A) JTM) JtA ) ^ (L (n )) N (M , n) | 3(M )S | 3(A) =•, S», <• r L(n)« {nc '} C F CS TM R E m u n)) D e fin itio n 2 . 1 .4 T h e o re m 2 . 1 .2 ff T h e o re m 2 . 1 .2 ff 2 . 1 .5 L e m m a 2 .2 .1 ff 2 .2.2 2 .2.2 2 .2 .3 L e m m a 2 . 2 .5 ff tf?(P (n),L (n)) 2 .3 .3 T h e o re m 2 .4 .4 T h e o re m 2 .4 .6 2 .5 .1 2 . 5 .5 3 .3 .1 3 .3 .2 I n te r p r e ta tio n {jj(M )| M € A} s e t a c c e p te d by M M) f M 6 A } c la s s o f L(n) s p a c e bounded tr a n s d u c e r s n u m b e r o f e q u iv a le n c e c l a s s e s of ID 's w ith in p u t of le n g th n {(fw l » l v ! ) f M(w) = v} {f 3(M )| I M 6 A} o r d e r in g r e la tio n s on fu n ctio n s s e t of fu n c tio n s on N x N : { { (n ,n c Ij(n ))fn<E N } |c € N} c la s s of P (n) p a s s , L(n) sp a c e b o u n d ed tr a n s d u c e r s c la s s o f C o n te x t F r e e la n g u ag es c la s s o f C o n te x t S e n s itiv e la n g u a g e s c la s s o f T u rin g M a ch in e s c la s s o f R e c u rs iv e ly E n u m e r a b le s e ts c la s s o f T(n) tim e bounded T u rin g M ach in e s c la s s o f L(n) s p a c e bounded pushdow n tr a n s d u c e r s 7 P r e lim in a r y D e fin itio n s P r e lim in a r y d e fin itio n s and n o ta tio n s u s e d th ro u g h o u t th is w o rk a r e now p r e s e n te d . W e u s e th e s ta n d a rd d e fin itio n of a fu n c tio n a s a s e t o f o r d e r e d p a ir s s a tis fy in g th e c o n d itio n th a t no tw o, d is tin c t p a ir s h a v e th e s a m e f i r s t m e m b e r . T he s e t o f p a ir s a r e a s u b s e t o f th e s e t of a ll o r d e r e d p a ir s : th e c r o s s - p r o d u c t o f tw o s e t s . T he d o m a in , ra n g e , p r e - im a g e and im a g e o f a fu n ctio n a r e d e fin e d . T h e c o m p o sitio n o f fu n c tio n s and v a rio u s c lo s u r e p r o p e r tie s a r e a ls o d e fin e d . T h e d e fin itio n s a r e ex te n d e d to s e ts o f fu n c tio n s . M any o f th e r e s u lt s in th is w o rk a r e c o n c e rn e d w ith s e ts of s tr in g s o v e r so m e a lp h a b e t. S om e p r e lim in a r y d e fin itio n s ab o u t s tr in g s a r e a ls o p r e s e n te d . D e fin itio n 1 .1 : A fu n c tio n f on A x B is a s e t of o r d e r e d p a ir s : f = {(a, b)f a € A an d b € B} su c h th a t if (a, b) 6 f and (a, c) 6 f th e n b = c . If (a, b) 6 f th e n w e s o m e tim e s w r ite f ( a ) = b . T h e d o m a in of f (w ritte n 3~(f)) is d efin e d : J( f) = {af th e r e is a b su ch th a t (a, b) € f } . T h e ra n g e o f f (w ritte n /?(f)) is d efin e d : ^ (f) = {b [th e re is a n a su c h th a t (a, b) € f} . D e fin itio n 1 .2 : L e t f b e a fu n c tio n on A X B . L e t A 'c A , T he im a g e o f A* u n d e r f (w ritte n f(A ')) is d efin e d : £(A*) = { b fth e re is a n a € A 7 su c h th a t (a, b) € f} . 8 L e t B ' c B , T he p r e - im a g e o f B* u n d e r f (w ritte n f '^ B ') ) is d efin e d : f ^(B ; ) = {a| th e r e is a b € B ; su c h th a t (a, b) € f} . N ote th a t if f is a fu n c tio n on A x B th e n 3~(i) = f ^(B) and R{i) = f(A ). D e fin itio n 1 .3 : L e t f b e a fu n c tio n on A X B an d g a fu n c tio n on B x C . T he c o m p o sitio n o f f an d g (w ritte n gof) is d efin e d : gof = {(a, c)f th e r e is a b € B su ch th a t (a, b) € f an d (b ,c ) € g} . W e s o m e tim e s in d ic a te th e c o m p o sitio n o f a fu n c tio n w ith it s e lf a s fo f = f^ = {(a, c)f th e r e is a b su c h th a t (a, b), (b, c) 6 f } . D e fin itio n 1 .4 : L e t F c {ff f is a fu n c tio n on A x A} . F is c lo s e d u n d e r c o m p o s itio n if fo r a ll f, g € F th e r e is a n h € F su ch th a t h = g of. If F and G a r e c la s s e s o f fu n c tio n s w e s a y FoG = { fo g ff € F an d g € G} . H e n c e , F is c lo se d u n d e r c o m p o sitio n if and o n ly if F o F c F . D e fin itio n 1 .5 : L e t F an d G b e s e ts o f fu n c tio n s . J{ F) is s a id to b e c lo s e d u n d e r G p r e - im a g e if fo r a ll f 6 F and g € G th e r e is an h € F su ch th a t ^ (h ) = g” ^(J"(f)). D e fin itio n 1 .6 : L e t f b e a fu n c tio n . T he in v e r s e o f f, d e n o te d f” *, is d e fin e d : f” * = {(y, x)| (x, y) 6 f} . D e fin itio n 1 .7 : L e t F b e a c la s s of fu n c tio n s . F is s a id to b e c lo s e d u n d e r in v e r s e if fo r a ll f € F , th e n f” * € F . 9 D e fin itio n 1 .8 : L e t X b e a c la s s of s e ts , an d le t F b e a c la s s of fu n c tio n s . X is s a id to b e c lo s e d u n d e r tr a n s d u c tio n by F if fo r a ll f € F an d a ll A S X , th e n f(A) € X . * D e fin itio n 1 .9 : L e t E b e a s e t o f s y m b o ls . T h en E is s o m e tin e s c a lle d a n a lp h a b e t. D efin tio n 1 .1 0 : L e t E b e an a lp h a b e t. A s tr in g o r w o rd w o v e r th e a lp h a b e t E is a se q u e n c e o f e le m e n ts e a c h o f w h ich b e lo n g s to E : w = a , a 0 • • • a , w h e re a. S E fo r a ll i . 1 2 n i D e fin itio n 1 .1 1 : L e t w b e a s tr in g . B y fw f is m e a n t th e le n g th o f th e s tr in g w , o r th e n u m b e r of d is tin c t o c c u r r e n c e s o f s y m b o ls . If w = • • • a n , th e n [w f = n . D e fin itio n 1 .1 2 : T h e n u ll s tr in g S is th e s tr in g c o n s is tin g o f no e le m e n ts . D e fin itio n 1 .1 3 : B y E'1 ' is m e a n t th e s e t o f a ll s tr in g s o v e r th e a lp h a b e t E, in c lu d in g th e n u ll s tr in g . D e fin itio n 1 .1 4 : L e t E b e so m e a lp h a b e t an d le t w^ an d W£ b e s tr in g s o v e r E : s a y w . = a . a 0 *«»a and w- = b .b 0 »»»b , w h e re ° 7 1 1 2 n 2 1 2 m ’ fo r a ll i, a.,b. 6 E. T he c o n c a te n a tio n of w , an d w 0 , w r itte n w ,»w „ l i ............... 1 2 * 1 2 is d efin e d : w . • w - = a . a_ • • • a b , b ~ • • • b . 1 2 1 2 n l 2 m L e t A, B c E *. T h en w e d e fin e A • B = {w j • W 2 I w^ € A and W2 € B }. 10 D e fin itio n 1 .1 5 : L e t X b e a c la s s o f s e ts of s tr in g s o v e r so m e a l p h a b e t s : X g {a ! A c E*}. X is s a id to b e c lo s e d u n d e r c o n c a te n a tio n if fo r a ll A , B 6 X, th e n A • B € X . D e fin itio n 1 .1 6 : L e t f b e a fu n c tio n on E * x £ * . T h e n f is s a id to b e a h o m o m o rp h is m if f(v • w) = f(v) * f(w ), fo r a ll v, w € £ * . D e fin itio n 1 .1 7 : L e t X b e a c la s s of s e ts o f s tr in g s o v e r so m e a lp h a b e t S: X c [a | A c E * } . X is s a id to b e c lo s e d u n d e r h o m o m o r p h is m i f fo r a ll A € X an d a n y h o m o m o rp h is m f, th e n f(A) € X . D e fin itio n 1 .1 8 : A n n - a r y p re d ic a te o v e r th e a lp h a b e t E is a s e t o f n - tu p le s : P = {(W j, • • •, wn)f € £ 'r fo r a ll 1 ^ i ^ n} . L e t th e c o m m a sy m b o l ( ,) b e su c h th a t , £ E, and le t P b e a p r e d ic a te o v e r th e a lp h a b e t E. T h en P m a y b e tr e a te d a lte r n a tiv e ly a s a s e t o f n - tu p le s o f s tr in g s o v e r th e a lp h a b e t E, o r a s a s e t of s tr in g s o v e r th e a lp h a b e t E U {, } . If P is a p r e d ic a te th e n w e s o m e tim e s w r ite P(w ) fo r w € P . C H A P T E R 2 M U L T IP L E PA SSES W ITH S P A C E BOUN DED TRA N SD U CERS In th is c h a p te r, th e th e o ry of m u ltip le p a s s e s w ith sp a c e b o u n d ed tr a n s d u c e r s is d e v e lo p e d . F i r s t , (s in g le p a s s , ) sp a c e b o u n d ed tr a n s d u c e r s a r e d e fin e d . T h en a m e th o d fo r c h a r a c te r iz in g th e le n g th s o f tra n s d u c tio n s fo r c la s s e s of m a c h in e s is in tro d u c e d . It is th e n p ro v e d th a t th e c la s s o f log(n) s p a c e bounded tr a n s d u c e r s is c lo s e d u n d e r c o m p o s itio n . N ex t, th e d e fin itio n of s p a c e b o u n d ed tr a n s d u c e r s is ex p an d ed to in c lu d e m u ltip le p a s s e s . It is show n how to c h a r a c te r iz e th e le n g th s o f th e o u tp u ts of th e s e m u ltip le p a s s tr a n s d u c e r s u sin g th e s a m e te c h n iq u e in tro d u c e d fo r th e sin g le p a s s a u to m a ta . In p a r tic - u la t, a c h a r a c te r iz a tio n of m u ltip le p a s s , log(n) sp a c e bounded t r a n s d u c e r s is g iv e n . U sin g th is c h a r a c te r iz a tio n , i t is th e n show n how m u ltip le p a s s e s m a k e log(n) tr a n s d u c e r s a s p o w e rfu l a s l a r g e r s p a c e b o u n d ed a u to m a ta . F in a lly , i t is a rg u e d th a t w h en c o n s id e re d a s a c o m p u ta tio n a l r e s o u r c e , p a s s e s s e r v e a s a c o m p le x ity m e a s u r e in th e s e n s e of th e B lu m a x io m s . S e c tio n 2 .1 T r a n s d u c e r s : B a s ic D efin itio n s T h is s e c tio n e s ta b lis h e s a fo u n d a tio n . T r a n s d u c e rs and th e ir s p a c e b o u n d ed s u b c la s s e s a r e d e fin e d . T h e ir fo rm a l m e a n s of 11 12 o p e ra tio n is g iv e n . T ra d itio n a lly , a u to m a ta th e o ry fo c u s e s p r i m a r ily on m a c h in e s a s d e v ic e s fo r a c c e p tin g s e ts of in p u t s tr in g s . In c o n tra s t, w e b e g in o u r stu d y w ith tr a n s d u c e r s — d e v ic e s w h ich co n v e r t in p u t s tr in g s to o u tp u t s tr in g s — an d d is c u s s th e fu n ctio n s th e y d e fin e . T h en w e id e n tify th e d o m a in of a tr a n s d u c e r -d e f in e d fu n c tio n w ith th e s e t o f s tr in g s a c c e p te d b y th a t tr a n s d u c e r . T h is a p p ro a c h p ro v id e s a c o n v e n ie n t an d c o m p re h e n s iv e fra m e w o rk fo r d is c u s s in g c la s s e s o f a u to m a ta and fo r m a k in g c o m p a ris o n s b etw ee n c l a s s e s . T r a n s d u c e r s , Y ield s and T r a n s d u c e r D efin ed F u n c tio n s D e fin itio n 2 . 1 . 1 : A tr a n s d u c e r M is a n 8 -tu p le M = (K, E, £, $, r , 6, pQ, F ), w h e re : a) K is a fin ite , n o n -e m p ty s e t (th e s e t o f s t a t e s ); b) E is a fin ite , n o n -e m p ty s e t (th e in p u t/o u tp u t a lp h a b e t); c) £ and $ (th e end m a r k e r s ) a r e sy m b o ls su c h th a t { £ ,$ } n £ = 0; w e le t Eq = z u { £ ,$ } ; d) r is a fin ite , n o n -e m p ty s e t (th e w o rk in g a lp h a b e t); w e le t r Q = rU {14}, w h e re U is th e " b la n k " ta p e sy m b o l; e) 6 is a m a p p in g (the tr a n s itio n fu n c tio n ) 6 : K x E 0 x r 0 -» K x r x { - l , 0 , + l } 2 X(E0 U{e}) f) Pq € K (th e s t a r t s t a t e ); g) F c K (th e s e t o f fin a l s t a t e s ) . In a d d itio n w e r e q u ir e th e follow ing r e s tr ic tio n s to h o ld fo r a ll 6(p, a , b) — (q, c , d j , ^ 2 ’ ^ * 1 * P = P q if a n ( i o n ly if e = £ . (T he f i r s t o u tp u t sy m b o l is th e £ sig n , and th e £ s ig n a p p e a rs o n ly a t th e f i r s t tr a n s itio n .) 2 . q 4 P q . (T h e re is no tr a n s itio n in to th e s t a r t s ta t e .) 3 . If a = £ (re s p e c tiv e ly , $), th e n d j ^ -1 (re s p e c tiv e ly , +1) (T h e in p u t r e a d h e a d d o es n o t m o v e p a s t th e en d s of th e in p u t.) 4 . q 6 F if and o n ly if e = $ . (T he fin a l o u tp u t sy m b o l is th e rig h t en d m a r k e r .) 5 . p|{ F . (T h e re is no tr a n s itio n o u t o f th e s ta te s o f F .) M u s e s th r e e ta p e s . 1. M r e a d s , b u t d o e s n o t w r ite on the in p u t ta p e . 2 . M w r ite s , b u t d o es n o t r e a d th e o u tp u t ta p e . 3 . M b o th re a d s and w r ite s on th e a u x ilia ry , w o rk in g ta p e . M w ill b e d e fin e d to a c t a s fo llo w s . S u p p o se th a t M 's in p u t r e a d h e a d is sc a n n in g so m e sy m b o l a € E q , th a t M 's w o rk in g ta p e r e a d / w r i te h e a d is sc a n n in g so m e sy m b o l b 6 Tq and th a t M is in s ta te p € K . L e t 6 ( p ,a ,b ) = (q, c, d j , dg, e ) . T hen^the follow ing o c c u r . 1. M m o v e s in to s ta te q . 2 . M 's w o rk in g ta p e r e a d / w r i te h e a d r e p la c e s sy m b o l b 14 3 . M 's in p u t ta p e h e a d m o v e s in th e d ir e c tio n d j . T h a t is , if M 's in p u t ta p e r e a d h e a d h a s b e e n sc a n n in g s q u a re m ^ th e n th a t h e a d m o v e s to s q u a re m ^ + d j. 4. In th e s a m e w ay , M 's w o rk in g ta p e h e a d m o v e s a c c o rd in g to • 5 . M ap p en d s th e sy m b o l e to th e r ig h t end o f th e o u tp u t s tr in g . D e fin itio n 2 . 1 . 2 : L e t M = (K, E, £, $, r , 6, P q , F) b e a tr a n s d u c e r . A n In s ta n ta n e o u s D e s c rip tio n (o r ID) o f M is a q u in tu p le (p, m , £w$, u / v ) , w h e re : 1. p € K is th e c u r r e n t s t a t e ; 2 . m = ( m p m ^ J S N x N in d ic a te s , re s p e c tiv e ly , th e p o s i tio n s of M 's in p u t r e a d h e a d an d w o rk in g ta p e r e a d / w r i te h e a d ; (w e c o n s id e r th e in p u t ta p e n u m b e re d b eg in n in g w ith s q u a re n u m b e r z e r o and the w o rk in g ta p e n u m b e re d b e g in n in g w ith s q u a re n u m b e r o n e ). 3 . £w$ is th e in p u t s tr in g , w h e re w € S* is th e in p u t; 4 . u € Tq is th e w o rk in g ta p e s tr in g ; an d 5 . £v, w h e re v € ( S * U 2 * * {$} ) is th e o u tp u t s t r i n g . D e fin itio n 2 . 1 . 3: L e t ij an d i 2 b e ID 's of M = (K, E, £, $, r , 6, Pq, F ) . W e s a y th a t i j y ie ld s i 2 w ith r e s p e c t to M (o r s im p ly i j y ie ld s i 2 if M is u n d e rsto o d ) if: 15 1. = (P, (m j.ir^ h ^ w I.U p ^ v ); 2 . sy m b o l of £w$ is th e sy m b o l a; 3 . sy m b o l m g of u^ is th e sy m b o l b; 4 . 6(p, a, b) = (q, c , d j , d2> e); and 5 . ig = (q, ( m j + d^, m g + dg), £ w $ ,U g ,v e ), w h ere sq u are rrig o f u2 is the sym b ol c . If m 2+ d2 > f u j f , then sym b ol m->+ d2 o f ^ is the blank tape sym b ol > 5 . If i j y ie ld s i 2 w ith r e s p e c t to M w e s o m e tim e s w r ite i . f i~ , o r s im p ly i . f— i 9 is M is u n d e rs to o d . W e le t 1 M * 1 * M ( r e s p e c tiv e ly , F -> b e th e r e fle x iv e , tr a n s itiv e c lo s u r e of |---- M (r e s p e c tiv e ly , f — ). D e fin itio n 2 . 1 . 4 : L e t M = (K, S, £, $, P, 6, Pq, F) b e a tr a n s d u c e r . a. If (Pn> (0» !)» £w $, 1 6 , e) (q, m , £w $, u , £v$), w h e re q € F , th e n w e u M s a y th a t M h a lts on w , and w e w r ite M(w) = v . N ote th a t w ith q € F , M h a lts b e c a u s e th e r e a r e no tr a n s itio n s f r o m s ta te s in F . W e le t ^(M ) = {(w, v).|M (w ) = v}, an d i f A is a c la s s o f tr a n s d u c e r s , w e le t 3(A) = {3(M)fM€ A } . W e s ta te w ith o u t p ro o f th e follow ing r e s u l t s . L e m m a 2 . 1 . 1 ; L e t M = (K, S, £, $, r, 5, P q , F) b e a tr a n s d u c e r . T h en 3(M ) is a fu n c tio n . T h e o re m 2 . 1 . 2 : 1) L e t M j an d Mg b e tr a n s d u c e r s and 3(M ^) = 3 (M g ). T h e n J*(3(M j)) = ,T(3(M g)). (R e c a ll th a t J(i), w h e re f is a 16 fu n c tio n , is th e d o m a in o f th a t fu n c tio n .) 2 . L e t A an d B b e c l a s s e s o f tr a n s d u c e r s and 3>(A) = 5 (B ). T h en r(3f(A)) =,y(3f(B)). T r a d itio n a lly th e n o ta tio n J"(M) is u s e d fo r th e s e t o f s tr in g s a c c e p te d b y th e a u to m a to n M . A lth o u g h , in th is p a p e r th a t s e t is p r o p e r ly in d ic a te d b y ,7(5(M )), w h en no co n fu sio n r e s u l t s , w e w r ite ,T(M) fo r J"(3*(M)) an d J(A) fo r J"(5(A)), w h e re M is a tr a n s d u c e r an d A a c la s s o f t r a n s d u c e r s . N o te th a t th e c o n v e rs e to T h e o re m 2.1 .2 is n o t v a lid . S e c tio n 3 .3 id e n tifie s tw o s e ts A an d B su c h th a t 3~(A) = J( B) b u t 5(A) ^5(B ). W e now d e fin e c l a s s e s o f tr a n s d u c e r s w h ich o p e ra te w ith in s p e c ific s p a c e l i m i t s . D e fin itio n 2 . 1 . 5 : L e t M b e a tr a n s d u c e r an d le t L b e a fu n c tio n on N x N . W e s a y th a t M is L(n) s p a c e bou n d ed if fo r a ll (w, v) € 5(M ), if (P0 * (°» i)» ^w $» e) (P> m » £w $* u> x ) f—- (q» m '» u/>v) M M th e n | u | ^ L ( f w |). W e le t ?ft(L(n)) = fM |M is a n L(n) s p a c e bounded t r a n s d u c e r ] . N o te th a t th e s p a c e bound is r e q u ir e d to h o ld o n ly in th o se c a s e s in w h ich M h a l t s . If M d o e s n o t h a lt, th e n M m a y u s e an u n lim ite d a m o u n t o f w o rk in g ta p e . S e lf-C o m p u ta b ility , M o n o to n ic ity and U n b o u n d ed n ess T h e r e a r e th r e e c o n d itio n s on s p a c e b ounds w h ich o c c u r v e ry fre q u e n tly in m a n y c o m p le x ity th e o ry w o r k s . T h ey a r e (i) s e lf 17 c o m p u ta b ility , (ii) m o n o to n ic ity and (iii) u n b o u n d e d n e ss. F o r s im p lic ity , w e d e fin e th e s e h e r e and th e n m a k e th e u m b re lla a s s u m p tio n th a t a l l sp a c e b ounds s a tis f y th e s e c o n d itio n s u n le s s e x p lic itly s ta te d o th e rw is e . D e fin itio n 2 . 1 . 6 : L e t L(n) b e a fu n c tio n on N x N . I t is s a id th a t L(n) is s e lf-c o m p u ta b le if th e r e is a n M 6 7 ft (L (n)) su c h th a t | M (w )| = L([w|), fo r a ll in p u t w . S e lf-c o m p u ta b ility of a s p a c e bound p e r m its th e c o n s tru c tio n o f tr a n s d u c e r s w h ich " m a r k o ff" th e ir w o rk in g ta p e a r e a a t th e s t a r t of th e ir c o m p u ta tio n . D e fin itio n 2 . 1 . 7 : L e t L(n) b e a fu n c tio n on N x N . L(n) is s a id to b e m o n o to n ic a lly in c re a s in g if L(n) ^ L (n+1), fo r a ll n€N. M o n o to n ic ity o f a s p a c e bound g u a ra n te e s th a t a s th e in p u t s tr in g s g ro w in s iz e , th e a m o u n t o f w o rk in g ta p e s to ra g e p e rm itte d a ls o g ro w s . D e fin itio n 2 . 1 . 8 ; L e t L(n) b e a fu n c tio n on N x N . L(n) is s a id to b e un b o u n d ed if fo r an y c € N th e r e is a n n € N su c h th a t L(n) > c . T h ro u g h o u t th e r e m a in d e r o f th is w o rk , u n le s s th e c o n tr a r y i s s ta te d e x p lic itly , a ll s p a c e b o u n d s w h ich a r e o th e rw is e u n d efin e d a r e a s s u m e d to b e s e lf-c o m p u ta b le , m o n o to n ic a lly in c re a s in g and u n b o u n d ed . L e m m a 2 . 1 . 3 ; T h e fu n c tio n s L(n) = log(n) an d L(n) = n a r e s e lf- c o m p u ta b le , m o n o to n ic a lly in c re a s in g an d u n b o u n d ed . 18 P r o o f : C le a r ly L(n) = n is s e lf-c o m p u ta b le , m o n o to n ic a lly i n c r e a s in g an d u n b o u n d ed . C o n s id e r L(n) = lo g (n ). It too is c le a r ly m o n o - to n ic a lly in c re a s in g an d u n b o u n d e d . To s e e th a t log(n) a s a s p a c e bound is a ls o s e lf-c o m p u ta b le , c o n s id e r th e tr a n s d u c e r M w h ich o p e ra te s a s fo llo w s fo r an y in p u t w . S te p 1 . M m a rk s off a n a r e a on its w o rk in g ta p e c o n s is tin g of one s q u a r e . S te p 2 . U sin g th e w o rk in g ta p e , M co u n ts f ro m 0 to c - 1 , w h e re th e s iz e o f th e w o rk in g ta p e a lp h a b e t is c and th e n u m b e r o f s q u a re s m a rk e d off on th e w o rk in g ta p e is k . W hile co u n tin g on th e w o rk in g ta p e , M m o v e s th e in p u t r e a d h e a d o ne s q u a re to th e r ig h t fo r e a c h s te p o f th e c o u n t. S te p 3 . If th e c o u n t on th e w o rk in g ta p e te r m in a te s b e fo re th e in p u t r e a d h e a d r e a c h e s th e r ig h tm o s t in p u t sy m b o l, M g o es to S te p 4 . If th e c o u n t on th e w o rk in g ta p e ta k e s a t le a s t a s m a n y s te p s a s th e le n g th of th e in p u t ta p e , th e n th e w o rk in g ta p e is o f le n g th l o g ( |w |). M p r in ts a s o u tp u t th e c o n te n ts o f th e w o rk in g ta p e . H en ce M (w ) = v w h e re f v( = log(fwf). S te p 4 . M r e s e t s th e c o n te n ts of th e w o rk in g ta p e to a ll 0's. M a d d s one a d d itio n a l s q u a re to th e w o rk in g ta p e . M r e tu r n s th e in p u t r e a d h e a d to th e le ftm o s t in p u t sy m b o l an d r e tu r n s to S tep 2 . ■ 19 S e c tio n 2 .2 . C lo s u re o f th e C la s s of log(n) T r a n s d u c e rs u n d e r C o m p o sitio n T h e m a in r e s u lt in th is s e c tio n is th a t th e c la s s o f fu n ctio n s 3!( tyf(L(n))), w h e re L(n) = log(n), is c lo s e d u n d e r c o m p o sitio n . To g e t th is r e s u lt a m e th o d fo r d is c u s s in g th e m a x im u m le n g th of th e o u tp u t o f a tr a n s d u c e r is d e v e lo p e d . W e u s e th is m e c h a n is m in l a t e r s e c tio n s , an d fo r th a t r e a s o n , d e v e lo p it so m e w h a t a b s tr a c tly . S ta te E q u iv a le n c e D e fin itio n 2 . 2 . 1 : (a) L e t I b e a s e t of ID 's fo r tr a n s d u c e r M = (K, E , £ ,$ , I \ 6, P q , F) and le t d and s b e fu n c tio n s d e fin e d a s fo llo w s: (i) L e t d b e th e fu n c tio n on IX I d e fin e d b y th e y ie ld r e l a tio n s h ip : d = {(i, j)f i, j € I and i |----- j} . (N ote th a t th e M y ie ld r e la tio n s h ip d e fin e s a fu n c tio n s in c e th e 6 m ap p in g p ro v id e s a t m o s t one tr a n s itio n fo r e a c h ID . T h a t is , a l l tr a n s d u c e r s in th is p a p e r a r e d e te r m in is tic .) (ii) L e t s b e th e fu n c tio n on IX K w h ic h p a ir s a n ID w ith its f i r s t co m p o n e n t, th e s ta te of th e tr a n s d u c e r , s = {(i, p)fi = (p, m , w , u, v) 6 I} . (b) L e t i, j € I . T h en i an d j a r e s ta te e q u iv a le n t if fo r a ll ngl-, e ith e r s(d n (i)) and s(d n (j)) a r e b o th u n d efin e d o r th e y a r e b o th d e fin e d an d a r e e q u a l. (R e c a ll th a t fo r an y fu n c tio n f, th e n o ta tio n f1 1 in d ic a te s th e c o m p o s itio n o f f w ith it s e l f n ti m e s .) 20 L e m m a 2 . 2 . 1 : S ta te e q u iv a le n c e is a n e q u iv a le n c e r e la tio n . P r o o f : O b v io u s. ■ L e t [x] = {yfy is s ta te e q u iv a le n t to x } . W e s h a ll b e c o n c e rn e d w ith th e n u m b e r of e q u iv a le n c e c l a s s e s o f ID 's fo r in p u ts o f so m e le n g th n . L e t N (M , n) = f { [x ]|x is an ID fo r tr a n s d u c e r M w ith in pu t w , w h e re fw | ^ n } | . T h a t is , N (M ,n ) is th e n u m b e r o f e q u i v a le n c e c la s s e s o f ID 's fo r tr a n s d u c e r M w ith in p u ts o f le n g th no g r e a t e r th a n n . U p p er b ounds fo r N (M ,n ) a r e d e riv e d b e lo w . L e m m a 2 . 2 . 2 : F o r a ll M = (K, E, £, $, T, 8, Pq, F) 6 ^ (L (n )), th e r e is a c o n s ta n t c su c h th a t N (M , n) l n * c k(n)^ P r o o f : Tw o ID 's a r e s ta te e q u iv a le n t if: (1) th e y h a v e th e s a m e s ta te ; (2) th e p o s itio n s of th e h e a d s on th e in p u t ta p e a r e th e s a m e ; (3) th e p o s itio n s of th e h e a d s on th e r e a d / w r i te ta p e s a r e th e s a m e ; an d (4) th e c o n te n ts of th e r e a d / w r i te ta p e s a r e th e s a m e . T h is is n o t to s a y th a t tw o s ta te e q u iv a le n t ID 's a r e n e c e s s a r i ly id e n tic a l, o n ly th a t if th e y a r e id e n tic a l th e n th e y a r e s ta te e q u iv a le n t. H en ce if w e d e te rm in e th e n u m b e r o f d is tin c t ID 's , w e h a v e an u p p e r bound on th e n u m b e r o f s ta te e q u iv a le n c e c l a s s e s . F o r (1) th e r e a r e | k | = k p o s s ib ilitie s . F o r (2) th e r e a r e n p o s s ib ilitie s . F o r (3) th e r e a r e L(n) p o s s ib ilitie s . F o r (4) th e re a r e t ^ n ^ p o s s ib ilitie s , w h e re f rf = t . T h u s th e r e a r e no m o r e th a n k • n • L(n) • t L (n )* k • n • ( tL (n))2 = k • n • (t2 )L (nW • ( k . t 2 )L(n) = n • c s ta te e q u iv a le n c e c l a s s e s . ■ 21 L e t ??/(L(n)) b e so m e c la s s of sp a c e b ounded tr a n s d u c e r s , and le t M €???(L (n)). C o n s id e r th e fu n ctio n on N X N d e fin e d a s fo llo w s . D e fin itio n 2 . 2 . 2 : L e t f!5(M)f = { (|w | , f v| )f M(w) = v}, and le t I 3>(^(L(n)))f = {[ 3>(M)f f M G #f(L(n))} . T h a t is , if th e sy m b o ls in th e in p u t and o u tp u t s tr in g s a r e d is r e g a r d e d an d o n ly th e le n g th s o f th e s tr in g s c o n s id e re d , th e n M d e fin e s th e fu n c tio n on N X N g iv e n b y [3>(M)| . S u b se q u e n t r e s u lt s r e q u ir e a c h a r a c te r iz a tio n o f th e s e fu n ctio n s fo r v a rio u s c l a s s e s o f t r a n s d u c e r s . T he n e x t few d e fin i tio n s an d r e s u lt s p ro v id e su c h a c h a r a c te r iz a tio n . L e m m a 2 . 2 . 3 ; L e t M = (K ,2 , £, $ ,T, 8, pQ, F) € ^ ( L ( n ) ) . F o r a ll w € | M (w)| g N (M , f w | ). P ro o f: L e t (pQ, (0, 1), £w $, U, e) = i Q f— i j f— i 2 f— • • • f — in = (q, m , £w $, u, £v$), w h e re q € F . T h en fo r a ll 0 ^ j g n an d 0 < k, L is n o t s ta te e q u iv a le n t to F o r a s s u m e o th e rw is e . T h en b y th e d e fin itio n o f s ta te e q u iv a le n c e , fo r a ll h s 0, L is s ta te e q u iv a le n t to I*1 th a t c a s e M n e v e r h a lts on w . So fo r a ll O g n , [L ] i s d is tin c t. A ls o , M p ro d u c e s a t m o s t on e o u tp u t sy m b o l a t e a c h t r a n s i tio n . H e n c e | M(w)f S N (M , |w f ) . ■ C h a r a c te r is tic C la s s e s D e fin itio n 2 . 2 . 3 ; L e t f an d g b e fu n c tio n s on N X N . T h en w e sa y th a t f(n)^»g(n) (o r g(n)5»f(n)) if th e r e is a c o n s ta n t c > 0 su c h th a t fo r a ll x § 0, f ( x ) ^ c » g ( x ) . W e s a y th a t f(n )< » g (n ) (o r g(n) >• f(n)) 22 i f f(n) g(n) an d g (n )^ * f(n ). If f(n) g(n) and g(n) f(n), w e s a y f(n) =• g (n ). D e fin itio n 2 . 2 . 4 : L e t A b e a c la s s o f fu n c tio n s on N X N . A s e t of fu n c tio n s F o n N X N is a c h a r a c te r i s t ic c la s s fo r A if: 1. fo r e v e r y fu n c tio n g 6 A th e r e is a fu n c tio n f € F su c h th a t g(n) f(n); and 2 . fo r e v e r y fu n c tio n f 6 F th e r e is a fu n c tio n h € A su c h th a t f(n) = ■ • h (n ). L e m m a 2 . 2 . 4 : T he c h a r a c te r i s t ic c la s s r e la tio n s h ip is a n e q u iv a le n c e r e la tio n . P r o o f : It is show n th a t th e c h a r a c te r i s t ic c la s s r e la tio n s h ip is (i) r e fle x iv e , (ii) s y m m e tr ic an d (iii) tr a n s i tiv e . (i) A ny s e t o f fu n c tio n s A on N XN is a c h a r a c te r i s t ic c la s s fo r i t s e l f . F o r le t f € A b e a r b i t r a r y . T h e n s in c e f ^ » f c o n d itio n s 1 a n d 2 of th e d e fin itio n s of c h a r a c te r i s t ic c la s s a r e s a tis f ie d b y f i t s e l f . (ii) L e t A an d F b e s e ts o f fu n c tio n s on N XN an d le t F b e a c h a r a c te r i s t ic c la s s fo r A . B ut th e d e fin itio n of c h a r a c te r i s t ic c la s s is c o m p le te ly s y m m e tr ic b e tw e e n A an d F . So A is a ls o a c h a r a c t e r i s t i c c la s s f o r F . (iii) L e t A , F an d G b e s e ts Of fu n ctio n s on N x N ; le t F b e a c h a r a c te r i s t ic c la s s fo r A an d le t G b e a c h a r a c te r i s t ic c la s s fo r F . T h en it is show n th a t G is a c h a r a c te r i s t ic c la s s fo r A . (1) C o n s id e r a n y h € A . T h en s in c e F is a c h a r a c t e r i s ti c c la s s fo r A th e r e is a fu n c tio n f € F su ch th a t h ? f . B ut a ls o , s in c e G is a c h a r a c te r i s t ic c la s s fo r F th e r e is a fu n c tio n g S G su c h th a t f ^ » g . B ut c le a r ly , th e r e la tio n is tr a n s i tiv e . H e n c e h g . (2) S im ila r ly c o n s id e r an y g 7 € G . S in ce G is a c h a r a c t e r i s t i c c la s s fo r F an d s in c e F is a c h a r a c te r i s t ic c la s s fo r A, th e n b y th e s e c o n d p a r t of th e d e fin itio n o f c h a r a c te r i s t ic c la s s th e r e a r e fu n c tio n s f 7 € F and h 7€ A s u c h th a t g 7S » f7 and f7 s » h 7. A g a in , s in c e th e r e la tio n is tr a n s itiv e , g 7^ * h 7. ■ L e m m a 2 . 2 . 5 : L e t C b e a c h a r a c te r i s t ic c la s s fo r A an d le t C j c C su c h th a t fo r a ll f £ C j th e r e is a g € C - C j su c h th a t f g . T h en C - C j is a c h a r a c te r i s t ic c la s s fo r A . P r o o f : I t is show n th a t C - C j is a c h a r a c te r i s t ic c la s s fo r C . P a r t 1 . C o n s id e r an y f € C - C j . T h en th e r e is a fu n c tio n g € C su c h th a t f ^ » g . F o r c o n s id e r th e fu n c tio n f i t s e l f . C le a r ly f € C and f s . f. P a r t 2 . C o n s id e r a n y fu n c tio n f/ € C . T h en e ith e r f7 € C j o r f 7 € C - C ^ . In th e f i r s t c a s e , th e le m m a h y p o th e s is g u a ra n te e s th a t th e r e is a fu n c tio n g7 € C - C j su c h th a t f7 g 7. In th e se c o n d c a s e , 24 s in c e a g a in f# I1 an d f' € C - C j th e c o n d itio n is s a tis f ie d . T h u s C - C j is a c h a r a c te r i s t ic c la s s fo r C . B u t th e c h a r a c t e r i s t i c c la s s re la tio n s h ip is tr a n s itiv e by L e m m a 2 . 2 . 4 . So C - C j is a c h a r a c te r i s t ic c la s s fo r A . ■ F o r th e m o s t p a r t, th e c h a r a c te r i s t ic c l a s s e s m a d e u s e of b elo w a r e of a s p e c ia l c h a r a c t e r . T h e s e c l a s s e s c o n s is t of s e ts o f fu n c tio n s w h ich a r e p a r a m e te r iz e d b y a c o n s ta n t. F o r e x a m p le , th e s e t o f fu n c tio n s { {(n, n c ^ n ^)f n 6 N} f c € N} a p p e a rs fre q u e n tly a s a c h a r a c te r i s t ic c l a s s . In th o s e c a s e s in w h ic h no c o n fu sio n a r i s e s , w e w r ite { n c ^ ^ } to r e p r e s e n t th e c la s s {{(n, n c ^ n ^)f n 6 N }| c €N } T h a t is , w e r e p r e s e n t a c la s s of fu n c tio n s b y a c la s s o f fo rm u la e fo r th e m a p p in g s th e fu n c tio n s id e n tify . F o r th e m o s t p a r t c h a r a c te r i s t ic fu n c tio n s fo r c l a s s e s of tr a n s d u c e r d e fin e d fu n c tio n s a r e of c o n c e rn in th is p a p e r . T h a t is , w e a r e in te r e s te d in a c h a r a c te r i s t ic c la s s fo r f< J(^ (L (n )))| fo r so m e L (n ). F o r n o ta tio n a l c o n v e n ie n c e , th e follow ing c o n v en tio n is a p p lie d h e r e a f t e r . If C is a c h a r a c te r i s t ic c l a s s fo r f 3f( ^ (L (n ))) | w e w r ite th a t C is a c h a r a c te r i s t ic c la s s fo r ^ (L ( n ) ) . T h e o re m 2 . 2 . 6 : F o r an y L (n), th e c la s s [ n c ^ n ^] is a c h a r a c te r i s ti c c la s s fo r #?(L (n)). P r o o f : It is r e q u ir e d h e r e to p ro v e tw o a s s e r t i o n s . 1 . F o r a n y g iv e n M € ^ ( L ( n ) ) th e r e is a c g N su c h th a t | M (w)f ^ • f wf c ^ f wf and 25 2 . fo r an y c o n s ta n t c 6 N th e r e is a n M € ?ft(L(n)) su ch th a t |w f c L (fW^ | M (w )| . P r o o f o f 1: W e know fro m L e m m a 2 . 2 .2 th a t fo r e a c h M € ty?(L(n)) T (n\ th e r e is a c o n s ta n t c 6 N su c h th a t N (M , n) i n c . L e m m a 2 .2 .3 g u a ra n te e s th a t fo r a ll M 6 ^ |(L (n )), f M (w )| 5N (M , f w | ), fo r a ll w . H e n c e , | M(w)[ s=N(M, [w | ) wf c L (M \ fo r a ll w . P r o o f of 2 ; L e t c € N b e an y c o n s ta n t. W e a s s e r t th e e x is te n c e of a tr a n s d u c e r M w ith c w o rk in g ta p e s y m b o ls , an d w h ich o p e r a te s a s fo llo w s. L e t th e in p u t s tr in g b e w , th e r e fw f = n . S te £ _ l. M m a rk s ou t L(n) s q u a r e s on its w o rk in g ta p e . M s e ts th e in p u t r e a d h e a d to th e le ftm o s t sy m b o l of th e in p u t, th e £ s y m b o l. S te p 2 . U sin g th e w o rk in g ta p e sy m b o ls a s d ig its , M c o u n ts fro m 0 T fn \ to c - 1 . A t e a c h s te p in th e co u n t, M o u tp u ts a n o u tp u t sy m b o l. S te p 3 . M m o v e s it s in p u t r e a d h e a d on e s q u a re to th e r ig h t. S te p 4 . If th e in p u t r e a d h e a d is sc a n n in g th e sy m b o l $, M te r m in a te s . If n o t, M r e tu r n s to S te p 2 . T (-n \ D u rin g e a c h p a s s a g e th ro u g h S tep 2 , M p ro d u c e s c o u t pu t s y m b o ls . B y S te p s 3 an d 4 , S tep 2 is r e p e a te d n ti m e s . H en ce fo r a n y in p u t w , w e h a v e f M(w)f s f w | c ^ f w f ). ■ T h e o re m 2 . 2 . 7 : If L(n) log(n) th e n ( c ^ n ^) is a c h a r a c te r i s t ic c la s s fo r #?(L (m )). 26 P r o o f : I t is r e q u ir e d to p ro v e th e s a m e tw o c o n d itio n s a s in th e T (r\\ p re v io u s th e o r e m . P ro o f o f 1. By T h e o re m 2 . 2 .6 , {nc } is a c h a r a c te r i s t ic c la s s fo r ?7J(L(n)). H en ce fo r an y M € ^ ( L ( n ) ) th e r e is a c € N su ch th a t | M (w )| =• n c ^ n ^ c (L (n)+log(n)) c (k I j(n)) s in c e L (n )g * log(n) = . (ck )L < n > =• b L < n ). T P ro o f o f P a r t 2 . T h is p a r t h a s a lre a d y b e e n e s ta b lis h e d fo r nc b y T h e o re m 2 . 2 . 6 . S in ce c ^ n ^ » n c ^ 1 ^ th e r e s u l t fo llo w s im m e d ia te ly . ■ C lo s u re o f % (log(n))) u n d e r C o m p o sitio n T h e o re m 2 . 2 . 8 : T he c la s s I?( ?ft(log(n))) is c lo s e d u n d e r c o m p o si tio n . P r o o f : L e t M j,M 2 6 ?7|(log(n)). A n M j is c o n s tru c te d w h ich s a t i s fie s th e follow ing: 1 . 3(M 3 ) = 3(M 1)o3!(M2 ); 2 . M3 € ^ (lo g (n )). It is in te n d e d th a t M3 s im u la te th e o p e ra tio n o f M 2 a c tin g on th e o u tp u t o f M j. M3 is c o n s tru c te d a s s u g g e s te d b y th e fo llo w in g : 27 1 . Mg s im u la te s . 2 . M 2 h a s a s its in p u t th e o u tp u t of M j . 3 . So to s im u la te M 2 , Mg m u s t p ro v id e it s e lf w ith th e o u tp u t of M j . 4 . T h e o u tp u t of M^ m a y b e too long fo r Mg to s to r e on its w o rk in g ta p e . 5 . W h e n e v e r, d u rin g th e s im u la tio n of M 2 , th e M 2 in p u t r e a d h e a d is m o v ed to p o sitio n j of M j(w ), M 2 in t e r ru p ts its s im u la tio n o f M 2 . 6 . Mg s im u la te s M^ u n til j o u tp u t sy m b o ls a r e p ro d u c e d . 7 . Mg th e n r e tu r n s to th e s im u la tio n of M 2 u n til a n o th e r sy m b o l fro m M j(w ) is r e q u ir e d . U sin g s ta n d a rd coding te c h n iq u e s (8), th e w o rk in g ta p e of Mg m a y b e su b d iv id e d in to fo u r t r a c k s . 1) T ra c k 1 s to r e s th e p o s itio n in M^(w) of th e re a d h e a d of M 2 . 2) T ra c k 2 c o n ta in s th e r e a d / w r i te ta p e fo r M 2 • 3) T ra c k 3 c o n ta in s th e r e a d / w r i te ta p e fo r M j . 4) T r a c k 4 co u n ts th e o u tp u t sy m b o ls g e n e ra te d b y M j . Mg o p e r a te s a c c o rd in g to th e follow ing a lg o rith m . S tep 1 . In itia liz a tio n . In itia liz e tr a c k 1 to 0 . Mg h a s its in p u t r e a d h e a d on s q u a re 0 o f its (im ag in ary ) in p u t ta p e . 28 I n itia liz e tr a c k 2 to a ll b la n k s . Mg b e g in s its c o m p u ta tio n w ith a b la n k r e a d / w r i te ta p e . M a rk th e le ftm o s t s q u a re a s th a t s c a n n e d by th e M 2 r e a d / w r i te h e a d . I n itia liz e tr a c k 3 to a ll b la n k s . M^ b e g in s its c o m p u ta tio n w ith a b la n k r e a d / w r i te ta p e . I n itia liz e tr a c k 4 to z e r o . T ra c k 4 is n o t in u s e . In itia liz e th e fin ite c o n tro l s to ra g e fo r th e Mg in p u t sy m b o l to th e sy m b o l T he Mg in p u t re a d h e a d is sc a n n in g th e £ sy m b o l. In itia liz e th e fin ite c o n tro l s to r a g e fo r th e Mg in te r n a l s ta te to pQ. S te p 2 . P e r f o r m one s te p in th e c o m p u ta tio n o f M g. U sin g th e in fo rm a tio n s to re d in fin ite c o n tro l (the s ta te of Mg and th e in p u t sy m b o l sc a n n e d b y th e Mg in p u t re a d h ead ) and th e in fo rm a tio n on t r a c k 2 of th e r e a d / w r i te ta p e (th e sy m b o l s c a n n e d b y th e Mg r e a d / w r i te h ead ) d e te rm in e th e a p p lic a b le Mg tr a n s itio n . A ss u m e th e a p p lic a b le Mg tr a n s itio n is 6 ( p ,a ,b ) = ( q , c , d p d g ,e ) . T h en M ^: (i) s to r e s q in fin ite c o n tro l s to ra g e a s th e Mg s ta te ; (ii) s to r e s th e sy m b o l c in p la c e of th e sy m b o l b on th e s q u a re s c a n n e d by th e Mg r e a d / w r i te h e a d in tr a c k 2; (iii) ad d s d j n u m e ric a lly to th e c o n te n ts o f tr a c k 1; (T ra c k 1 c o n ta in s th e lo c a tio n o f th e Mg in p u t re a d h e a d an d d^ in d ic a te s how th a t h e a d m o v e s a s a r e s u l t of th e e x e c u te d t r a n s i t i o n . ) 29 (iv) a d ju s ts tr a c k 2 to r e f le c t th e new ( a lte r e d b y d£> p o s itio n of th e M 2 r e a d / w r i te h ead ; (v) o u tp u ts th e sy m b o l e . T h a t is , Mg ap p en d s th e sy m b o l e to th e Mg o u tp u t s tr in g . If no M 2 tr a n s itio n is a p p lic a b le , Mg h a l t s . If th e M 2 s ta te is in th e s e t of a c c e p tin g s ta te s fo r M 2 , th e n Mg te r m in a te s b y e n te rin g an a c c e p tin g s ta t e . S tep 3 . G e n e ra te M j(w ). Mg s im u la te s M ^ . A s s u m e tr a c k 1 c o n ta in s th e in te g e r j . Mg u s e s tr a c k 3 o f its r e a d / w r i te ta p e a s th e r e a d / w r i te ta p e fo r M j . S in c e th e in p u t to Mg is th e in p u t to M j, Mg c a n s im u la te M^ d ir e c tly . In s te a d of w ritin g th e o u tp u t sy m b o ls f ro m M j on its o u tp u t ta p e , h o w e v e r, Mg s im p ly co u n ts th e s e s y m b o ls . Mg u s e s tr a c k 4 fo r th is c o u n t. W hen th e co u n t of th e M j o u tp u t re a c h e s j, i . e . w h en tr a c k 4 and tr a c k 1 b e c o m e id e n tic a l, Mg te r m in a te s th e s im u la tio n o f M j and s to r e s in its fin ite m e m o ry th e j - t h o u tp u t sy m b o l M j p ro d u c e d . S te p 4 . R e tu rn to S te p 2 . It is a s s e r te d th a t M g(w) = w . Now w e a rg u e th a t Mg m a y b e c o n s tru c te d to p e r f o r m a ll of th e ab o v e c o m p u ta tio n s w ith in a w o rk in g ta p e o f no m o r e th a n lo g (|w f) s y m b o ls . T he fo u r tr a c k s a r e c o n s id e re d in o r d e r . W e know f ro m T h e o re m 2 . 2 .7 th a t { c ^ ° ^ n ^} is a c h a r a c te r - M 2 <Mj. (w)), fo r e v e r y in p u t s tr in g 30 is ti c c la s s fo r ^ (lo g (n )). T h u s fo r an y in p u t w , w e know th a t fM j(w )| ^ c ^ °6 (fw f ) > fo r so m e c o n sta n t c . H en ce to s to r e th e po s itio n in M j(w ) of th e M 2 r e a d h e a d , Mg u s e s lo g (c^ °® ^ W^ ) ta p e s q u a r e s . B ut lo g (c^ °® ^ w ^ ) = lo g ( |w |) » k fo r so m e c o n s ta n t k . H e n c e b y su b -d iv id in g tr a c k 1 in to k tr a c k s , Mg m a y s to r e an y in te g e r b e tw e e n 1 and f M j(w )| . So Mg m a y s to r e th e p o s itio n o f th e M 2 r e a d h e a d on its r e a d / w r i te ta p e . How m u c h s p a c e is n e e d e d fo r th e M 2 r e a d / w r i te ta p e ? W e c o m p u te d th e m a x im u m le n g th fo r th e in p u t to M 2 ju s t a b o v e .. It i s |M j(w)| . S in ce M 2 € ^ (lo g (n )), th e M 2 r e a d / w r i te ta p e is lo g (| M j(w )f). B u t th a t is ju s t th e v a lu e co m p u ted fo r tr a c k 1. So Mg m a y s to r e th e p o s itio n of th e M 2 r e a d h e a d on its r e a d / w r i te ta p e . S in c e M^ 6 9ft(log(n)), th e n c le a r ly Mg m a y s to r e th e M^ r e a d / w r i t e ta p e on its ow n tr a c k 3 . T r a c k 4 o f th e Mg r e a d / w r i te ta p e r e q u ir e s th e s a m e sp a c e a s tr a c k 1. H e n c e Mg m a y s to r e a ll fo u r tr a c k s on a log(n) ta p e an d so Mg € ^ (lo g (n )). W e h a v e show n th a t fo r an y M j, M^ € ???(log(n)), th e r e is a n Mg € 77|(log(n)) su c h th a t M g(w) = M 2 (M ^(w)) fo r an y in p u t w . T h u s Mg = M 2 <>Mj. H en ce Wj(log(n)) is c lo s e d u n d e r c o m p o sitio n . ■ T h is r e s u l t g e n e r a liz e s to an y fin ite n u m b e r of c o m p o s itio n s . C o r o lla r y 2 . 2 . 9 : If M p M 2 , • • •, M ^ € ^ (L (n )), w h e re L (n), log(n), th e n th e r e is an M 6 ^ (L (n )) su ch th a t M = M J 0 M 2 O • • • o M ^ . 31 P r o o f : W e e lim in a te tr a n s d u c e r s o n e -b y -o n e . T h e re is a n j € ^ (L (n )) su c h th a t M^_ j(w ) = j (^ ( w ) ) . So M j(M 2 (»• '( M ^ ^ M ^ w ) ) ) • ••)) = j (w ))• • • ) ) . L ik e w ise w e s u b s titu te 2 fo r 2 and ^ . W e c o n tin u e th is w a y u n til fin a lly w e find an M j to r e p la c e M j an d M 2 . T h en M |(w ) = M j (M2 (« • ‘ (M ^w)) • • •)) an d w e ta k e M = M j . ■ S e c tio n 3 .2 show s how th e c lo s u r e of 3(A) u n d e r c o m p o sitio n im p lie s a n in te r e s tin g c lo s u r e r e s u l t a b o u t 3~(A). S e c tio n 2 .3 M u lti- P a s s T r a n s d u c e r s : B a s ic D e fin itio n s A m u lti- p a s s tr a n s d u c tio n in v o lv e s a sin g le tr a n s d u c e r w h ich u s e s its ow n o u tp u t a s in p u t fo r f u r th e r p r o c e s s in g . If M € ?^(L(n)), th e n fo r e x a m p le , c o n s id e r M (M (w )). T h is is a tw o p a s s c o m p u ta tio n . T h a t is , w e f i r s t ap p ly M to th e in p u t w . To th e r e s u l t M(w) w e th e n ap p ly M a g a in . W e t r e a t th e n u m b e r of p a s s e s a tr a n s d u c e r ta k e s a s a c o m p le x ity m e a s u r e . In th e p re v io u s s e c tio n , i t is show n th a t th e c la s s of log(n) tr a n s d u c e r s is c lo s e d u n d e r fin ite c o m p o sitio n . In th is s e c tio n w e d e fin e m u lti- p a s s tr a n s d u c e r s w ith o u t p la c in g a n a p r i o r i bound on th e n u m b e r of p a s s e s p e r m itte d . C h a r a c te r is tic c l a s s e s fo r v a rio u s c a te g o r ie s o f th e s e tr a n s d u c e r s a r e d e r iv e d . M u lti- P a s s T r a n s d u c e rs D e fin itio n 2 . 3 . 1 : A M u lti- P a s s T r a n s d u c e r M is a p a ir M = (M q, G), w h e re M ^ = (K, S, £, $, I \ 6, P q , F) is a tr a n s d u c e r and G c F . W e 32 s o m e tim e s w r ite M = (K, E, £, $ , r , 6, Pq, F , G ). A m u lti- p a s s tr a n s d u c e r M = (M q,G ) w ill b e d e fin e d to a c t id e n tic a lly to a tr a n s d u c e r w ith th e follow ing a d d itio n a l f e a tu r e . C on s id e r so m e in p u t w . If M q(w ) h a lts w ith th e fin a l s ta te q 6 F -G , th e n w e c o n s id e r M to h a v e fin ish e d its f i r s t p a s s , b u t n o t to h a v e c o m p le te d its c o m p u ta tio n on w . In th is c a s e , w e th e n c o n s id e r M q(M q(w )). If M q(M q(w )) h a lts a g a in in a s ta te in F -G , th e n w e ta k e th e r e s u lt a s th e o u tp u t o f th e se c o n d p a s s of M b u t n o t th e fin a l o u tp u t. W e c o n tin u e to ap p ly M q to th e o u tp u t o f s u c c e s s iv e p a s s e s u n til M q te r m in a te s in a s ta te p € G . In th is c a s e , w e co n s id e r M to h a v e c o m p le te d its c o m p u ta tio n and w e ta k e th is fin a l o u tp u t a s M (w ). T he o p e ra tio n o f M is fo rm a liz e d b elow : D e fin itio n 2 . 3 . 2 : L e t Mq = (K, 2 , £, $, r» 6, Pq, F) and le t M = (M q, G) b e a m u lti- p a s s tr a n s d u c e r . L e t w € 2 * . A ss u m e (P0 » (°» !)> £w $,1i, e) |— (q r m j , £w $, U j, V jJ .q j € F -G M 0 (P0 * (°» l) » ^ v 1$,l4, e) h r ( q 2 »m 2 » ^ v i$ » u2 » v2 ^ q 2 e F “ G M o (P0» (°» !)» ^ v2$,16, e) (q3 , m 3 , ^ v 3$, u3 , v3 ) ,q 3 € F -G M 0 T h en w e s a y th a t M h a lts on w an d w e w r ite M(w) = v . 33 In th e c o m p u ta tio n show n ab o v e, e a c h lin e show s a c o m p le te c o m p u ta tio n o f M q . E a c h o f th e s e c o m p u ta tio n s of M q is c a lle d a p a s s of M . D e fin itio n 2 . 3 . 3 : L e t M = (K, E, £ , $ , T\ 6, P q . F , G) b e a m u lti- p a s s tr a n s d u c e r . L e t M (w) = v , W e s a y th a t M m a k e s n p a s s e s o n w , if fo r so m e n § 1 . M h a lts on w an d M(w) = v . W e sa y th a t M is P (n) p a s s b o u n d ed (o r, th a t M is a P (n) p a s s tr a n s d u c e r ) if, fo r a ll in p u ts w , w h e n e v e r M h a lts on w , th e n M m a k e s no m o re th a n P (fw f) p a s s e s on w . W e d e a l b elo w w ith m u lti- p a s s tr a n s d u c e r s w h ich a r e a ls o sp a c e b o u n d e d . L e t tyf(P(n), L(n)) = { M |M is a P (n) p a s s , L(n) s p a c e b o u n d ed tr a n s d u c e r } . S e lf-C o m p u ta b ility o f P a s s B ounds D e fin itio n 2 . 3 . 4 : A p a s s bound P (n) is s a id to b e s e lf-c o m p u ta b le in c o n ju n c tio n w ith a sp a c e bound L(n) if th e r e is a n M 6 W (P(n), L (n)) su c h th a t w h e n e v e r M h a lts on w th e n M h a lts o n w in e x a c tly P ( |w |) p a s s e s . F o r c o n v e n ie n c e i t is a s s u m e d h e r e a f t e r th a t a ll p a s s b.ounds a r e s e lf-c o m p u ta b le in c o n ju n c tio n w ith th e s p a c e b ounds w ith w h ich th e y a r e p a ir e d . It is a ls o a s s u m e d th a t a ll p a s s b ounds P (n) a r e m o n o to n ic a lly in c re a s in g w ith n an d a r e u n b ounded (s e e D e fin itio n 2 . 1 .6 an d D e fin itio n 2 . 1 . 7 ) . C h a r a c te r is t ic C la s s e s fo r M u lti- P a s s T r a n s d u c e rs 34 C h a r a c te r is tic c la s s e s fo r m u lti- p a s s tr a n s d u c e r s a r e die- r iv e d . F i r s t w e find c h a r a c te r i s t ic c l a s s e s fo r ^ (P ( n ) , L (n)), w h e re L(n) lo g (n ). W e th e n e x a m in e tw o s p e c ific c a s e s : (1) L(n) = log(n) and (2) L (n )= n an d d e riv e c h a r a c te r i s t ic c l a s s e s . F in a lly c h a r a c - t e r t i s i c c l a s s e s a r e found fo r ^ ( P ( n ) , L (n)), w h ere L(n) < • lo g (n ). (N ote th a t w h en w e w r ite th a t C is a c h a r a c te r i s t ic c la s s fo r ??|(P(n), L(n)) w e a r e in te n d in g , in fu ll fo rm a lity , th a t C is a c h a r a c t e r i s t i c c la s s fo r f 3 ( ^ ;( P ( n ) , L (n)))j . W e a r e s im p ly c a r r y in g o v e r to th e m u lti- p a s s c a s e th e n o ta tio n a l a b b re v ia tio n in tro d u c e d fo r c h a r a c te r i s t ic c l a s s e s of tr a n s d u c e r s .) T h e o re m 2 . 3 . 1 ; L e t C = ff(n)} b e a c h a r a c te r i s t ic c la s s fo r ^ (L (n )), w h e re L(n) lo g (n ). T h en { f ^ n \n ) } is a c h a r a c te r i s t ic c la s s fo r ??((P(n), L (n )). (R e c a ll th a t f ^ n ^(n) in d ic a te s f c o m p o se d w ith i t s e l f P (n) t i m e s .) P r o o f : E a c h p a s s o f an y M € 7??(P(n), L(n)) is e q u iv a le n t to so m e M q 6 ??f(L(n)). So P (n) p a s s e s w ith M is e q u iv a le n t to th e c o m p o si tio n o f M q w ith i t s e l f P (n) t i m e s . T h u s, s in c e M is g u a ra n te e d to ta k e no m o r e th a n P (n) p a s s e s , w e h a v e : f M(w) | [ M ^ ^ ^(w)f fo r so m e f(n) € C . To c o m p le te th e p ro o f i t is r e q u ir e d to show th a t fo r an y f(n) € C th e r e is a n M € ?ft(P(n), L(n)) su c h th a t f ^ ^ w ^ ( | w f ) |M (w )( . B u t b y h y p o th e s is , th e r e is a n M j 6 7ft(L(n)) su c h th a t f ( [ w |) I M j(w )f , fo r a ll w . H en ce So le t M € 7ft(P(n), L(n)) b e d efin e d : M(w) = M ^ ^ w ^ (w ) fo r a ll w . T h e n f ^ W^ ( | w f ) f M (w )| . N ote th a t in th e d e fin itio n o f M a s th e c o m p o sitio n o f M j w ith it s e l f P ( f w |) tim e s , w e r e ly on th e im p lic it a s s u m p tio n m a d e a b o u t th e s e lf-c o m p u ta b ility of P (n) in c o n ju n c tio n w ith L (n ). ■ L e m m a 2 . 3 . 2 : F o r a ll a , b, c > 0 , a lo ®bc = c lo ®ba . P r o o f : a lo gbc = (b lo gba )lo gbc = (blo gbc )lo eb a = c lo gba . ■ k P (n >| C o r o lla r y 2 . 3 . 3 ; T h e s e t C = £n |k > 0 } is a c h a r a c te r i s t ic c la s s fo r 7 ft (P (n ), lo g (n )). P r o o f : T h e o re m 2 , 2 . 7 sh o w s th a t i f L(n) log(n), th e n { c ^ n ^} is a c h a r a c te r i s t ic c la s s fo r 7ft(L(n)). H en ce {c*0 ® ^ } is a c h a r a c t e r i s t i c c la s s fo r ?ft(log(n)). B y L e m m a 2 . 3 .2 , c * ° ^ n ^ = n*0 ® ^ = n^" fo r s o m e k > 0 . H en ce It {n } is a c h a r a c te r i s t ic c la s s fo r 7ft(log(n)), b y L e m m a 2 . 2 . 5 . T h e o re m 2 .3 .1 sh o w s th a t if {f(n)} is a c h a r a c te r i s t ic c la s s fo r 7ft(L(n)), th e n £ f ^ n ^(n)} is a c h a r a c te r i s t ic c la s s fo r 7ft(P(n), L (n )). B u t if f(n )= n , th e n f ' * (n )= n . So {n } is a c h a r a c te r i s t ic c la s s fo r 7ft(P(n), L (n )). ■ 36 C o r o lla r y 2 . 3 . 4 : If L (n )= n , th e n a c h a r a c te r i s t ic c la s s fo r ^ (P ( n ) ,L ( n ) ) is Ic P (n) tim e s P r o o f : B y a n a rg u m e n t alo n g th e s a m e lin e s a s in th e p re c e d in g c o r o lla r y , {c1 1 } is a c h a r a c te r i s t ic c la s s fo r #J(L(n)), w h e re L(n) = n . n j c y • B ut if f(n) = c 1 1 , th e n f ^ n ^(n) = c cy P (n) tim e s , p ro v in g th e c o r o ll a r y . ■ F o r c o m p le te n e s s , so m e r e s u lt s ab o u t c h a r a c te r i s t ic c l a s s e s fo r #?(L(n)) an d #?(P(n), L (n)), w h e re L(n) <• log(n) a r e d e r iv e d . T h e o re m 2 . 3 . 5 : L e t L (n)< » lo g (n ). T h e re is a c h a r a c te r i s t ic c la s s C = ff(n)} o f ^ (L (n )) s u c h th a t fo r a ll f(n) € C , f(n) s a tis f ie s : n < • f(n) < • n * * e , fo r a ll e > 0 . P r o o f : T h e o re m 2 .2 .6 p ro v e s th a t { n c ^ n ^} is a c h a r a c te r i s t ic c la s s fo r # f(L (n )). C le a r ly fo r c > l , w e h a v e th a t n < » n c ^ ^ l B ut a ls o , s in c e L (n)< » lo g (n ), th e n fo r a n y c, w e h a v e th a t n c ^ n ^ <• n * * e , fo r a ll e > 0 . H e n c e , th e c la s s C = £ n c ^ n ^| c > 1} s a tis f ie s th e r e q u ir e m e n ts o f th e th e o re m , an d b y L e m m a 2 . 2 . 5 is a c h a r a c t e r i s t i c c l a s s . ■ 37 1+e L e m m a 2 . 3 . 6 : L e t n § 1 and le t n< » g(n) <• n fo r a ll e > 0. i 1+e T h en n< * g (n)<» n , fo r a ll e > 0 an d fo r a ll i > 0 . P r o o f : T he p ro o f is in tw o p a r t s . P a r t 1 . C le a r ly , if l g n < » g(n), th e n gx(n) >• n , fo r a ll i . 1+e P a r t 2 . It w ill b e show n b y in d u c tio n o n i th a t if g(n)<» n , fo r a ll e > 0 , th e n g*(n)<» n * * e , fo r a ll e > 0 . 1 1+e L e t i = 1. T h e n g (n) = g (n )< » n b y h y p o th e s is . 1 1 + © 1+1 A s s u m e th a t gJ(n)< » n fo r so m e j an d c o n s id e r gJ (n ). A s s u m e fo r th e sa k e of c o n tra d ic tio n th a t th e r e is an e su ch th a t g ^ ^ (n ) >• 1+e n . B u t g^+ 1(n) = g (g j(n )). W h en ce g(g^(n)) >• n*+e a n d so g^(n) >• g“ *(n*"*"e ) >• (n**e ) ^ ^ c ^, f o r a l l c > 0 , s in c e g(n) <• n 1+e > • c 1+d fo r so m e d > 0 . B u t th is c o n tra d ic ts th e in d u c tio n h y p o th e sis ab o u t g^(n). l +i 1+© H e n c e th e r e is no e su c h th a t gJ (n) > • n , p ro v in g th e le m m a . ■! C o r o lla r y 2 . 3 . 7 : L e t L(n) <• lo g (n ). T h e n fo r a ll P (n) th e r e is a c h a r a c t e r i s t i c c la s s C = {f(n)} fo r ^ (P ( n ) ,L ( n ) ) su c h th a t n< » f(n )<• n 1+e, fo r a ll f(n) € C . 38 P r o o f : T h e r e s u l t fo llo w s im m e d ia te ly f ro m th e a p p lic a tio n of L e m m a 2 . 3 .6 to T h e o re m 2 . 3 .5 and T h e o re m 2 . 3 . 1 . ■ S e c tio n 2 . 4 . S u b stitu tin g P a s s e s fo r S p ace In th is s e c tio n w e u s e th e r e s u l t c h a r a c te r iz in g log(n) sp a c e b o u n d ed m u lti- p a s s tr a n s d u c e r s to show how a d d itio n a l p a s s e s w ith a log(n) tr a n s d u c e r c a n in c r e a s e its p o w e r. In S e c tio n 2 .2 i t w a s show n th a t th e c la s s ^ (lo g (n )) is c lo se d u n d e r c o m p o sitio n . W e u s e th is r e s u l t to p ro v e th e fo llo w in g co n s ta n t s p e e d -u p th e o re m fo r p a s s e s o f ??j(P(n), lo g (n )). T h e o re m 2 . 4 . 1 : L e t k € N . T h en ^ ( k • p(n), lo g ( n ) ) c ^ ( P ( n ) , lo g (n )). P r o o f : L e t M € ^ ( k • P (n ), lo g (n )). It is r e q u ir e d to show th a t th e r e is a n M j € ^ ( P ( n ) , log(n)) su ch th a t 3>(M) = 3*(Mj). L e t M j b e th a t tr a n s d u c e r w h ic h , o n e a c h p a s s , p e r f o r m s th e c o m p u ta tio n c o n s is t ing o f k of th e p a s s e s o f th e tr a n s d u c e r M . C o r o lla r y 2 .2 .9 e s ta b lis h e s th e e x is ta n c e o f su c h a tr a n s d u c e r M j . ■ T he follow ing th e o re m p ro v id e s one of th e m a in r e s u lt s of th is w o rk . I t e s ta b lis h e s a r a te o f e x c h a n g e fo r s u b s titu tin g a d d i tio n a l p a s s e s in p la c e o f co m p u tin g s p a c e . T h e o re m 2 . 4 . 2 : L e t 1 . L 2 (n)g» L j(n ); 2 . {f(n)} b e a c h a r a c te r i s t ic c la s s fo r 7?j(Lj(n)); and 3 . fP (n )( n ) S . L j ^ L ^ n ) ) . 39 T h en 5 ( ^ ( L 2 (n)) c 3 ( ^ ( P ( n ) , L ,(n ))). P r o o f : L e t M2 G ^i(L 2 (n )). F o r e a c h su c h M 2 an a u to m a to n € ^ ( L j( n ) ) is c o n s tr u c te d . M j o p e ra te s in tw o p h a s e s . D u rin g p h a se 1, M j ta k e s m u ltip le p a s s e s to ex p an d its in p u t ta p e f ro m £w$ to £w bb * * * b $ , w h e re w is th e o rig in a l in p u t s tr in g and b is a s p e c ia l b la n k ta p e sy m b o l. c o n tin u e s in p h a se 1 u n til th e ta p e is L j* ( L 2 (|w f )) s q u a re s long, i . e . u n til | £wbb»»» 16$f 5 L j 1(L 2 ( |w f ) ) . In p h a s e 2, M j s im u la te s M2 d ir e c tly . T he in p u t ta p e to p h a s e 2 is L j* ( L 2 ( f w f )) s q u a r e s long; so M j h a s L j( L j* ( L 2 ( |w |))) = L 2 ( f w |) s q u a r e s o f r e a d / w r i te s to ra g e in w h ich to w o rk s in c e is L j(n ) sp a c e b o u n d ed . S in c e is L2 (n) sp a c e bo u n d ed , m a y b e c o n s tru c te d to s im u la te M2 d ir e c tly . P h a s e 1. E x p an d th e ta p e . M j is c o n s tru c te d so th a t w h ile in p h a s e 1, e a c h p a s s ex p an d s th e ta p e to th e m a x im u m e x te n t p o s s i b le . M j is c o n s tru c te d f i r s t to d u p lic a te th e in p u t to e a c h p a s s in p h a s e 1 onto th e o u tp u t ta p e — up to , b u t n o t in c lu d in g , th e sy m b o l $ , th e r ig h t end m a r k e r . T h en c y c le s th ro u g h a ll p o s s ib le in te r n a l s ta te s , in c o n ju n c tio n w ith a ll p o s s ib le c o n fig u ra tio n s o f th e w o rk ta p e an d th e in p u t r e a d h e a d . T h is te c h n iq u e is m o r e fu lly s e t ou t in T h e o re m 2 . 2 . 6 . A t e a c h tr a n s itio n , M j is c o n s tru c te d to ap p en d a n a d d itio n a l sy m b o l, s a y b, to th e o u tp u t ta p e . (It is a s s u m e d th a t th e sy m b o l b is n o t p a r t of th e in p u t a lp h a b e t of M2 . O th e rw is e M j is c o n s tru c te d to u s e so m e o th e r sy m b o l w h ich is n o t 40 p a r t of th a t a lp h a b e t.) S in ce {f(n)} is a c h a r a c te r i s t ic c la s s fo r ???j(Li(n)), M j m ay b e c o n s tru c te d so th a t |M j(w )f f(fw f) fo r a ll w , on e a c h p a s s . T h is follow s f r o m th e d e fin itio n of c h a r a c te r i s t ic c la s s , D e fin itio n 2 . 2 . 4 . B y T h e o re m 2 . 3 .1 , m a y b e c o n s tru c te d so th a t a f te r P (fw f) p a s s e s , |M j ( w ) |^ » f ^ W^ ( f w [ ) . M j te rm in a te s p h a se 1 a f te r P ( |w |) p a s s e s , w h e re fp (fw| )(j w | ) L j 1(L2(fwf)). N o te, th a t it is n o t in d ic a te d in th e p ro o f of th is th e o re m how to c o n s tr u c t to c o m p u te P ( f w f ) . T h a t is , th e m e c h a n is m th a t p e r m its M j to te r m in a te p h a s e 1 a f te r th e p r o p e r n u m b e r o f p a s s e s is n o t d is c u s s e d . F o r th is th e o re m , w e r e ly on o u r u m b r e lla a s su m p tio n th a t P (n) is s e lf-c o m p u ta b le in c o n ju n c tio n w ith L^(n) (s e e D e fin itio n 2 . 3 . 4 f f .) . F o r a p p lic a tio n of th is th e o re m to p a r tic u la r sp a c e b ounds L j(n ) and L 2 (n), it is r e q u ir e d to p ro v e th e s e lf-c o m p u ta b ility of L j^ ( L 2 (n)) in c o n ju n c tio n w ith L ^ (n ). P h a s e 2 . S im u la te M ^ w ) . A t th e end of p h a se 1, th e in p u t ta p e h a s L i 1(L2(|w!)) s q u a r e s . S in ce Mj is L j(n ) sp a c e bounded Mj has ¥ " 1 1 a w o rk in g ta p e o f ( L ^ f w l ) ) ) s q u a r e s . B ut L j(-^1 ^ 2 (fw f))) = L 2 ( |w f ). H en ce h a s e x a c tly a s m a n y w o rk in g ta p e s q u a r e s a s w ould M 2 i f g iv e n th e in p u t w . T hus c a n b e c o n s tru c te d to s im u la te M 2 o p e ra tin g on w . So fo r an y M 2 € 7/{(L 2 (n)) th e r e is a n M j € ^ (P ( n ) , L j(n )), w h e re (f(n)} is a c h a r a c te r i s t ic c la s s fo r ty|(L j(n)) an d f ^ n ^(n) = L j ^ L ^ n ) ) , su ch th a t M j(w ) = M2 (w) fo r a ll in p u t w . T h is p ro v e s th e th e o r e m . ■ M u ltip le P a s s e s w ith L og(n) T r a n s d u c e rs 41 T h e o re m 2 . 4 . 3 ; L e t L^(n) =• log(n) an d L 2 (n) 5* lo g (n ). A lso le t P(n) a . lo g (L 2 (n )/lo g (n )). T h en 3 (tf|(L 2 (n))) c 3( ^ ( P ( n ) , lo g (n ))). P r o o f : By th e p re c e d in g th e o re m it is r e q u ir e d to show th a t P (n) lo g (L 2 (n )/lo g (n )) s a tis f ie s : f P ^ ( n ) lo g ” 1(L 2 (n)), w h e re {f(n)} is a c h a r a c te r i s t ic c la s s fo r ???(log(n)). t / rj \ T h e o re m 2 . 2 . 7 a s s e r t s th a t {c } is a c h a r a c te r i s t ic c la s s fo r ^ (L (n )) if L(n) lo g (n ). In th is c a s e L(n) = L j(n ) =• lo g (n ). B y L e m m a 2 . 3 .2 , c ^°S(n ) = = g 0 L e m m a 2 . 2 .5 , {nk } is a c h a r a c te r i s t ic c la s s fo r ^ (lo g (n )). L e ttin g f(n) = n , o r m o re k p r e c is e ly le ttin g ^ ( n ) = n , w e h a v e th a t (f^(n)^ is a c h a r a c te r i s t ic c la s s fo r ^ (lo g (n )). It is r e q u ir e d to show th a t fP (n )(n) g . lo g " 1(L 2 (n)). ■ p , fP (n ), . k P(n) B ut f^ '(n) = n So i t is r e q u ir e d to show th a t n k P (n) a . lo g " 1(L 2 (n)), w h e re P(n) lo g (L 2 (n )/lo g (n )). lo g (L 2 (n )/lo g (n )) L e t g(n) = nk 42 l°g(L2(n)/log(n)) T h en log(g(n)) = log(n) • k lo g (L 2 (n )/lo g (n )) and lo g (g (n ))/lo g (n ) = k w h e n c e lo g (lo g (g (n ))/lo g (n )) = lo g (L 2 (n )/lo g (n )) • log(k) f r o m w h ich it fo llo w s th a t k 2 (n) lo g (g (n )). So, g(n) lo g " 1(L 2 (n)) . H en ce if P (n) lo g (L 2 (n )/lo g (n )), i t fo llo w s th a t g(n) lo g _1(L 2 (n )). ■ A g ain in th is th e o re m , w e r e ly on o u r u m b r e lla a s s u m p tio n th a t so m e P (n) lo g (L 2 (n )/lo g (n )) is s e lf-c o m p u ta b le in co n ju n c tio n w ith lo g (n ). F o r p a r tic u la r L < 2 (n), th a t s e lf-c o m p u ta b ility m u s t b e d e m o n s tr a te d . Now w e d e riv e so m e a p p lic a tio n s . It is d e te rm in e d how m a n y p a s s e s w ith a log(n) tr a n s d u c e r a r e r e q u ir e d to p e r f o r m v a rio u s c o m p u ta tio n s know n p e rfo rm a b le b y m o r e p o w e rfu l sin g le p a s s t r a n s d u c e r s . T he c la s s o f c o n te x t f r e e la n g u a g e s a p p e a rs fre q u e n tly th ro u g h o u t th e l i te r a t u r e o f c o m p u te r s c ie n c e . F o r a c o m p le te d e fin itio n s e e [ 8 ] , [1 3 ], H e re w e le t C F b e th e c la s s of c o n te x t f r e e la n g u a g e s . S im ila rly , w e u s e CS fo r th e c la s s o f C o n tex t S e n s itiv e la n g u a g e s [ 8 ] , [1 3 ], 43 T he follow ing th e o re m c h a r a c te r i z e s th e c o n te x t f r e e la n g u ag es s u ffic ie n tly fo r th is p a p e r . 2 T h e o re m 2 . 4 .4 ; F o r an y L € C F , th e r e is an M € ???((log(n)) ) su ch ; th a t L = P r o o f : (See [1 3 ].) ■ C o r o lla r y 2 . 4 . 5 : F o r an y L € C F th e r e is a n M € 7?f(log(log(n)),log(n)) su c h th a t L = ,r(M ). P r o o f : T ak e L gf1 1 ) = log(n) • log(n) in T h e o re m 2 . 4 . 3 . ■ It is r e q u ir e d to show th a t P (n) = log(log(n)) is s e lf-c o m p u ta b le in c o n ju n c tio n w ith lo g (n ). B ut log(n) is s e lf-c o m p u ta b le b y L e m m a 2 . 1 .3 ; and b y th e s a m e a rg u m e n t u s e d in th a t le m m a , w h en a p p lie d to log(n) its e lf , log(log(n)) is c o m p u ta b le in log(n) s p a c e . T he follow ing w e ll know n th e o re m c h a r a c te r iz e s CS fo r th is p a p e r . T h e o re m 2 . 4 . 6 : L e t L(n) = n . T h en C S c J"(?|(L (n ))). P r o o f : (See [1 3 ].) ■ C o r o lla r y 2 . 4 . 7 : F o r an y L € CS th e r e is a n M € ??f(log(n/log(n)), log(n)) su c h th a t L = J"(M). P r o o f : T a k e L£(n) = n in T h e o re m 2 . 4 . 3 . A g ain , it m u s t b e show n th a t P (n) = lo g (n /lo g (n )) is s e lf - c o m p u ta b le in c o n ju n c tio n w ith lo g (n ). In th is c a s e w e u s e a d iffe re n t 44 te c h n iq u e . R e fe r b a c k to th e p ro o f o f T h e o re m 2 . 4 . 3 . R e c a ll th a t th e m u lti- p a s s tr a n s d u c e r c o n s tru c te d th e r e o p e ra te d in tw o p h a s e s . In th e f i r s t p h a s e , th e tr a n s d u c e r (in th is c o r o lla r y c a lle d M) ex p an d s th e ta p e u n til th e ex p an d ed ta p e w * s a tis f ie s log ^ ( |w / f)*a» f w f , w h e re w is th e o rig in a l in p u t ta p e . T h a t is , th e ta p e is ex p an d ed u n til th e a m o u n t of w o rk in g s to r a g e a v a ila b le to a log(n) s p a c e bou n d ed tr a n s d u c e r m a tc h e s th e a m o u n t a v a ila b le to an n s p a c e b o u n d ed tr a n s d u c e r g iv e n th e o rig in a l in p u t. So M is co n s tr u c te d to o p e ra te a s fo llo w s . A t th e s t a r t of e a c h p a s s o f p h a se 1, M m a rk s off log(n) s q u a r e s o f w o rk in g ta p e , w h e re n is th e le n g th of th e in p u t ta p e to th a t p a s s . L e m m a 2 .1 .3 e s ta b lis h e s th e s e lf-c o m p u ta b ility of lo g (n ). M is th e n c o n s tru c te d to c o m p a re th e le n g th o f th e w o rk in g ta p e fo r th a t p a s s w ith th e le n g th of th e o rig in a l ta p e . (T h e o rig in a l in p u t is s til l a v a ila b le a s th e s t a r t of th e in p u t to e a c h p a s s .) If th e w o rk in g ta p e is a t le a s t a s long a s th e o rig in a l in p u t, th e n M te r m in a te s p h a se 1 and m o v e s on to p h a se 2 . In th is w ay P(n) = lo g (n /lo g (n )) is s e lf-c o m p u ta b le in c o n ju n c tio n w ith a log(n) sp a c e b o u n d . ■ P a s s B ound N e c e s s ity W e now know th a t fo r an y L (n), 3 > ( 9?;(L(n))) c 3(tyf(log(L (n)/log(n)), lo g (n ))). W e do n o t e s ta b lis h th e r e v e r s e con ta in m e n t. W e do, h o w e v e r, show th a t P (n) is m in im a l. T h a t is , w e fin d fu n c tio n s c o m p u ta b le b y a n M € ty((L(n)) w h ich r e q u ir e 45 lo g (L (n )/lo g (n )) p a s s e s b y an y M € ??f(log(n)). O ne su c h is th e tr a n s - k L(n) d u c tio n w h ic h m a p s 1 in to 1 L e m m a 2 . 4 . 8 : T h e re is a n M € #j(L (n)) su c h th a t 3>(M) = ,L (n ) f ( l n , 1 ) |n € N } fo r an y L (n ). P r o o f : L e t MG ?^(L(n)) b e th e tr a n s d u c e r w h ich o p e r a te s a s fo llo w s S tep 1 . C h eck th e in p u t. If i t is in th e fo rm l n , go on to S tep 2 . S tep 2 . M a rk off L(n) s q u a r e s on th e r e a d / w r i te ta p e . M m a y b e c o n s tru c te d to m a r k off its w o rk in g ta p e a r e a o n th e b a s is of o u r a s s u m p tio n o f th e s e lf-c o m p u ta b ility o f L (n ). S te p 3 . On th e w o rk ta p e , co u n t fro m 0 to k ^ n ^ - l . A ss u m in g th a t M h a s k sy m b o ls in its w o rk in g ta p e a lp h a b e t, M m a y b e c o n s tr u c te d to co u n t on its w o rk in g ta p e fro m 0 to k ^ n ^ - l . A t e a c h s te p in th e co u n t M is c o n s tru c te d to ap p en d th e sy m b o l 1 to th e o u tp u t s t r i n g . , L(n) T h u s M (ln ) = 1K . ■ c L(n) L e m m a 2 . 4 . 9 : L e t M € ?ft(P(n), log(n)) and le t M(1 ) = 1 fo r so m e L(n) lo g (n ). T h en P (n) lo g (L (n )/lo g (n )). P ro o f: S in c e M € ty?(P(n), log(n)), C o r o lla r y 2 .3 .3 g u a ra n te e s th a t k P(n) | M( w) | fw | , fo r s o m e k . So if w = l n , th e n fw f = n and P (n) w e h a v e n ^ | M (ln )| = c*"^1 1 ^ . T ak in g lo g s of b o th s id e s tw ic e an d e lim in a tin g c o n s ta n ts , on th e a u th o rity o f T h e o re m 2 . 4 .1 , w e fin d P (n) lo g (L (n )/lo g (n )). ■ 46 L e m m a 2 . 4 .1 0 : F o r a n y L(n) th e re is an M € tyf(L(n)) su c h th a t if M '€ ^ ( P ( n ) , log(n)) and 3?(M) = JJ(M/ ) th e n P (n) lo g (L (n )/lo g (n )). L(n) P r o o f : T ak e M (ln ) = l c and ap p ly L e m m a 2 . 4 . 9 . ■ T h is th e o re m s u m m a r iz e s p a s s and sp a c e r e la tio n s h ip s b e tw e e n th e tw o c la s s e s 3F(#!(L(n))) an d 3?(^(P(n), log(n))) w h e re P (n) lo g (L (n )/lo g (n )). T h e o re m 2 . 4 . 1 1 ; 3(??j(L(n))) c 3>(^(P(n), log(n))) if and o nly if P (n) lo g (L (n )/lo g (n )). P r o o f : T h is fo llo w s im m e d ia te ly fro m T h e o re m 2 .4 .3 and L e m m a 2 . 4 .1 0 . ■ S e c tio n 2 .5 M u ltip le P a s s e s a s a C o m p le x ity M e a s u re W e c o n clu d e th is c h a p te r on m u lti- p a s s tr a n s d u c e r s b y show ing th a t o u r p a s s m e a s u r e s a tis f ie s B lu m 's a x io m s fo r g e n e ra l c o m p le x ity m e a s u r e s [2 ], W hile th is is n o t a s u r p r is in g r e s u lt, it d o e s e s ta b lis h th e p a s s m e a s u r e a s h a rm o n io u s w ith o th e r a u to m a ta th e o ry a p p ro a c h e s to c o m p le x ity . In a d d itio n , in o r d e r to show th e B lu m a x io m s , w e m u s t d efin e and d is c u s s T u rin g M a c h in e s. In th e n e x t c h a p te r w e m a k e f u r th e r u s e of T u rin g M a c h in e s. T u rin g M a c h in e s and R e c u rs iv e E n u m e ra b ility : B a s ic D e fin itio n s D e fin itio n 2 . 5 . 1 : A T u rin g M ach in e M is a q u in tu p le M = (K, S, 6, Pq, F) w h e re : 47 1. K is a fin ite , n o n -e m p ty s e t (the s e t o f s t a t e s ); 2 . E is a fin ite , n o n -e m p ty s e t (th e in p u t/o u tp u t/c o m p u ta - tio n a l a lp h a b e t); E fl K = 0; 3 . 6 is a m ap p in g (th e tr a n s itio n m a p p in g ): 6 :(K-F)x(ZU {#))■* KxEX {+1,0,-1}; 4 . pQ € K is th e s t a r t s ta t e ; 5 . F c z K is th e s e t o f a c c e p tin g s t a t e s . W e le t TM = {M| M is a T u rin g M achine} . D e fin itio n 2 . 5 . 2 : L e t M = (K, E, 6, Pg, F) b e a T u rin g M a c h in e . A n ID fo r M is a s tr in g upv w h e re : 1. u € E * ; 2 . p € K; and 3 . v € E * . D e fin itio n 2 . 5 . 3 : L e t i = u ap b v , w h e re u, v € E , p € K and a , b € E U (tf}> b e a n ID fo r T u rin g M a ch in e M = (K ,E , 6, Pg, F ) . L e t j b e an ID fo r M . T h en it is s a id th a t i y ie ld s i w ith r e s p e c t to M (w ritte n i j) i f one o f th e follow ing h o ld s: 1. 6(p, b) = (p ', b ' , -1 ) an d j = u p 'a b 'v ; 2 . 6(p, b) = (p , » b ', 0) a n d j = u a p 'b 'v ; o r 3 . 6(p, b) = (p ', b ', +1) and j = u a b 'p 'v . We le t f-rrr b e th e re fle x iv e , tr a n s itiv e c lo s u r e o f I — - . M M D e fin itio n 2 . 5 . 4 : L e t M = (K, E, 6» Pq» F) b e a T u rin g M a c h in e . If PqW upv, w h e re p € F , th e n M is s a id to h a lt on w . In th is c a s e w e w r ite M(w) = u v . A s u s u a l, w e le t 3>(M) = {(w, v)| M(w) = v) . D e fin itio n 2 . 5 . 5 : A s e t A is c a lle d R e c u rs iv e ly E n u m e ra b le if th e r e is a T u rin g M ach in e M su c h th a t A = T (M ). W e le t R E = {A |A is R e c u rs iv e ly E n u m e ra b le ) . N o te, th e n , th a t R E = J (T M ). T h e o re m 2 . 5 . 1 : L e t L(n) 5 c, a c o n s ta n t, th e n fo r an y M € TM th e r e is an M^ € # |(P (n ), L(n)) su ch th a t 3<(M) = 3<(Mj), fo r so m e P (n ). P r o o f : W e c o n s tru c t M^ to s im u la te M . F o r e a c h tr a n s itio n in M 's c o m p u ta tio n , M^ m a k e s one p a s s . C le a r ly th e r e is no d iffic u l ty in c o n s tru c tin g su c h a tr a n s d u c e r . In th is c a s e w e h a v e th a t th e p a s s bound P (n) = m ax( {m | M(w) = v in m tr a n s itio n s an d | wf = n ) ). M j m a y b e c o n s tru c te d to u s e no w o rk in g s p a c e . A ll in fo rm a tio n m a y b e s to r e d in fin ite m e m o ry . So M j 6 ^ (L (n )), w h e re L(n) a c, a c o n s ta n t. N ote th a t M is an a r b i t r a r y T u rin g M ach in e an d is n o t g u a r a n te e d to h a lt on e v e ry in p u t. ■ D e fin itio n 2 . 5 . 6 : L e t f b e a fu n c tio n on E * x E * . T he fu n c tio n f is c a lle d p a r ti a l r e c u r s iv e if th e r e is a T u rin g M a ch in e M su c h th a t 49 f = 3>(M). T he fu n c tio n f is c a lle d to ta l r e c u r s iv e if f is p a r tia l r e c u r s iv e w ith f = 3*(M), an d in a d d itio n , M h a lts on w fo r a ll w € E * . N ote th a t th e s e t of p a r ti a l r e c u r s iv e fu n ctio n s is th e s e t 3(T M ) an d th a t th e s e t of to ta l r e c u r s iv e fu n c tio n s is a s u b s e t of th e s e t o f p a r tia l r e c u r s iv e fu n c tio n s . B lu m 's A x io m s M . B lu m [2] d e v e lo p e d tw o v e ry w e a k and g e n e r a l a x io m s to c h a r a c te r i z e c o m p le x ity m e a s u r e s . W e d u p lic a te B o ro d in 's s ta t e m e n t of th e A x io m s [3 ], A n e ffe c tiv e l i s t (c P q»cp^, » ***) °* Pa r ti a l r e c u r s iv e fu n c tio n s is p o s tu la te d su c h th a t th is l i s t s a tis f ie s th e S ^ 1 th e o re m an d th e u n iv e rs a l T u rin g m a c h in e th e o r e m . T he l i s t {cp.} sh o u ld b e th o u g h t o f a s a l i s t o f a lg o r ith m s o r d e v ic e s fo r c o m p u tin g th e p a r tia l r e c u r s iv e fu n c tio n s . A p a r ti a l fu n c tio n is a s s o c ia te d w ith e a c h a lg o rith m cp^. T he s e t o f fu n c tio n s s a t i s fie s th e fo llo w in g a x io m s . A x io m 1: $^(x) is d e fin e d »cpj(x) is d e fin e d . A x io m 2: T h e fu n c tio n C (i, x , m ) = 0 o th e rw is e is to ta l r e c u r s iv e The fu n c tio n $> . c o u n ts th e a m o u n t of r e s o u r c e u se d i 50 by c d . in its com p u tation . P a s s e s S a tisfy th e B lu m A x io n s fo r C o m p le x ity M e a s u re s To show th a t th e n u m b e r o f p a s s e s ta k e n by m u lti- p a s s t r a n s d u c e r s c a n s e r v e a s a c o m p le x ity m e a s u r e in B lu m 's s e n s e w e ta k e th r e e s t e p s . 1 . A s e t E is c o n s tru c te d an d show n to b e a n e ffe c tiv e l i s t of a ll p a r tia l r e c u r s iv e fu n c tio n s . E s a tis f ie s th e S ^ 1 th e o re m an d th e U n iv e rs a l T u rin g M ach in e T h e o re m . 2 . A s e t M / of p a s s co u n tin g T u rin g M a c h in e s is c o n s tru c te d . T h a t s e t is show n to s a tis f y a x io m 1 of B lu m 's s c h e m e . It is a ls o show n th a t fo r e a c h m u lti- p a s s tr a n s d u c e r M ., if M. h a lts on in p u t w in n p a s s e s th e n M /(w ) = n . 3 . A T u rin g M a c h in e M is c o n s tru c te d su ch th a t %(M) is to ta l r e c u r s iv e and M c o m p u te s th e fu n c tio n in d ic a te d in B lu m 's a x io m 2 . C o n s id e r th e s e t o f a ll m u lti- p a s s t r a n s d u c e r s . S ta n d a rd te c h n iq u e s [ 5 ] , [27] p e r m it th e en co d in g o f d e s c rip tio n s of th e s e m u lti- p a s s tr a n s d u c e r s in to an y a lp h a b e t o f 2 sy m b o ls o r m o r e . D e fin itio n 2 . 5 .7 : L e t E = f c d f /c o _ * € E* is an en co d in g o f s o m e m u lti - — — — — — — — j x p a s s tr a n s d u c e r } . T he c p ^ m a y b e a s s u m e d o r d e r e d by: 1 . f<p.| 3 fcpi+ 1 | ; and 2 . if | cp-f = !cpi+ 1 l th e n cp . p re c e d e s cp.^j a c c o rd in g to so m e a g r e e d le x ic o g ra p h ic o r d e r in g o f th e sy m b o ls o f E . 51 It is a s s e r te d w ith o u t p ro o f: T h e o re m 2 . 5 . 2 : 1 . E is R e c u rs iv e ly E n u m e ra b le ; 2 . E s a tis f ie s th e S ^ 1 th e o re m ; 3 . E s a tis f ie s th e U n iv e rs a l T u rin g M ach in e T h e o re m ; and 4 . F o r e a c h m u lti- p a s s tr a n s d u c e r M ., th e r e is a c p ^ € E su c h th a t cp . is a n en co d in g of . F o r d e ta ils on how th e s e a s s e r tio n s m ig h t b e p ro v e d , s e e [~ J, [19] o r [2 7 ], w h e re n e a r ly id e n tic a l a s s e r tio n s a r e p ro v e d . T h e o re m 2 . 5 . 3 : E l i s t s a ll p a r tia l r e c u r s iv e fu n c tio n s . P ro o f: E a c h e le m e n t c d . of E id e n tifie s so m e tr a n s d u c e r M .. W e ■ ■ i le t cpj r e p r e s e n t th e fu n c tio n co m p u te d b y a s w e ll a s th e e n c o d in g o f M j. T h a t i s , w e d e fin e cp^(w) = (jgfM -jjw)• E v e r y m u lti- p a s s tr a n s d u c e r is in c lu d e d in th e l i s t E . So by T h e o re m 2 . 5 .1 , e v e r y p a r ti a l r e c u r s iv e fu n c tio n is in c lu d e d in th e l i s t E . ■ T h e o re m 2 . 5 . 4 : L e t {M.} b e th e s e t o f a ll t r a n s d u c e r s . T h en th e r e is a s e t [ M / | m / 6 T M ] su c h th a t fo r a ll w : 1. M /(w ) is d e fin e d if and o n ly if M^(w) is d efin e d ; and ! n , if M. h a lts on w in n p a s s e s , 1 u n d e fin e d o th e rw is e . P ro o f: M / is c o n s tru c te d to s im u la te M .. H o w e v e r, M / is a x x * 1 52 T u rin g M ach in e an d n o t a tra n s d u c e r., So M / is c o n s tru c te d so th a t a ll th e in p u t and o u tp u t ta p e s g e n e ra te d b y M. d u rin g its p a s s e s r e m a in on th e one T u rin g M ach in e ta p e . L e t th e in p u t to M. b e £w$ and th e o u tp u t of th e f i r s t p a s s b e £ w j$ . T h e in p u t to M / is a ls o £ w $ . A fte r sim u la tin g th e f i r s t p a s s of M j, th e T u rin g M ach in e ta p e of M / a p p e a rs £ w $ £ w j$ . S u b se q u e n t p a s s e s follow th e s a m e p a tte r n . If M. h a lts , th e n M / is c o n s tru c te d to te rm in a te th e s im u la tio n of M. an d co u n t th e n u m - 1 b e r o f p a s s e s m a d e b y M .. It is t r i v ia l to m a k e th a t co u n t, sin c e th e in p u t and o u tp u t fro m e a c h p a s s is s til l a v a ila b le on th e ta p e fo r M / . M / is c o n s tru c te d th e n to le a v e on th e ta p e o nly th e co u n t ju s t m a d e , e r a s in g a ll o th e r in fo rm a tio n . M / s a tis f ie s a x io m 1. F o r if M. h a lts th e n M / h a lts and if 1 X I M . d o es n o t h a lt th e n M / d o es n o t h a l t, m / c o m p u te s th e fu n ctio n : l i i ! n, if M. h a lts on w in n p a s s e s 1 u n d efin e d o th e rw is e C le a r ly th e r e a r e su c h T u rin g M ach in e s M /. ■ N o rm a lly a T u rin g M ach in e is c o n s id e re d to c o m p u te a fu n c tio n w ith one a r g u m e n t. I t is p o s s ib le , h o w e v e r, to c o n s id e r so m e p a r tic u la r sy m b o l of th e in p u t a lp h a b e t to b e a p u n c tu a tio n m a rk — e . g . th e c o m m a sy m b o l — an d h e n c e to s e e a n in p u t s tr in g a s an n -tu p le s e p a r a te d b y c o m m a s . In th e follow ing th e o re m , a T u rin g M a c h in e is c o n s id e re d to h a v e tr i p le s a s in p u ts . A ll d e fin itio n s a r e a s s u m e d to c a r r y o v e r in th e n a tu r a l w ay to th is c a s e . 53 T h e o re m 2 . 5 . 5 : L e t { M /} b e th e s e t of T u rin g M a c h in e s d e fin e d in T h e o re m 2 . 5 . 4 . T h en , th e r e is a T u rin g M a ch in e M su ch th a t: 1. 5?(M) is to ta l r e c u r s iv e ; and P r o o f : M o p e r a te s b y s im u la tin g M / fo r n p a s s e s . S tep 1 . R e c o n s tru c t an d s im u la te cp^, M. and M / . S in ce th e l i s t E of D e fin itio n 2 . 5 .7 is r e c u r s iv e ly e n u m e ra b le , b y T h e o re m 2 . 5 .2 , i t is p o s s ib le to c o n s tr u c t a n M to r e c o n s tr u c t cp.. S in ce b y th e s a m e d e fin itio n an d th e o re m , E s a tis f ie s th e S ^ 1 th e o re m an d th e U n iv e rs a l T u rin g M ach in e th e o re m , it is p o s s ib le to b u ild M to s im u la te M / sim u la tin g M ^. H en ce in S tep 1 M s im u la te s M / o p e ra tin g on th e in p u t w . S te p 2 . If M /(w ) = n th e n M te rm in a te s w ith a 1 on its ta p e . If M /(w ) = m ^ n th e n M te r m in a te s w ith a 0 on its ta p e . If M / d o e s n o t te r m in a te on w , th e n M m u s t s till te r m in a te w ith a n o u tp u t o f 0. M is c o n s tru c te d so th a t if, in th e s im u la tio n of M /, M / ta k e s l ’ i m o r e th a n n s im u la te d p a s s e s , th e n M te r m in a te s th e s im u la tio n an d h a lts w ith an o u tp u t of 0 . ■ 0 o th e rw is e 1 if M .'(w) = n C H A P T E R 3 SIN G L E PASS TRA N SDU CERS In C h a p te r 3, p r o p e r tie s o f sin g le p a s s tr a n s d u c e r s a r e in v e s tig a te d . F i r s t , in S e c tio n 3 .1 , a n in te r e s tin g r e la tio n s h ip b e tw e e n tr a n s d u c e r s an d th e r e c u r s iv e ly e n u m e ra b le s e ts is d e v e lo p e d . L e t L(n) log(log(n)) and le t A b e an y r e c u r s iv e ly e n u m e ra b le s e t. T h en th e r e is a tr a n s d u c e r M € #f(L(n)) su c h th a t A is th e ra n g e of M . T h a t is , M (E*) = /?(M) = A . F r o m th is r e s u lt it fo llo w s th a t c e r ta in p r e d ic a te s on s u b c la s s e s of tr a n s d u c e r s a r e n o t d e c id a b le . In S e c tio n 3 .2 c lo s u r e of p r o p e r tie s of tr a n s d u c e r s a r e ex a m in e d . It is show n th a t 3'(7)\(log(n))) is c lo se d u n d e r p r e - im a g e b y an y M € ^ (lo g (n )), an d h e n c e by an y h o m o m o rp h is m . I t is a ls o show n th a t ^ (^ (lo g fa )) is n o t c lo s e d u n d e r h o m o m o rp h is m . P e r h a p s m o r e in te r e s tin g , an a lg e b r a ic fo rm u la tio n o f c lo s u r e is d e v e lo p e d . L e t F b e an y c la s s o f fu n c tio n s . It is show n th a t: 1. c lo s u r e o f ,T(F) u n d e r p r e - im a g e b y m e m b e r s o f F f o l low s fro m th e c lo s u r e o f th e c la s s F it s e lf u n d e r co m p o s itio n w ith its e lf ; and 2 . c lo s u r e o f J"(F) u n d e r s e lf - tr a n s d u c tio n — i . e . m a p p in g s b y m e m b e r s o f F — follow s fro m th e c lo s u r e of th e c la s s F it s e lf u n d e r c o m p o sitio n w ith in v e r s e s o f m e m b e rs of F . 54 55 T h a t is , one m a y p ro v e a c lo s u r e r e s u l t b y c o n s tru c tin g a fu n c tio n w h ich is o p p o site to th e d e s ir e d d ir e c tio n of th e c lo s u r e . F o r e x a m p le , to show c lo s u r e u n d e r h o m o m o rp h ic p r e - im a g e , one m a y c o n s tr u c t a fu n c tio n to ta k e a h o m o m o rp h is m . T he fin a l s e c tio n in v e s tig a te s th e r e la tio n s h ip b e tw e e n tim e , s p a c e and p u sh dow n s to r e s w ith r e s p e c t to sin g le p a s s tr a n s d u c e r s . It h a s b e e n an o p en q u e s tio n w h e th e r th e a d d itio n o f a p u sh dow n s to r e ad d s an y c o m p u ta tio n a l p o w er to sin g le p a s s a u to m a ta [ l ] . T h is la s t s e c tio n e s ta b lis h e s th a t th e a d d itio n of a p u sh dow n s to r e d o es ad d tra n s d u c tio n p o w e r. W h e th e r o r n o t a p u sh dow n s to r e in c r e a s e s th e a c c e p tin g p o w er of sin g le p a s s tr a n s d u c e r s r e m a in s unknow n. T he p ro o f te c h n iq u e w h ic h e s ta b lis h e s th e m a in r e s u l t of S e c tio n 3 .3 is a ls o o f i n t e r e s t . It sh o w s how to c o n s tr u c t a n a u to m - T ( t i \ a to n w ith a n L(n) s p a c e bound an d a c sp a c e bou n d ed p u sh c L(n) dow n s to r e and w h ich c a n co u n t fro m 1 to d , fo r a r b i t r a r y c . d S N . S e c tio n 3 .1 T r a n s d u c e r s and R e c u r s iv e E n u m e ra b ility T h is s h o r t s e c tio n in v e s tig a te s th e r e la tio n s h ip b e tw e e n t r a n s d u c e rs and th e R E s e t s . F i r s t w e d e fin e d e c id a b ility . T h en w e show th a t fo r an y s p a c e bound L (n), th e r e is an M S 7ft(Li(n)) w h ich a c c e p ts a v a r ia n t o f th e a c c e p tin g ID se q u e n c e fo r an y T M . It is a ls o show n th a t fo r an y A S R E th e r e is a n M S ?ft(Li(n) su ch th a t £ (M ) = M (E*) = A . R e c a ll th a t ^ (M ) d e n o te s th e ra n g e o f th e fu n c tio n d e fin e d b y M . D e fin itio n 3 . 1 . 1 : A p r e d ic a te P o v e r a n a lp h a b e t E is s a id to be r e c u r s iv e ly d e c id a b le , o r ju s t d e c id a b le , i f th e r e is a n M € TM su ch th a t P(w ) iff w S J"(M) fo r a ll w € E *. If a p r e d ic a te is n o t d e c id a b le it is s o m e tim e s c a lle d u n d e c id a b le . R e c a ll fro m th e p r e lim in a r ie s th a t a p r e d ic a te m a y b e c o n s id e r e d e ith e r a s a s e t of n - tu p le s of s tr in g s o v e r so m e a lp h a b e t E, w h e re th e c o m m a sy m b o l , (£ E, o r a s a s e t o f s tr in g s o v e r th e a lp h a b e t E U {, } • In S e c tio n 2 .4 , d e s c r ip tio n s o f tr a n s d u c e r s a r e co d ed in to s t r i n g s . H e re w e g e n e r a liz e th a t te c h n iq u e to in c lu d e en c o d in g s o f T u rin g M a c h in e s . F o r c o m p le te d e ta ils s e e [ 5 ] , [2 7 ], W ith T u r in g M a c h in e s a v a ila b le a s in p u ts , w e m a y d is c u s s p r e d ic a te s on T u rin g M a c h in e s an d tr a n s d u c e r s an d d e te r m in e th e d e c id a b ility of th e s e p r e d ic a te s . It is w e ll know n th a t th e r e is n o r e c u r s iv e p ro c e d u re to d e te r m in e , fo r a n a r b i t r a r y T u rin g M a c h in e M an d a r b i t r a r y in p u t w , w h e th e r w € ,T (M ), T h is r e s u l t is c a lle d th e u n d e c id a b ility o f th e H a ltin g P r o b le m . T h e o re m 3 . 1 . 1 ; L e t P (M , w) b e th e p r e d ic a te : w € T (M ). T h e n P is n o t d e c id a b le . P r o o f : T h is is a w e ll know n r e s u l t . S ee, fo r e x a m p le , [5 ], ■ 57 W e now show th a t th e sp a c e bou n d ed a u to m a ta h a v e m a n y p r o p e r tie s w h ich a r e n o t d e c id a b le . S u p p o se th a t th e sy m b o l / is n o t p a r t of th e a lp h a b e t of so m e T u rin g M ach in e M . (If i t is , th e n w e c a n p ic k a n o th e r sy m b o l w h ich is n o t.) A s s u m e th a t fo r a g iv e n in p u t w , th e ID 's a r e : PqW = Xq,X j , • • • ,X j , • • • , w h e re x^ x .^ j fo r a ll i g 0 . W e c a n s tr in g th e ID 's to g e th e r, s e p a r a te d b y s la s h e s , to fo rm a n ID S e q u e n c e : x = Xq/ x j / • • • / x . / • • • , w h e re th e ID S eq u en ce x is one long s tr in g r a t h e r th a n m a n y s h o r t s t r i n g s . T h is m a y b e fo rm a liz e d a s fo llo w s . D e fin itio n 3 . 1 . 3 : F o r M = (K, S, 5, PqF) € TM , a n . ID se q u e n c e is a s tr in g x = X q /x j/« » » /x n w h e re : (i) Xj is a n ID fo r M; (ii) Xj x .+1, fo r O i i g n - 1 ; (iii) th e sy m b o l / £ (E U {16}); an d (iv) x Q = pQ w . L e t xn = u p v , w h e re u , v € E* an d p€K. I f p € F th e n x is s a id to b e a n a c c e p tin g ID s e q u e n c e . T h e o re m 3 . 1 . 2 : If L(n) s . log(n) th e n fo r e v e r y M € TM th e r e is a n M '€ ? ft(L (n )) su c h th a t T (M #) = {xf x is a n a c c e p tin g se q u e n c e fo r M} . P r o o f : T h is th e o re m h a s b e e n know n fo r y e a r s . S ee, fo r e x a m p le [2 3 ]. ■ 58 T h e o re m 3 . 1 . 3 ; F o r a n y M 6 TM an d an y L (n ),th e re is a n M 7 € ^ L ( n ) ) su c h th a t ^ (M ') = X , w h e re X = fOax |x = X Q /x j/» » » /x n is a n a c c e p tin g ID s e q u e n c e fo r M and L (a + |x f) & m a x ( lo g ( |x .f ))} . i P r o o f : L e t M 7 b e th e a u to m a to n w h ich o p e r a te s a c c o rd in g to th e fo llo w in g a lg o rith m . S te p 1 . D e te rm in e w h e th e r th e in p u t is of th e fo rm 0a x j / x ^ / • • • / x ^ , w h e re x. is a n ID fo r T u rin g M ach in e M . S te p 2 . F o r e a c h i, 0 £ i £ n - 1 , d e te rm in e w h e th e r x. is c o n s tr u c te d to c o m p a re x ., sy m b o l b y sy m b o l, w ith S te p 3 . D e te rm in e w h e th e r th e s ta te of ID x n is a n a c c e p tin g s ta te fo r T u rin g M ach in e M . S te p 4 . If th e th r e e s e ts of c o n d itio n s s e t o u t in th e th r e e s te p s a r e m e t, th e n M 7 is to a c c e p t th e in p u t . It is c la im e d th a t M 7 m a y b e c o n s tru c te d to p e r f o r m S te p s 1 an d 3 w ith no a u x ilia r y s to r a g e . To e x e c u te S te p 2, M 7 is co n s tr u c te d to k e e p on its w o rk in g ta p e , th e p o s itio n s in x^ and o f th e s y m b o ls s c a n n e d . S in ce M 7 is c o n s tru c te d to c o m p a re x. and x j +i sy m b o l b y sy m b o l, one p o s itio n fo r e a c h of th e s e two s tr in g s is a ll th a t is m a in ta in e d a t one tim e . M 7 is b u ilt to s to r e th e s e p o s itio n s i n l o g ( f x .f ) and lo g (fx ^ +j f ) s p a c e . H en ce in e x e c u t in g a ll th r e e s te p s M 7 u s e s o n ly m a x (lo g (x .)) w o rk in g ta p e s q u a r e s . i S in c e b y o u r u m b r e lla a s s u m p tio n L(n) is s e lf-c o m p u ta b le , i t is p o s s ib le to c o n s tr u c t M 7 n o t to u s e m o r e th a n L ( [ w |) w o rk in g 59 ta p e s q u a r e s . H e n c e , if th e in p u t is o f th e p r o p e r fo rm an d if M* o p e r a te s a s in d ic a te d ab o v e in th e a llo w e d sp a c e th e n L (a+ fx f ) a m a x (lo g (f X jf)). ■ i C o r o lla r y 3 . 1 . 4 : F o r an y M S TM , th e r e is an M* € #f(L(n)) su ch th a t M '(E * ) = f f ( M ') =J"(M ). P r o o f : W e a l t e r M* fro m T h e o re m 3 .1 .3 so th a t th e o u tp u t is xn if th e in p u t is a c c e p te d . ■ W e s tre n g th e n a th e o re m of R o s e n b e rg 's [2 3 ], C o r o lla r y 3 . 1 . 5 : F o r so m e M 6 ? [(L (n )) th e follow ing a r e a ll u n d e- c id a b le fo r J"(M) and /^(M ). (L e t X s ta n d fo r e ith e r J"(M) o r /p(M ).) (1) X = 0; (2) X = E*; (3) X is fin ite ; (4) X is r e g u la r ; (5) X is c o n te x t f r e e . P r o o f : (1) fo llo w s im m e d ia te ly fro m C o r o lla r y 3 . 1 .4 and th e u n d e c id a b ility of th e h a ltin g p ro b le m T h e o re m 3 . 1 . 1 . S ta n d a rd r e d u c tio n te c h n iq u e s (s e e [23] fo r m o r e d e ta ils ) y ie ld (2), (3), (4) an d (5) d ir e c tly . ■ I t is n o te w o rth y th a t in n o n e o f th e th r e e p re c e d in g r e s u lts w a s a lo w e r lim it p la c e d on th e s p a c e bound L (n ). T h is p ro d u c e s a n a p p a re n t c o n flic t w ith a r e s u l t of S te a rn s e t . a l . [2 8 ]. T h ey found th a t w h e n e v e r L(n) <• log(log(n)) th e n J ( ^ ( L ( n ) ) ) is a w e ll know n c la s s c a lle d th e R e g u la r S e ts . I t fo llo w s fro m th e ir w o rk th a t 3>(^(L(n))) is w h a t is know n a s a fin ite s ta te tra n s d u c tio n and th a t /? (77l(h(n))) is a ls o a s e t o f r e g u la r s e t s . W hy, th e n w a s it 60 n o t n e c e s s a r y to h y p o th e s iz e , in th e th r e e p re c e d in g r e s u lt s th a t L(n) lo g (lo g (n ))? O u r u m b r e lla a s s u m p tio n s e r v e d th e s a m e fu n c tio n . W e h a v e a s s u m e d th a t L(n) is s e lf-c o m p u ta b le . C o r o lla r y 3 . 1 . 6 : If L(n) <• lo g (lo g (n )), th e n L(n) is n o t s e lf - c o m p u ta b le . P r o o f : I m m e d ia te . ■ S e c tio n 3 .2 C lo s u re R e s u lts T h is s e c tio n in v e s tig a te s c lo s u r e p r o p e r ti e s . It is show n th a t J-(^ (lo g (n ))) is c lo s e d u n d e r h o m o m o rp h ic p r e - im a g e b u t n o t c lo s e d u n d e r h o m o m o rp h is m . T h e f i r s t r e s u lt is n o t new and th e se c o n d i s . P e r h a p s m o r e in te r e s tin g th a n e ith e r s p e c ific r e s u lt, a lg e b r a ic p r o p e r tie s a r e g iv e n w h ic h g u a ra n te e v a rio u s ty p e s o f c lo s u r e an d w h ic h e lu c id a te th e m e th o d s b eh in d m a n y a u to m a ta c lo s u r e p r o o f s . S u ffic ie n c y C o n d itio n fo r C lo s u re u n d e r P r e - I m a g e R e c a ll th a t if F and G a r e tw o s e ts of fu n c tio n s, th e n J"(F) — th e c la s s o f d o m a in s of m e m b e rs f € F — is s a id to b e c lo s e d u n d e r G p r e - im a g e if fo r a l l f € F an d g 6 G th e r e is a n h € F su c h th a t J1(h) = g 'V ( f ) ) . T h e o re m 3 . 2 . 1 : L e t F an d G b e s e ts o f fu n c tio n s . If FoG c F — i . e . if F is c lo s e d u n d e r c o m p o s itio n w ith G — th e n J"(F) is c lo s e d u n d e r G p r e - im a g e . 61 P r o o f : W e m u s t show th a t fo r a n y f € F , g € G, th e r e is a n h 6 F su c h th a t J(h) = g" ^(,T(f)). W e know by h y p o th e sis th a t th e r e is an h = fog € F . B ut th e n J"(h) = J{iog) = {xf 3(y, z) € f su ch th a t (x, y) € g} = {xf 3 y 6 j" (f) su c h th a t (x, y) € g) = {xf'3y€ J(i) su ch th a t g” *(y) = x} = g_1(X(f))-. ■ T h is r e s u l t a p p lie s to tr a n s d u c e r s d ir e c tly . W e know fro m T h e o re m 2 . 2 . 8 th a t th e c la s s «?(#j(log(n))) is c lo s e d u n d e r c o m p o sitio n . So w e m a y le t F = G = 3 ?(^ (lo g (n ))). C o r o lla r y 3 . 2 . 2 ; J"(?^(log(n))) is c lo s e d u n d e r p r e - im a g e . P r o o f : Im m e d ia te . ■ Now i t is im m e d ia te th a t ^ (lo g (n ))) is c lo s e d u n d e r h o m o m o rp h ic p r e - im a g e . C o r o lla r y 3 . 2 . 3 ; J( ^ (lo g (n ))) is c lo s e d u n d e r h o m o m o rp h ic p r e im a g e . P r o o f : L e t H b e th e c la s s o f h o m o m o rp h is m s . I t is a s s e r te d th a t H e 3(#?(L (n))), ^o r an Y L (n ). ■ A ny h o m o m o rp h is m h on E*X E* is c h a r a c te r iz e d b y a m a p ping h ' on E x E * . T h u s fo r an y s tr in g , a h o m o m o rp h is m h t r a n s la te s th a t s tr in g , sy m b o l b y sy m b o l, v ia h /, in to so m e o th e r s tr in g . H e n c e a h o m o m o rp h is m r e q u ir e s no c o m p u ta tio n a l m e m o ry , and 62 a ll c l a s s e s o f sp a c e b o u n d ed tr a n s d u c e r s • p e r f o r m a ll h o m o m o r- p h is m s . So s in c e H c5 (W i(lo g (n ))) and s in c e TCWilog(n))) is c lo se d u n d e r p r e - im a g e , b y C o r o lla r y 3 .2 .2 th e n «7*(^(log(n))) is c lo se d u n d e r h o m o m o rp h ic p r e - im a g e , ■ T h e o re m 3 .2 .1 an d th e tw o c o r o lla r ie s w h ich fo llo w f o r m a l iz e and g e n e r a liz e a p ro o f te c h n iq u e found fre q u e n tly in a u to m a ta th e o r y . C o n s id e r a s e t o f la n g u a g e s L and a s e t of a c c e p to r s A su ch th a t L = T (A ). H ow d o es one show th a t L is c lo s e d u n d e r h o m o m o rp h ic p r e - im a g e ? A s s u m e one knew th a t A w e r e c lo s e d u n d e r h o m o m o rp h is m . T h u s fo r an y M € A and an y h € H , th e s e t of h o m o m o rp h is m s , th e r e is a n M # = M oh € A . T h en w h a t is w € JTfM ') iff w € J"(Moh) iff h(w) 6 J*(M) iff w 6 h “ 1(^'(M )). H en ce ^ (M ') = h"*(,T(M )), an d so i t is c o n clu d ed th a t J(A) is c lo se d u n d e r h o m o m o rp h ic p r e - im a g e . H e re to fo re , m o s t p ro o fs of h o m o m o rp h ic p r e - im a g e c lo s u r e h a v e p ro c e e d e d in th e r e v e r s e d ir e c tio n . T y p ic a lly o ne p ic k s a n a r b i t r a r y a c c e p to r M € A , and t r i e s to show th a t fo r an y h o m o m o r p h is m h , h € L . T o do so one show s how to c o n s tr u c t an M '€ A su ch th a t = h" *(,T(M)). B ut M ' is o ften th e a u to m a to n 63 w h ic h , fo r an y in p u t w , f i r s t fin d s h(w ) an d th e n s im u la te s M on th e r e s u l t h(w) = v . T h a t i s , M 7 a s k s if h(w) € T (M ). If so , w € h ” * (T (M )). T hus M 7 = M oh s in c e M 7 f i r s t p e r f o r m s h(w ) an d th e n M (h(w )). T he tr a d itio n a l te c h n iq u e d o e s n o t fo cu s on a u to m a ta a s t r a n s d u c e rs — o n ly a s a c c e p to r s . B y lo o k in g f i r s t a t a u to m a ta a s t r a n s d u c e r s , an d f ro m th e r e m o v in g on to a u to m a ta a s a c c e p to r s , w e c la r if y an d s im p lify th e c lo s u r e p ro o f fo r h o m o m o rp h ic p r e - im a g e . S u ffic ie n c y C o n d itio n fo r C lo s u re u n d e r T ra n s d u c tio n T h e c lo s u r e o f a c la s s o f fu n c tio n s F u n d e r c o m p o sitio n g u a r a n te e s th a t 3~(F) is c lo s e d u n d e r p r e - im a g e , is th e r e a s im ila r c o n d itio n to g u a ra n te e c lo s u r e u n d e r tra n s d u c tio n ? R e c a ll th a t a c la s s J ( F) is c lo s e d u n d e r tr a n s d u c tio n b y G if g(A) € .T(F) fo r a ll g 6 G an d a ll A € J"(F). T h e o re m 3 . 2 . 4 : L e t fog” '* ' 6 F fo r a ll f € F an d g € G . T h en J"(F) is c lo s e d u n d e r G tra n s d u c tio n . P r o o f : B y h y p o th e s is , fo r e v e r y f € F and g 6 G, th e r e is a n h = fo g " 1 € F . So J"(h) = >y(fog"1) = { y /(x , y) € g & (x, z) € f} = g W f)). ■ A s s u m e th a t th e s e t o f h o m o m o rp h is m s H is a s u b s e t o f th e s e t F an d fo r a ll f, g 6 F th a t fo g ” * - € F . T h en w e m a y co n clu d e th a t J"(F) is c lo s e d u n d e r h o m o m o rp h is m . In th is c a s e w e h a v e fo r e v e ry f, g 6 F th a t th e r e is a n h = fog * € F . H e n c e , 64 J th ) = ^ (fo g - 1 ) and w 6 J"(h) iff w € ^ (fo g " 1) iff g ' J (w )e J(f) if f w € g(T(f)). T h u s T (h) = g(J"(f)), fro m w h ic h i t fo llo w s th a t ?(F) is c lo s e d u n d e r s e lf - tr a n s d u c tio n an d h e n c e u n d e r h o m o m o rp h is m . T h e o re m s 3 .2 .1 and 3 . 2 . 4 f o r m a liz e tw o in te r e s tin g c lo s u r e c h a r a c te r i s t ic s fo r any- s e t o f fu n c tio n s F . 1 . C lo s u re of J"(F) u n d e r p r e - im a g e fo llo w s fro m c lo s u r e o f F u n d e r c o m p o sitio n w ith m e m b e r s o f F . 2 . C lo s u re o f ,7(F) u n d e r s e lf - tr a n s d u c tio n fo llo w s fro m c lo s u r e of F u n d e r c o m p o sitio n w ith in v e r s e s o f m e m b e r s o f F . T h a t is , to p ro v e a c lo s u r e r e s u lt , one c o n s tr u c ts an a u to m a to n w h ic h p e r f o r m s th e o p p o s ite o f th e c lo s u r e d e s ir e d : 1 . to p ro v e c lo s u r e u n d e r p r e - im a g e of m e m b e r s o f so m e s e t H , one show s c o m p o sitio n w ith m e m b e r s o f H its e lf ; w h ile 2 . to p ro v e c lo s u r e u n d e r tr a n s d u c tio n b y m e m b e r s o f so m e s e t H , o n e show s c o m p o sitio n w ith in v e r s e s o f m e m b e rs o f H . C o m b in in g th e s e r e s u lt s y ie ld s : C o r o lla r y 3 . 2 . 5 : If F , a c la s s o f fu n c tio n s, is c lo s e d u n d e r b o th c o m p o s itio n and in v e r s e th e n T{F) is c lo s e d u n d e r b o th p r e - im a g e an d s e lf - tr a n s d u c tio n . ty?(log(n))) is n o t C lo se d u n d e r H o m o m o rp h ism It h a s b e e n a n o p en q u e s tio n w h e th e r 7??(log(n)) is c lo s e d u n d e r h o m o m o rp h is m . U sin g a r e s u lt of G in sb u rg e t a l [9], w e now show th a t 3"{^ (lo g (n ))) is n o t c lo se d u n d e r h o m o m o rp h is m . R e c a ll th a t C F is th e s e t of c o n te x t f r e e la n g u a g e s an d R E th e s e t o f R e c u rs iv e ly E n u m e ra b le s e t s . G in sb u rg r e la te d th e c o n te x t f r e e la n g u a g e s, h o m o m o rp h is m s and r e c u r s iv e ly e n u m e r a b le s e t s . T h e o re m 3 . 2 . 6 : F o r an y A € R E th e r e a r e tw o s e ts B , C € C F an d a h o m o m o rp h is m h su ch th a t A = h(B fl C ). P r o o f : S ee [9 ]. ■ T h e re is a s u b c la s s of C F w h ic h is v e ry u s e fu l h e r e . T he D yck s e ts m a y b e r e la te d to th e c o n te x t f r e e la n g u a g e s v ia h o m o m o r p h is m s . D yck s e ts a r e n o t d e fin e d d ir e c tly s in c e th e follow ing s a tis f ie s o u r n e e d s . T h e o re m 3 . 2 . 7 : F o r an y A € C F th e r e is a D yck s e t D and two h o m o m o rp h is m s g an d h su ch th a t A = g *(h(D )). P r o o f : S ee [8 ]. ■ I t is a ls o w e ll know n th a t th e D yck s e ts a r e r e la te d to th e log(n) t r a n s d u c e r s . 66 L e m m a 3 . 2 . 8 : F o r an y D yck s e t D th e r e is a n M € #i(log(n)) su ch th a t D = M ). P r o o f : S ee [2 2 ], ■ A fin a l p r e lim in a r y to show ing th a t 3~ ( W?(log(n))) is n o t c lo s e d u n d e r tra n s d u c tio n b y h o m o m o rp h is m s is to show th a t ^ (lo g (n ))) is c lo s e d u n d e r in te r s e c tio n . L e m m a 3 . 2 . 9 : L e t M j, M 2 € ???(log(n))• T h en th e r e is an M 2 € ^ (lo g (n ))) su c h th a t T(M 3 ) = jf M j) n / ( M 2 ). P r o o f : L e t b e th e tr a n s d u c e r w h ic h s im u la te s f i r s t M j and th e n M 2 . C le a rly , if M j an d M 2 b o th h a lt in lo g (jw f) s p a c e on so m e in p u t w , th e n M 3 h a lts on w in lo g ( f w f ) s p a c e . L ik e w ise , if e ith e r M j o r M 2 fa ils to h a lt on w , th e n M^ d o es n o t h a lt on w . S in ce M j and M 2 a r e b o th log(n) tr a n s d u c e r s , n e ith e r h a lts on an in p u t in m o re th a n log(n) s p a c e . H en ce M 3 h a lts o n in p u ts in log(n) sp a c e o r n o t a t a ll, an d h e n c e M^ 6 ?^(log(n)L T h u s ^ (M j) n-T(M2 ) = J-(M3 ) € X (^ (lo g (n ))). ■ T h e o re m 3 . 2 .1 0 : ^ (lo g (n ))) is n o t c lo s e d u n d e r h o m o m o rp h is m . P r o o f : It is w e ll know n th a t R E # |(lo g (n ))). B u t fo r an y A € R E th e r e a r e tw o D yck s e ts B and C an d fiv e h o m o m o rp h is m s h j , h g , h 3 , h ^ and hg su c h th a t A = h j( h 2 *(h3 (B)) n h ^ ( h g ( C ) ) ) . T h is fo llo w s fro m T h e o re m s 3 .2 .6 an d 3 , 2 . 7 . B ut C o r o lla r y 3 . 2 .3 shows th a t «T(??j(log(n))) is c lo s e d u n d e r p r e - im a g e b y h o m o m o rp h is m s , 67 L e m m a 3 . 2 . 8 g u a ra n te e s th a t th e D yck s e ts a r e m e m b e rs of J"(#|(log(n))) an d L e m m a 3 . 2 .9 sh o w s th a t ^ (lo g (n ))) is c lo se d tin d e r in te r s e c tio n . H en ce if ?7j(log(n))) w e r e c lo s e d u n d e r h o m o m o r p h is m a s w e ll, w e w o u ld h a v e a c o n tra d ic tio n . F o r th e n , g iv e n A € R E , i t w ou ld follow th a t A € J"(^ (lo g (n ))), w h e n c e R E c J-(7?;(log(n))). So J"(^(lo g (n ))) is n o t c lo se d u n d e r h o m o m o rp h is m . ■ S e c tio n 3 .3 T im e , S p ace an d P u sh d o w n S to re s In th is s e c tio n w e in v e s tig a te th e r e la tio n s h ip b e tw e e n tim e , sp a c e an d pushdow n s t o r e s . O u r m a in r e s u lt p ro v id e s a p a r tia l a n s w e r to a n o p en q u e stio n : d o es th e a d d itio n o f a pushdow n s to r e to a sp a c e b o u n d ed tr a n s d u c e r add r e a l c o m p u ta tio n a l p o w e r? I t is show n th a t th e c la s s of L(n) s p a c e bou n d ed tr a n s d u c e r s is n o t e q u i- t \ v a le n t to th e c la s s of c tim e b ounded t r a n s d u c e r s . W e a lso d is p la y tw o s e ts o f tr a n s d u c e r s A and B su ch th a t J"(A) = ,T(B), b u t w h e re 3>(A) 4 5 (B ). T im e B ounded T u rin g M a c h in e s T h e n o tio n of a tim e bou n d ed c o m p u ta tio n is in tro d u c e d . D e fin itio n 3 . 3 . 1 : A T u rin g M ach in e MG T M is tim e bou n d ed b y a fu n c tio n T(n) of its in p u t, if M h a lts its c o m p u ta tio n on w in no m o r e th a n T ( |w |) s te p s w h e n e v e r M h a lts on w . W e le t J-^fT('tt')')"-' {Mf M is a T(n) tim e bounded T u rin g M achine} . 68 It h a s long b e e n know n th a t fo r an y L(n) sp a c e bounded co m - t /r% \ p u ta tio n th e r e is a n e q u iv a le n t c tim e b ounded c o m p u ta tio n . T h e o re m 3 . 3 . 1 ; F o r a ll L (n), 5 (^ (L (n ))) c 3 t ( ^ ( c L (n ))). P r o o f ; T r iv ia l. ■ T r a n s d u c e rs w ith A u x ilia ry P u sh d o w n S to re s S . C ook [4] r e c e n tly in tro d u c e d th e n o tio n of a n a u x ilia r y p u sh dow n s t o r e . C o n s id e r an y sp a c e bound L(n) and im a g in e a t r a n s d u c e r M S Wf(L(n)). Now su p p o se w e add an unbounded pushdow n s to r e to M . H ave w e in c r e a s e d M 's co m p u tin g p o w e r? C ook [4] show ed th a t th e s e a u x ilia r y pushdow n m a c h in e s w e re e q u iv a le n t to c e r ta in c l a s s e s of tim e b ounded a u to m a ta . D e fin itio n 3 . 3 . 2 ; A n a u x ilia r y pushdow n tr a n s d u c e r M is a 9 -tu p le M = (K, 2, £, $, T, A, 6, p0 , F) w h e re ; 1. K, E, £ q , £ , $, r , Pq and F a r e a s in a tr a n s d u c e r (s e e D e fin itio n 2 .1 .1 ) ; 2 . A is a fin ite , n o n -e m p ty s e t (th e pushdow n a lp h a b e t); 3 . 6 : (K-F)xE0 xrx A-»KX { - 1 ,0 ,+ 1 } 2 x A*x(EU{e}). If M u s e s no m o r e th a n L ( |w f ) s q u a re s of its w o rk ta p e w h en e v e r M h a lts on w , th e n M is s a id to be L(n) sp a c e b o u n d e d . T he T he pushdow n ta p e is n o t lim ite d . W e le t ^ ^ (L (n )) = {m | m is an L(n) s p a c e b o u n d ed , a u x ilia ry pushdow n tra n s d u c e r} . A n M € ^>^(L(n)) o p e ra te s a s fo llo w s. G iven a c u r r e n t s ta te 69 and s e t o f sy m b o ls sc a n n e d b y th e in p u t r e a d h e a d , th e w o rk ta p e h e a d an d th e pushdow n s to r e h e a d , M 's tr a n s itio n fu n c tio n g iv e s a new s ta te , a new sy m b o l fo r th e w o rk ta p e an d a new s tr in g fo r th e p u sh d o w n s t o r e . T h e old pushdow n s to r e sy m b o l is re m o v e d and th e new s tr in g r e p la c e s i t . T he tr a n s itio n fu n c tio n a ls o p ro v id e s d ir e c tio n s fo r th e h e a d s o n th e in p u t ta p e and th e w o rk ta p e . T he p u shdow n s to r e h e a d is a lw a y s a s s u m e d to b e sc a n n in g th e r ig h t m o s t o r to p sy m b o l o f th e pushdow n s t o r e . A fo r m a liz a tio n c a n b e found in C o o k [4 ], C o o k 's m a jo r r e s u lt is th e fo llo w in g . T h e o re m 3 . 3 . 2 : If L(n) log(n) th e n c L ^ ) ) = L (n ))). P r o o f : S ee C ook [4 ], ■ B y T h e o re m 3 .3 .1 3 (^ (L (n ))) c 3 O T |( c L (n )) ) . So ^ (^ (L (n ))) c T (r\\ )). C o m b in in g th is r e s u lt w ith th a t of th e p re v io u s th e o re m w e h a v e : J ( ^ ( L ( n ) ) ) c 7 ( ^ ( c L (n))) c J ( ^ ( L ( n ) ) ) . H en ce J(W |(L(n))) c ^'(^) ^ (L i(n )). Is th e c o n v e rs e a ls o v a lid ? T h is is a n o p en q u e s tio n . I f i t w e r e v a lid , th e n th e c o n v e rs e to T h e o re m 3 .3 .1 w ould a ls o b e v a lid . A ls o , w e w o u ld h a v e th e s u r p r is in g r e s u l t th a t ad d in g a p u sh dow n s to r e to a s p a c e b o u n d ed tr a n s d u c e r a d d s no a c c e p tin g p o w e r. W e do n o t a n s w e r th e c o m p le te o p en q u e s tio n . T h is s e c tio n c lo s e s w ith th e r e s u l t th a t 3<(^) ^ (L (n ))) ^ 3(??|(L(n))), a p a r tia l a n s w e r . ? W ( L ( n ) ) ) d g ( W L ( n ) ) ) c L(n) L e m m a 3 . 3 .3 ; T h e re is a n M € ^ ? ^ (L (n )) su c h th a t M (ln ) = l^ , 70 fo r a ll n . P r o o f : A tr a n s d u c e r M € &7I{{L (n)) is c o n s tru c te d to u s e its p u sh e d dow n s to r e an d its w o rk ta p e to c o u n t fro m 0 to d . M is c o n s tru c te d to co u n t on th e pushdow n ta p e . It is a s s u m e d th a t th e p u shdow n a lp h a b e t A h a s d s y m b o ls . It is th e in te r a c tio n b e tw e e n th e pushdow n ta p e an d th e w o rk ta p e th a t p ro v id e s th e k e y to th e p ro o f. T h e re a r e tw o a r e a s w h e re th is in te r a c tio n is im p o r ta n t. F i r s t , th e w o rk ta p e is u s e d to in itia liz e th e p u shdow n ta p e b e fo re th e s t a r t o f th e c o u n t. T he w o rk ta p e a lp h a b e t is a s s u m e d to T h a v e c s y m b o ls . M is c o n s tru c te d to co u n t fro m 1 to c on th e w o rk ta p e . A t e a c h s te p o f th e co u n t, a z e r o is in s e r te d in to th e pushdow n ta p e . A t th e end of th e in itia liz a tio n , th e p u sh d o w n ta p e c o n ta in s a s tr in g o f c ^ n ^ z e r o s . S eco n d , d u rin g th e cotint, th e w o rk ta p e s e r v e s to r e c o r d in f o rm a tio n lo s t fro m th e pushdow n ta p e . W h e n e v e r M " c a r r i e s " in to th e p u shdow n ta p e , M u s e s th e w o rk ta p e to r e c o r d th e n u m b e r o f z e r o s to r e tu r n to th e pushdow n s to r e a t th e end o f th e s e r i e s o f c a r r i e s . M o re d e ta ils o f th e a lg o rith m fo llo w . S te p 1 . M w r ite s Z 0 ^ n ^Z on its w o rk ta p e . M m a y p e r f o r m th is s te p s in c e L(n) is s e lf-c o m p u ta b le . T he sy m b o l Z is u s e d h e r e to m a r k th e en d s o f th e w o rk ta p e . c L (n > Ste P 2 - M w r ite s Z0 in to its p u shdow n s t o r e . M p e r f o r m s th is s te p b y u sin g its w o rk ta p e a s a c o u n te r. W e a s s u m e th a t M 's 71 w o rk ta p e a lp h a b e t h a s c s y m b o ls . M is c o n s tru c te d to co u n t fro m T / T1 ^ 1 to c o n th e w o rk ta p e . A t e a c h s te p o f th e co u n t, M w r ite s one a d d itio n a l 0 in to th e pushdow n s t o r e . L(n) d S te p 3 . M w r ite s 1 on its o u tp u t ta p e . M p e r f o r m s th is s te p b y u sin g its pushdow n s to r e a s a c o u n te r . W e a s s u m e th a t M 's p u shdow n a lp h a b e t h a s d s y m b o ls . M c o u n ts on th e pushdow n s to r e in th e fo llo w in g m a n n e r . T he c o n te n ts of th e p u shdow n s to r e a r e c o n s id e re d to b e an c L(n) in te g e r b e tw e e n 0 and d - 1. T he le a s t s ig n ific a n t d ig it is th e to p m o s t e le m e n t o f th e pushdow n s t o r e . A t th e s t a r t o f e a c h co u n t c y c le , th e w o rk ta p e is a ll z e r o s . T h e re a r e tw o c a s e s . C a s e 1 . T he to p m o s t e le m e n t of th e pushdow n s to r e is no t th e h ig h e s t v a lu e d of th e pushdow n d ig its . T h a t is , no c a r r y o c c u rs w hen a 1 is ad d ed to th e co u n t in th e p u shdow n s t o r e . In th is c a s e , M is b u ilt s im p ly to in c re m e n t th e r ig h tm o s t (top) sy m b o l in th e p u sh dow n s t o r e . C a s e 2 . T h e to p m o s t e le m e n t of th e pushdow n s to r e is th e h ig h e st v a lu e d d ig it o f th e pushdow n a lp h a b e t. T h a t is , a c a r r y is re q u ir e d to add 1 to th e pushdow n c o n te n ts . In th is c a s e th e w o rk in g ta p e is u s e d . A s s u m e th a t no t o n ly th e to p e le m e n t b u t th e to p k e le m e n ts o f th e p u shdow n ta p e a r e th e s a m e h ig h e s t v a lu e d d ig it. T h a t is , to ad d 1 to th e pushdow n s to r e , M m u s t b e c o n s tru c te d to c a r r y o v e r k d ig its in to th e pushdow n ta p e . S in ce th e pushdow n ta p e lo s e s in fo rm a tio n to th e r ig h t o f th e r e a d h e a d — th a t is th e r e a d h e a d 72 m a y n o t " e n te r " th e pushdow n ta p e w ith o u t e r a s in g th e sy m b o ls p a s s e d — th a t in fo rm a tio n m u s t b e s to r e d e ls e w h e r e . M is b u ilt to s to r e th e in te g e r k on th e w o rk ta p e . A fte r th e c a r r y is c o m p le te d , M r e s t o r e s k z e r o s to th e pushdow n ta p e . T he w o rk ta p e is e x a c tly long enough to s to r e th e co u n t of th e m a x im u m n u m b e r o f d ig its e r a s e d d u rin g an y co u n t c y c le . F o r e a c h in c re m e n t in th e co u n t c y c le , M is c o n s tru c te d to ap p en d a 1 to th e o u tp u t ta p e . T hus a t th e end of th e co u n t, th e c L < n > d o u tp u t is 1 . ■ L e m m a 3 . 3 . 4 : L e t L(n) b e an y s p a c e bound, and le t f = L(n) C(ln , 1 )fngl}. T h e n th e fu n c tio n f £ . 3 S ( ^ (L ( n ) ) ) . P r o o f : A s s u m e fo r th e s a k e of c o n tra d ic tio n th a t f £ 2J(7ft(L(n))) fo r so m e L(n) — i . e . , th a t th e r e is an M € ^ (L (n )) su ch th a t f= 3>(M). T B y T h e o re m 2 . 2 .6 , th e c la s s {nb '} is a c h a r a c te r i s t ic c la s s fo r *^(L (n)). B y th e d e fin itio n o f c h a r a c te r i s t ic c la s s , D e fin i tio n 2 . 2 . 4 , th e r e is a b . su c h th a t | M(w)f fw fb !" '^ ™ ^ . B ut if L (| w f ) , ,» | v 5(M ) = f, th e n fM (w )f = d >• fwfbj ' w * fo r e v e r y c h o ic e b j . T h is is a c o n tra d ic tio n ; h e n c e f i. ? ( W (L(n))) fo r an y L (n ). ■ T h e o re m 3 . 3 . 5 : #(■(?#!(L (n))) < £ 3 ?( 97|(L(n))). P r o o f : Im m e d ia te fro m th e tw o p re c e d in g r e s u l t s . ■ C o r o lla r y 3 . 3 . 6 : 3 ( ^ ( c JLi(n))) / a W i ( L ( n ) ) ) . P r o o f : T ak e f a s in L e m m a 3 . 3 . 4 . T h en f € 7(4?W ( (L (n))) b u t f £ 3 W ( c L (n ))). F o r c le a r ly in s te p s no tr a n s d u c e r ca n c L(n) g e n e ra te d o u tp u t s y m b o ls . ■ T hus w e h a v e fu lfille d th e p ro m is e m a d e in S e c tio n 2 .1 to e x h ib it tw o c la s s e s A an d B su ch th a t A) = J"(B) and 7(A) 4 7 (B ). F o r a n y L(n) log(n), le t A = T 7 ? ( c ^ n ^) and le t B = £>^(L(n)). B y T h e o re m 3 .3 .2 , S(A) = J(B) and b y C o r o lla r y 3 .3 .6 , 7 ( A ) ^ 7 ( B ) . B IB L IO G R A P H Y [1] A ho, A . V ., U llm a n , J . D . an d H o p c ro ft, J . E . , "O n th e C o m p u ta tio n a l P o w e r of P u sh d o w n A u to m a ta , " JC SS, v o l. 4, 1970, p p . 1 2 9 -1 3 6 . [2] B lu m , M ., "A M a c h in e -In d e p e n d e n t T h e o ry of th e C o m p le x ity of R e c u rs iv e F u n c tio n s, " JA C M , v o l. 14, 1967, p p . 3 2 2 -3 3 6 . [3] B o ro d in , A ., "C o m p u ta tio n a l C o m p le x ity and th e E x is te n c e of C o m p le x ity G ap s, " JA C M , v o l. 19, 1972, p p . 1 5 8 -1 7 4 . [4] C ook, S . A ., " C h a r a c te r iz a tio n o f P u sh d o w n M a c h in e s in T e r m s o f T im e -B o u n d e d C o m p u te r s ," JA C M , v o l. 18, 1971, p p . 4 - 1 8 . [5] D a v is, M ., C o m p u ta b ility and U n so lv a b ility , N ew Y ork: M c G ra w -H ill, 1958. [6] F is c h e r , P . O . and R o s e n b e rg , A . L ., " M u lti-T a p e O n e-W ay N o n -W ritin g A u to m a ta , " JC S S, v o l. 2, 1968, p p . 8 8 -1 0 1 . [7] G ilb e rt, P . , "O n th e S y n tax of A lg o rith m ic L a n g u a g e s ," JA C M , v o l. 13, 1966, p p . 5 1 -5 5 . [8] G in sb u rg , S ., T he M a th e m a tic a l T h e o ry of C o n te x t- F r e e L a n g u a g e s , N ew Y ork: M c G ra w -H ill, 1966. [9] G in sb u rg , S ., G re ib a c h , S . an d H a r r is o n , M . A ., "O n e-W ay S ta c k A u to m a ta , " JA C M , v o l. 14, 1967, p p . 3 8 9 -4 1 8 . [10] G in s b u rg , S . and R o se , G . F . , " P r e s e r v a tio n of L an g u a g es by T r a n s d u c e r s , " I & C , v o l. 9, 1966, p p . 1 5 3 -1 7 6 . [11] G ra y , J . , H a r r is o n , M . A . and I b a r r a , O . H ., "T w o -W ay P u sh d o w n A u to m a ta , " I & C , v o l. 11, 1967, p p . 3 0 -7 0 . [12] H a r r is o n , M . A . and I b a r r a , O . H ., " M u lti-T a p e and M u lti- H ead P u sh d o w n A u to m a ta , " I_&_C, v o l. 13, 1968, p p . 4 3 3 -4 7 0 . [13] H o p c ro ft, J . E . and U llm a n , J . D ., F o r m a l L a n g u a g e s and th e ir R e la tio n to A u to m a ta , R e a d in g , M a s s a c h u s e tts : A d d iso n - W e sle y , 1969. 74 75 I b a r r a , O . H ., " C h a r a c te r iz a tio n of S om e T a p e and T im e - C o m p le x ity C la s s e s o f T u rin g M a c h in e s in T e rm s o f M u lti h e a d and A u x ilia ry S tac k A u to m a ta , " JC S S , v o l. 5, 1971, p p . 8 8 -1 1 7 . K am ed a , T . , "P u sh d o w n A u to m a ta w ith C o u n te rs , " JC SS, v o l. 6, 1970, p p . 1 3 8 -1 5 0 . K r e id e r , D . L . and R itc h ie , R . W ., "A B a s is T h e o re m fo r a C la s s of Tw o W ay A u to m a ta , " Z e i ts c h r . f . M a th . L o g ik . und G ru n d . d . M a th ., v o l. 12, 1966, p p . 2 4 3 -2 5 5 . L u c h a m , D . C . , P a r k , D . M . R . an d P a te r s o n , M . S ., "O n F o r m a liz e d C o m p u te r P r o g r a m s , " JC SS, v o l. 4 , 1970, p p . 2 2 0 -2 4 9 . M e y e r, A ., "A u to m a ta T h e o ry , " th e u n p u b lish e d n o te s of M IT c o u rs e 1 8 .9 0 , S p rin g 1970. M in sk y , M ., C o m p u tatio n : F in ite an d In fin ite M a c h in e s , E nglew ood C liffs, N ew J e r s e y : P r e n tic e - H a ll, 19^7. P a te r s o n , M . S ., " T a p e B ounds fo r T im e -B o u n d e d T u rin g M a c h in e s, " JC S S , v o l. 6, 1972, p p . 1 1 6 -1 2 4 . R ab in , M . O . and S c o tt, D ., " F in ite A u to m a ta an d T h e ir D e c is io n P r o b le m s , " IBM J o u r n a l o f R e s e a r c h an d D ev elo p - m e n t, v o l. 3, 1959, p p . 1 1 4 -1 2 5 . R itc h ie , R . W . and S p rin g s te e l, F . N ., " R e c o g n itio n of L a n g u a g e s by M a rk in g A u to m a ta , " P r o c . of H aw aii In t. C o n f. on Sys . S c i . , H onolulu: U n iv e rs ity of H aw aii P r ., 1968, p p . 2 0 -2 2 . R o s e n b e rg , A . L . , "O n M u lti-H e a d F in ite A u to m a ta , " IBM J o u r n a l of R e s e a r c h and D ev elo p m en t, v o l. 10, 1966, p p . 3 8 8 -3 9 3 . R o s e n b e rg , A . L . , "M u ltita p e F in ite A u to m a ta w ith R ew ind I n s tr u c tio n s , " JC S S , v o l. 1, 1967, p p . 2 9 9 -3 1 5 . S a v itc h , W . J . , " R e la tio n s h ip B etw ee n N o n d e te rm in is tic and D e te r m in is tic T ap e C o m p le x itie s , " JC S S , v o l. 4 , I9 6 0 , p p . 1 1 7 -1 9 2 . S h e p e rd so n , J . C . an d S tu rg is , H . E . , "C o m p u ta b ility of R e c u rs iv e F u n c tio n s, " JA C M , v o l. 10, 1963, p p . 2 1 7 -2 5 5 . 76 [27] S m u lly a n , R . , T h e o ry o f F o r m a l S y s te m s , A n n a ls o f M a th . S tu d ie s, N o . 4 7 , t-’r in c e to n , N ew J e r s e y : P r in c e to n U n iv e r s ity P r e s s , 1961. [28] S te a rn s , R . , H a r tm a n is , J . and L e w is, P . ,11, " H ie r a r c h ie s o f M e m o ry L im ite d C o m p u ta tio n s, " IE E E S ix th A n n u al S y m p o siu m on S w itch in g T h e o ry an d L o g ic a l D e s ig n , 1965, p p . 1 7 9 -1 9 0 . [29] S u d b o ro u g h , I . H ., " C o m p u ta tio n s b y M u lti-H e a d F in ite A u to m a ta , " SW AT, 1971, p p . 1 0 5 -1 1 3 . A b b re v ia tio n s o f P u b lic a tio n N a m e s I & C In fo rm a tio n and C o n tro l JA C M J o u r n a l of th e A s s o c ia tio n fo r C o m p u tin g M a c h in e ry JC SS J o u r n a l of C o m p u te r an d S y s te m S c ie n c e s SW AT IE E E A n n u al S y m p o s iu m on S w itch in g T h e o ry and L o g ic a l D e sig n

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Creator Abbott, Russell Joseph (author)

Core Title Some Properties Of String Transducers

Contributor Digitized by ProQuest (provenance)

Degree Doctor of Philosophy

Degree Program Computer Science

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Computer Science,OAI-PMH Harvest

Language English

Advisor Gilbert, Phillip (committee chair), Chandler, William J. (committee member), Marston, Albert R. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-813057

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Rights Abbott, Russell Joseph

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Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

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