Hardess Results for Itersectio No-Emptiess Michael Wehar Uiversity at Buffalo mwehar@buffalo.edu Jauary 16, 2015 Abstract We carefully reexamie a costructio of Karakostas, Lipto, ad Viglas (2003) to show that the itersectio o-emptiess problem for DFA s (determiistic fiite automata) characterizes the complexity class NL. I particular, if restricted to a biary work tape alphabet, the there exist costats c 1 ad c 2 such that for every k itersectio o-emptiess for k DFA s is solvable i c 1 k log() space, but is ot solvable i c 2 k log() space. We optimize the costructio to show that for a arbitrary umber of DFA s itersectio o-emptiess is ot solvable i o( log() log(log()) ) space. Furthermore, if there exists a fuctio f(k) = o(k) such that for every k itersectio o-emptiess for k DFA s is solvable i f(k) time, the P NL. If there does ot exist a costat c such that for every k itersectio o-emptiess for k DFA s is solvable i c time, the P does ot cotai ay space complexity class larger tha NL. 1 Itroductio Let A deote a class of machies. The itersectio o-emptiess problem for A, deoted by IE A, cosists of all fiite lists of machies i A whose uderlyig laguages have a o-empty itersectio. By fixig the umber of machies i the iput to k, oe obtais itersectio o-emptiess for k machies which we deote by k-ie A. Itersectio o-emptiess problems ca be motivated by the followig sceario. Cosider that you are tryig to costruct a object x for a particular applicatio. You propose a fiite list of coditios for x to satisfy such that each 1
2 Michael Wehar coditio ca be decided by a machie i A. A algorithm that solves itersectio o-emptiess for A provides a method for checkig if there exists a object x satisfyig the proposed coditios. Let IE D deote the itersectio o-emptiess problem for DFA s. Oe ca solve IE D by checkig reachability i a product machie. Give a iput cosistig of k machies each of size at most m, the product machie has size at most m k. Therefore, checkig reachability takes at most m O(k) time. Is it possible to solve IE D more efficietly? I [5], it was show that IE D is PSPACE-complete. Cosider restrictig the umber of machies i the iput of IE D by a fuctio g() where is the total iput legth. I [6], it was show that if g() is subliear ad log-space-costructible, the such a restrictio yields a complete problem for NSPACE(g() log()). I [4], it was show that the existece of a more efficiet algorithm for IE D would imply a separatio result. I particular, if there exists a fuctio f(k) = o(k) such that IE D is solvable i m 1 m f(k) 2 time where m 1 is the size of a desigated largest machie ad all other machies have size at most m 2, the NL P. I this paper, we carefully reexamie ad optimize the costructio from [4] i order to prove ew results. We show that if restricted to a biary work tape alphabet, the there exist costats c 1 ad c 2 such that for every k, k-ie D NSPACE(c 1 k log()) ad k-ie D / NSPACE(c 2 k log()). The, we itroduce a optimized costructio to show that IE D / NSPACE(o( )). Fially, log() log(log()) we combie these results with a diagoalizatio argumet to show that if there exists a fuctio f(k) = o(k) such that for every k, k-ie D DTIME( f(k) ), the P NL. If there does ot exist a costat c such that for every k, k-ie D DTIME( c ), the NSPACE(f()) P for all f() = ω(log()) such that f is space-costructible. 2 Notatio ad Covetios The iput for IE D is a ecodig of a fiite list of DFA s. For each ecodig, will deote the legth ad k will deote the umber of machies that are represeted. For each atural umber k, k-ie D deotes a restrictio of the IE D problem such that we oly accept iputs that ecode at most k machies. Wheever we use the term Turig machie, we refer to a determiistic or o-determiistic machie with a two-way read oly iput tape ad a two-way
Hardess Results for Itersectio No-Emptiess 3 read/write work tape. For our purposes, we will oly cosider Turig machies where the work tape alphabet is biary. A work tape over a biary alphabet will be referred to as a biary work tape. A cell o a biary work tape will be referred to as a bit cell. For each k, there are acceptace problems for space ad time bouded Turig machies deoted by Nk S log ad DT, respectively. N S k k log refers to the problem where we are give a ecodig of a o-determiistic Turig machie M with a biary work tape ad a iput s. We accept (M, s) if ad oly if M accepts s usig at most k log() work tape bit cells where deotes the legth of s. D T is defied similarly k for k determiistic time. We deote by NSPACE 2 (h()) the set of problems solvable by a o-determiistic Turig machie usig at most h() work tape bit cells. Such classes are used to measure the biary space complexity of problems [2]. We associate Nk S log with NSPACE2 (k log()) ad D T with DTIME( k ). k 3 Biary Space Complexity We itroduce a fuctio S NL (k) that measures the actual space complexities of the Nk S log problems. I particular, S NL(k) is defied as follows: S NL (k) := mi{ d N Nk S log NSPACE 2 (d log()) }. (1) I this sectio, we sketch how oe could apply stadard techiques from the space hierarchy theorem to prove that there exist costats c 1 ad c 2 such that for every k sufficietly large, N S k log NSPACE2 (c 1 k log()) ad N S k log / NSPACE2 (c 2 k log()). Usig the fuctio S NL (k), we express this result as S NL (k) = Θ(k). Propositio 1. S NL (k) = O(k). Sketch of proof. Usig the simulatio foud i ay commo proof of the space hierarchy theorem, oe shows that Nlog S NL. Further, oe shows S NL(k) = O(k) by usig paddig to reduce Nk S log to N log S for every k. Propositio 2. S NL (k) = Ω(k). Sketch of proof. Usig the stadard diagoalizatio argumet foud i ay commo proof of the o-determiistic space hierarchy theorem, oe shows S NL (k) = Ω(k). Notice that i order to carry out the diagoalizatio oe eeds to show
4 Michael Wehar there exists c such that for all k, NSPACE 2 (k log()) co -NSPACE 2 (ck log()). (2) First, oe applies the result NL = co -NL to show that there exists c such that Nlog S co -NSPACE2 (c log()). Further, oe shows (2) by usig paddig to reduce Nk S log to N log S for every k. Corollary 3. S NL (k) = Θ(k). 4 Reductios We itroduce a fuctio S IE (k) that measures the actual space complexities of the k-ie D problems. I particular, S IE (k) is defied as follows: S IE (k) := mi{ d N k-ie D NSPACE 2 (d log()) }. (3) I this sectio, we carefully reexamie the costructio from [4] to show that there exist costats c 1 ad c 2 such that for every k sufficietly large, k-ie D NSPACE 2 (c 1 k log()) ad k-ie D / NSPACE 2 (c 2 k log()). Usig the fuctio S IE (k), we ca express this result as S IE (k) = Θ(S NL (k)) = Θ(k). Propositio 4. S IE (k) = O(k). Sketch of proof. As was previously discussed, oe ca solve IE D by checkig reachability i a product machie. A state of the product machie ca be stored as a strig of k log() bits. Give such a state, we ca o-determiistically guess which state comes ext. There exists a path from a iitial state to a fial state if ad oly if there exists a path from a iitial state to a fial state of legth at most k. costat c. Therefore, k-ie D is solvable usig at most ck log() bits for some Theorem 5. S IE (k) = Ω(S NL (k)). Proof. We will describe a reductio from N S k log to k-ie D. The, we will discuss ecodig details to show that this is a log-space reductio. Let a k log() space bouded o-determiistic Turig machie M of size M ad a iput strig s of legth s be give. Together, a ecodig of M ad s
Hardess Results for Itersectio No-Emptiess 5 represet a arbitrary iput for Nk S log. Let deote the total size of M ad s combied i.e. := M + s. Our first task is to costruct k DFA s, deoted by < D i > i [k], each of size at most p() for some fixed polyomial p such that M accepts s if ad oly if i [k] L(D i) is o-empty. The DFA s will read i a strig that represets a computatio of M o s ad verify that the computatio is valid ad acceptig. The work tape of M will be split ito k sectios each cosistig of log( s ) sequetial bits of memory. The ith DFA, D i, will keep track of the ith sectio ad verify that it is maaged correctly. I additio, all of the DFA s will keep track of the iput ad work tape head positios. We will achieve a better simulatio i Theorem 7 where we split up the maagemet of the tape head positios to separate DFA s. The followig two cocepts are essetial to our costructio. A sectio i cofiguratio of M is a tuple of the form (state, iput positio, work positio, ith sectio of work tape). A forgetful cofiguratio of M is a tuple of the form (state, iput positio, work positio, write bit). The states of D i are idetified with sectio i cofiguratios. The alphabet characters are idetified with forgetful cofiguratios. Ituitively, D i reads i forgetful cofiguratios that represet where to move the tape heads ext ad how the curret bit cell should be maipulated. Formally, the trasitios for the DFA D i are defied as follows. Let a forgetful cofiguratio a ad sectio i cofiguratios r 1 ad r 2 be give. It s possible that either the work tape positio of r 1 is i the ith sectio, or the work tape positio is i aother sectio. I the first case, there is a trasitio from state r 1 with alphabet character a to state r 2 if (1a) goig from r 1 to r 2 represets a valid trasitio of M o iput s, (1b) the ith sectio of r 2 appropriately chages accordig to the write bit of a, ad (1c) a ad r 2 agree o state, iput positio, ad work positio. I the secod case, there is a trasitio from state r 1 with alphabet character a to state r 2 if (2a) r 1 ad r 2 agree o the ith sectio of the work tape, ad (2b) a ad r 2 agree o state, iput positio, ad work positio. We assert without proof that for every strig x, x represets a valid acceptig computatio of M o s if ad oly if x i [k] L(D i). Therefore, M accepts s if
6 Michael Wehar ad oly if i [k] L(D i) is o-empty. We show that the D i s have size at most p() for some fixed polyomial p. Each D i cosists of a start state, a list of fial states, ad a list of trasitios where each trasitio cosists of two states ad a alphabet character. Each state is represeted by a sectio i cofiguratio ad each alphabet character is represeted by a forgetful cofiguratio. Therefore, i total there are M s k log( s ) 2 log(s) sectio i cofiguratios ad M s k log( s ) forgetful cofiguratios. Hece, there exists a fixed two variable polyomial q such that each D i has size at most q(, k). Sice k is fixed, oe ca blow up the degree of q to get a polyomial p such that p does t deped o k ad each D i has size at most p(). It should be clear from the precedig that there is a fixed polyomial t() such that for every k, N S k log is t()-time reducible to k-ie D. However, we wat to show that there is a costat c such that for every k, Nk S log is c log()-space reducible to k-ie D. We accomplish this by describig how to prit the strig ecodig of the D i s to a auxiliary write oly output tape usig at most c log() space for some costat c. We will describe how to prit the trasitios for each D i ad leave the remaiig ecodig details to the reader. We use a bit strig i to represet the curret DFA ad two bit strigs j 1 ad j 2 to represet sectio i cofiguratios. We iterate through every combiatio of i, j 1, ad j 2. If D i has a trasitio from j 1 to j 2, the we prit (i, j 1, a, j 2 ) where a is the forgetful cofiguratio that agrees with j 2. We assert that checkig whether to prit (i, j 1, a, j 2 ) requires o more tha d log(k) + d log() bits for some costat d. Therefore, i pritig the ecodig of the D i s, we use o more tha c log(k) + c log() bits for some costat c. For each k, whe is sufficietly large, the log(k) term goes away. It follows that for every k, N S k log is c log()-space reducible to k-ie D. Corollary 6. S IE (k) = Θ(S NL (k)) = Θ(k). Proof. By Corollary 3, we have S NL (k) = Θ(k). Applyig Propositio 4 ad Theorem 5, we get that S IE (k) = Θ(S NL (k)) = Θ(k). Theorem 7. IE D / NSPACE(o( log() log(log()) )). Proof. By the o-determiistic space hierarchy theorem, we may choose a problem Q such that Q NSPACE(), but Q / NSPACE(o()). Choose c N ad a o-determiistic Turig machie M that solves Q usig at most c bit cells.
Hardess Results for Itersectio No-Emptiess 7 We optimize the costructio from the proof of Theorem 5 to show that if IE D NSPACE(o( )), the Q NSPACE(o()). Sice we kow that Q / log() log(log()) NSPACE(o()), it follows that IE D / NSPACE(o( log() log(log()) )). Let a iput strig s for M of legth be give. Our task is to costruct (c+1) DFA s each with at most d log() states for some costat d such that M accepts s if ad oly if the DFA s have a o-empty itersectio. The DFA s will read i a bit strig that represets a computatio of M o s ad verify that the computatio is valid ad acceptig. I this costructio, we split up the maagemet of the tape head positios to separate DFA s. There are DFA s, deoted by < I i > i [], that maage the iput tape ad there are c DFA s, deoted by < W i > i [c], that maage the work tape. The followig cocept is essetial to our costructio. A iformative cofiguratio of M is a tuple of the form (state, iput positio, curret iput bit, work positio, curret work bit). The DFA s will read i a sequece of iformative cofiguratios that are ecoded as bit strigs. I cotrast to the previous costructio, the DFA s will have a biary iput alphabet. Each DFA is assiged to maage a bit positio of either the iput tape or work tape. Each I i stores the ith iput tape bit ad operates as follows. It reads each iformative cofiguratio ad checks if it represets the iput positio i. If it does ot, the it igores the iformative cofiguratio ad moves o to the ext oe. However, if it does represet the iput positio i, the it checks that the stored bit matches the curret iput bit ad uses the curret work bit to check that the iput positio ad state validly trasitio to the ext iformative cofiguratio. Each W i stores the ith work tape bit ad operates as follows. It reads each iformative cofiguratio ad checks if it represets the work positio i. If it does ot, the it igores the iformative cofiguratio ad moves o to the ext oe. However, if it does represet positio i, the it checks that the stored bit matches the curret work bit ad uses the curret iput bit to both modify the stored bit ad check that the work positio ad state validly trasitio to the ext iformative cofiguratio. It s importat to remark that DFA s for boudary positios such as I 1, I, W 1, ad W c caot allow the iput positio or work positio to go outside [] or [c], respectively. We assert without proof that for every bit strig x, x represets a valid acceptig computatio of M o s if ad oly if x i [] L(I i) ad x i [c] L(W i).
8 Michael Wehar Therefore, M accepts s if ad oly if there exists a strig x such that x i [] L(I i) ad x i [c] L(W i). A DFA with log(c) states ca be costructed to recogize a fixed biary umber i [c]. Sice a tape positio i could oly trasitio to i 1, i, or i + 1 i oe step, it follows that a DFA with d log() states for some costat d ca be costructed to check the validity of trasitioig to the ext iformative cofiguratio. Therefore, we ca costruct each DFA with at most d log() states for some costat d. We described how to costruct (c + 1) DFA s each with at most d log() states for some costat d whose itersectio is o-empty if ad oly if M accepts s. Sice the total legth of the strig ecodig of < I i > i [] combied with < W i > i [c] is at most log() log(log()), it follows that IE D NSPACE(o( log() log(log()) result because Q / NSPACE(o()). )) implies Q NSPACE(o()). We obtai the desired 5 Space vs Time We itroduce fuctios R NL (k) ad R IE (k) that measure the actual time complexities of Nk S log ad k-ie D, respectively. I particular, R NL (k) ad R IE (k) are defied as follows: R NL (k) := mi{ d N Nk S log DTIME( d ) } (4) R IE (k) := mi{ d N k-ie D DTIME( d ) }. (5) I this sectio, we show that if there exists a fuctio f(k) = o(k) such that for every k, Nk S log DTIME(f(k) ), the P NL. Usig the fuctio R NL (k) we ca express this result as if R NL (k) = o(k), the P NL. Notice that by usig the reductio from Theorem 5, we also have R IE (k) = Θ(R NL (k)). It follows that if R IE (k) = o(k), the P NL. Propositio 8. R IE (k) = Θ(R NL (k)). Theorem 9. If R NL (k) = o(k), the NL P. Proof. Suppose that NL = P. Sice D T P, we have D T NL. Choose d N such that D T NSPACE 2 (d log()). Further, by usig paddig to reduce D T k to D T for every k, oe ca show that there exists d such that for all k, D T k
Hardess Results for Itersectio No-Emptiess 9 NSPACE 2 (d k log()). Choose such a costat d satisfyig for all k, D T k NSPACE 2 (d k log()). Suppose for sake of cotradictio that R NL (k) = o(k). By Propositio 2, we may choose c such that for all k sufficietly large Sice R NL (k) = o(k), for all k sufficietly large k Nk S log / NSPACE 2 ( log()). (6) c R NL (k) < k. (7) cd Choose m satisfyig Nm S log / NSPACE2 ( m c log()) ad RNL (m) < m cd. Therefore, N S m log DTIME(o( m cd )). (8) Sice D T k NSPACE 2 (d k log()) for all k, m m D T m NSPACE2 (d log()) NSPACE 2 ( log()). (9) cd cd c Sice we ca trivially reduce every problem i DTIME(o( m cd )) to D T m cd, Nm S log DTIME(o( m m cd )) NSPACE 2 ( log()) (10) c which is a cotradictio because Nm S log / NSPACE2 ( m c log()). Corollary 10. If R IE (k) = o(k), the NL P. Next, we show that if R NL (k) is ubouded, the P does ot cotai ay space complexity class larger tha NL. Sice R IE (k) = Θ(R NL (k)), it follows that if R IE (k) is ubouded, the P does ot cotai ay space complexity class larger tha NL. For every fuctio f, let N S f bouded o-determiistic Turig machies. Nf S is o-determiistically solvable i f() space. deote the acceptace problem for f()-space is of particular iterest to us if it Theorem 11. If R NL (k) is ubouded, the N S f / P for all fuctios f() = ω(log()).
10 Michael Wehar Proof. We will prove the cotrapositive. Suppose that Nf S P for some fuctio f() = ω(log()). By assumptio, we may choose c N ad a determiistic Turig machie T such that T solves Nf S i at most O(c ) time. Let k N be give. Choose a o-determiistic Turig machie M that solves Nk S log usig at most O(log()) bit cells. We ca determiistically solve Nk S log i at most O(c ) time by feedig T a ecodig of M ad the iput strig. Sice k is arbitrary, Nk S log is solvable i O(c ) time for every k. It follows that R NL (k) is bouded. Corollary 12. If R NL (k) is ubouded, the NSPACE(f()) P for all f() = ω(log()) such that f is space-costructible. Proof. Suppose R NL (k) is ubouded. Let a fuctio f() = ω(log()) such that f is space-costructible be give. Apply the precedig theorem to get that N S f / P. Sice f is space-costructible, oe ca use the simulatio foud i ay commo proof of the space hierarchy theorem to show that Nf S NSPACE(f()). Sice Nf S / P ad Nf S NSPACE(f()), it follows that NSPACE(f()) P. Corollary 13. If R IE (k) is ubouded, the NSPACE(f()) P for all f() = ω(log()) such that f is space-costructible. 6 Coclusio I Sectio 4, we showed that S NL (k) = S IE (k) = Θ(k). Therefore, we thik of itersectio o-emptiess for DFA s as characterizig the complexity class NL. Further, we showed that IE D / NSPACE(o( log() log(log()) )). I Sectio 5, we showed that if R IE (k) = o(k), the NL P ad if R IE (k) is ubouded, the NSPACE(f()) P for all f() = ω(log()) such that f is space-costructible. Therefore, the asymptotic complexity of R IE (k) determies the relatioship betwee space ad time complexity classes. There are several related problems that appear to be harder tha k-ie D, but easier tha Nk S log. For example, cosider itersectio o-emptiess for k NFA s, o-emptiess for k-tur 2DFA s, ad itersectio o-emptiess for k DFA s ad a oe-couter automato. We ca use S IE (k) = Θ(S NL (k)) ad R IE (k) = Θ(R NL (k)) as squeeze theorems to show that all of these problems are of equivalet difficulty. Also, oe could defie a fuctio that maps the k-ie D problems to their actual circuit complexities. The asymptotic complexity of such a fuctio could determie the relatioship betwee NL vs NP ad P/poly vs space complexity classes [4].
Hardess Results for Itersectio No-Emptiess 11 Several related itersectio o-emptiess problems have bee studied. There are two such problems that we would like to metio. I [10], itersectio oemptiess for acyclic DFA s, which are DFA s without directed cycles, was show to be NP-complete. We assert that oe could modify the costructio from the proof of Theorem 5 to reduce the acceptace problem for -time ad k log()- space bouded o-determiistic Turig machies to itersectio o-emptiess for k acyclic DFA s. Also, i [11], itersectio o-emptiess for tree automata was show to be EXPTIME-complete. I a upcomig paper, the author ad Joseph Swerofsky itroduce time complexity lower bouds for itersectio oemptiess for tree automata. Ackowledgmets I greatly appreciate all of the help ad suggestios that I received. I particular, I would like to thak Christos Kapoutsis for suggestios related to the costructios, Joseph Swerofsky for proof readig ad may discussios, Richard Lipto ad Keeth Rega for callig attetio to my results i a article o their blog [8], ad the may aoymous referees. I would especially like to thak all those at Caregie Mello Uiversity who offered their help ad support for my hoors thesis o the same topic. I particular, I would like to thak my thesis advisor, Klaus Suter, ad my thesis committee members, Mauel Blum ad Richard Statma. Refereces [1] Michael Blodi, Adreas Krebs, ad Pierre McKezie. The complexity of itersectig fiite automata havig few fial states. Computatioal Complexity (CC), 2014 (to appear). [2] Oded Goldreich. Computatioal Complexity: A Coceptual Perspective. Cambridge Uiversity Press, New York, 2008. [3] Neil D. Joes, Y. Edmud Lie, ad William T. Laaser. New problems complete for odetermiistic log space. Mathematical Systems Theory 10, 1976. [4] G. Karakostas, R. J. Lipto, ad A. Viglas. O the complexity of itersectig fiite state automata ad NL versus NP. Theoretical Computer Sciece, 302:257 274, 2003.
12 Michael Wehar [5] Dexter Koze. Lower bouds for atural proof systems. Proc. 18th Symp. o the Foudatios of Computer Sciece, pages 254 266, 1977. [6] Klaus-Jör Lage ad Peter Rossmaith. The emptiess problem for itersectios of regular laguages. Lecture Notes i Computer Sciece, 629:346 354, 1992. [7] R. J. Lipto. O the itersectio of fiite automata. Gödel s Lost Letter ad P=NP, August 2009. [8] R. J. Lipto ad K. W. Rega. The power of guessig. Gödel s Lost Letter ad P=NP, November 2012. [9] M. O. Rabi ad D. Scott. Fiite automata ad their decisio problems. IBM Joural, 1959. [10] Narad Rampersad ad Jeffrey Shallit. Detectig patters i fiite regular ad cotext-free laguages. Iformatio Processig Letters, 110, 2010. [11] Margus Veaes. O computatioal complexity of basic decisio problems of fiite tree automata. UPMAIL Techical Report 133, 1997. [12] Michael Wehar. Itersectio emptiess for fiite automata. Hoors thesis, Caregie Mello Uiversity, 2012.