Nondeterministic Space is Closed Under Complementation Neil Immerman Computer Science Department Yale University New Haven CT SIAM Journal of Computing Intro duction In this pap er we show that nondeterministic space sn is closed under com plementation for sn greater than or equal to log n It immediately follows that the contextsensitive languages are closed under complementation thus settling a question raised by Kuro da in See Hartmanis and Hunt for a discussion of the history and imp ortance of this problem and Hop croft and Ullman for all relevantbackground material and denitions The history b ehind the pro of is as follows In we showed that the set of rstorder inductive denitions over nite structures is closed under complementation This holds with or without an ordering relation on the structure If an ordering is present the resulting class is PMany p eople exp ected that the result was false in the absence of an ordering In we studied rstorder logic with ordering with a transitive closure op erator We showed that NSPACElog n is equal to FO p os TC ie rstorder logic with ordering plus a transitive closure op eration in which the transitive closure op erator do es not app ear within any negation symbols Nowwehave returned to the issue of complementation in the lightof recent results on the collapse of the log space hierarchies Wehave shown that the class FO p os TC is closed under complementation Our Research supp orted by NSF Grant DCR main result follows In this pap er wegive the pro of in terms of machines and then state the result for transitive closure as Corollary The question of whether FO posTCwithout ordering is closed under complementation remains op en Our work in rstorder expressibility led to our pro of that nondetermin istic space is closed under complementation However b ecause rstorder expressibility classes are not directly relevant to the pro ofs in this pap er we omit those denitions here The interested reader should refer to for all these denitions Note that the pro of of Theorem in is more complicated than the pro of of Theorem but quite similar to it The same is true of the pro of in that the rstorder inductive formulas are closed under complementation Results Theorem For any sn log n NSPACEsn coNSPACEsn Pro of We do this bytwo lemmas We will show that counting the exact numb er of reachable congurations of an NSPACEsn machine can b e done in NSPACEsn Lemma Lemma says that once this number has b een calculated we can detect rejection as well as acceptance Note the similaritybetween Lemma and a similar result ab out census functions in Lemma Suppose we are given an NSPACEsn machine M a size sn initial conguration START and the exact number N of congurations of size sn reachable by M from START Then we can test in NSPACEsn if M rejects Pro of Our NSPACEsn tester do es the following It initializes a counter to and a target conguration to the lexicographically rst string of length sn For each such target either we guess a computation path of M from START to target and increment b oth counter and target or we simply The conguration ofaTuring machine is the contents of its work tap es the p ositions of its heads and its state Note that for sn log nthenumb er of p ossible congurations sn is less than c for some constant candthus can b e written in O sn space increment target For each target that wehave found a path to if it is an accept conguration of M then we reject Finally if when we are done with the last target the counter is equal to N we accept otherwise we reject Note that we accept i wehave found N reachable congurations none of which is accepting Supp ose that M accepts In this case there can b e at most N reachable congurations that are not accepting and our machine will reject On the other hand if M rejects then there are N nonaccepting reachable congurations Thus our nondeterministic machine can guess paths to each of them in turn and accept That is we accept i M rejects Lemma Given START as in Lemma we can calculate N the total number of congurations of size sn reachable by M from STARTin NSPACEsn Pro of Let N b e the numb er of congurations reachable from START d in at most d steps The computation pro ceeds by calculating N N and so on By induction on d weshow that each N may b e calculated in d NSPACEsn The base case d is obvious Inductive step Given N weshowhow to calculate N AsinLemma d d wekeep a counter of the number of d reachable congurations and we cycle through all the target congurations in lexicographical order For each target we do the following Cycle through all N congurations reachable in d at most d steps again we nd a path of length at most d for eachreachable one and if we dont nd all N of them then we will reject For eachof d these N congurations check if it is equal to target or if target is reachable d from it in one step If so then increment the counter and start on target If we nish visiting all N congurations without reaching target then just d start again on target without incrementing the counter When weve completed this algorithm for all targets our counter contains N Since N d sn is b ounded ab oveby c for some constant c the space needed is Osn To complete the pro of of the lemma and the theorem note that N is equaltotherstN suchthatN N d d d Remark In our original statement of Theorem we made the assump tion that sn is space constructible However the standard denition of a nondeterministic Turing machine having space complexity sn is that no sequence of choices enables it to scan more than sn cells Thus the ab ove pro of works even if sn is not space constructible We just let sn increase as needed The following corollary is immediate Corollary The class of context sensitive languages is closed under com plementation Pro of Kuro da showed in that CSL NSPACEn th The k level of the log space alternating hierarchy ALOG is dened k to b e the set of problems accepted by alternating log space Turing machines that make at most k alternations and b egin in an existential state Re cently Lange Jenner and Kirsig showed that this hierarchy collapsed to the second level ALOG This result was then extended by several authors who showed that the log space oracle hierarchy collapses to NL L Here LDSPACElog n and NL NSPACElog n The logspace or OLOG k acle hierarchyisgiven by OLOG NLand OLOG NL k In the case of the p olynomial time hierarchy the oracle and alternating hier archies are identical but they app eared to b e dierent in the log space case We knew that the logspace oracle hierarchy is equal to FO TC This together with the ab ove results led us to exp ect Theorem The following is again immediate Corollary The LogSpaceAlternating Hierarchy and the LogSpaceOracle Hierarchy both col lapse to NSPACElog n In we showed that NL is equal to FO p os TC In Theorem of wealsoshowed that any problem in NL may b e expressed in the form max where is a quantier free rstorder formula and and TC max are constantsymb ols It now follows that the same is true for the class FOTC Corollary NSPACElog nFOpos TC FOTC Any formula in FOTCmaybe expressed in the form TC max where is a quantier free rstorder formula Michael Fischer has observed that one can now diagonalize nondeter ministic space and thus easily proveatight hierarchy theorem for nonde terministic space Although Corollary is not new our techniques givea much simpler pro of than was previously known See Chapter in for the old pro of Corollary For any tapeconstructible sn log n tn lim n sn implies NSPACE tn NSPACE sn Conclusions and Directions for Future Work Most of the interesting questions concerning the p ower of nondeterminism remain op en We still do not know whether nondeterministic space is equal to deterministic space or whether Savitchs Theorem is optimal It is interesting to consider whether our pro of metho d can b e extended to answer these questions or to tell us anything new ab out nondeterministic time So on after weproved Theorem Tompa et al gavetwo extensions they proved that LOGCFL the set of problems log space reducible to a context free language is closed under complementation and they showed that Symmetric Log Space cf is contained in ZPLP the class of errorless probabilistic Turing machines running in O log n space and p olynomial exp ected time We suggest the following op en problems Is FO without p os TC closed under complementation Is Symmetric Log Space equivalently FO p os STC closed under complementation Is NL equal to a complexity class that was previously known to b e closed under complementation eg L AC or DSPACElog n In the pro of of Theorem we made use of the linear space compression theorem Theorem in Our actual construction multiplies the space b ound by ab out eight It is interesting to ask howmuch this can b e reduced Note in particular that if we could complementlogn times while only increasing the space b ound by a constant factor then it would followthatNLAC Acknowledgements Thanks to Sam Buss MikeFischer Lane Hemachan dra Steve Mahaney and Jo el Seiferas who contributed comments and cor rections to this pap er References A Boro din SA Co ok PW Dymond WL Ruzzo and M Tompa Two Applications of Complementation via Inductive Counting this volume S R Buss SA Co ok P Dymond and L Hay The Log Space Oracle Hierarchy Collapses in preparation SA Co ok A Taxonomy of Problems with Fast Parallel Algorithms Information and Control J Hartmanis and HB Hunt I I I The LBA