Languages That Capture Complexity Classes y Neil Immerman Intro duction Wepresent in this pap er a series of languages adequate for expressing exactly those prop erties checkable in a series of computational complexity classes For example weshow that a prop erty of graphs resp ectively groups binary strings etc is in p olynomial time if and only if it is expressible in the rst order language of graphs resp ectively groups binary strings etc together with a least xed p oint op erator As another example a prop erty is in logspace if and only if it is expressible in rst order logic together with a deterministic transitive closure op erator The ro ots of our approach to complexity theory go back to when Fagin showed that the NP prop erties are exactly those expressible in second order existential sentences It follows that second order logic expresses exactly those prop erties which are in the p olynomial time hierarchy Weshow that adding suitable transitive closure op erators to second order logic results in languages capturing p olynomial space and exp onential time resp ectively The existence of such natural languages for each imp ortant complexity class sheds a new light on complexitytheory These languages rearm the imp or tance of the complexity classes as much more than machine dep endent issues Furthermore a whole new approach is suggested Upp er b ounds algorithms can b e pro duced by expressing the prop ertyofinterest in one of our languages Lower b ounds maybedemonstratedbyshowing that such expression is imp os sible For example from the ab ovewe knowthatP NP if and only if every second order prop erty is already expressible using rst order logic plus least xed p oint Similarly nondeterministic logspace is dierent from P just if there is some sentence using the xed p oint op erator which cannot b e expressed with a single application of transitive closure SIAM J of Computing A preliminary version of this pap er app eared as y Research supp orted by an NSF p ostdo ctoral fellowship Current address Computer Science Dept UMass Amherst MA In previous work Im Imb weshowed that the complexity of a prop erty is related to the number of variables and quantiers in a uniform sequence of sentences where each expresses the prop erty for structures of n size n Our present formulation is more pleasing b ecause it considers single sentences in more p owerful languages The rst order expressible prop erties at rst seemed to o weak to corresp ond to any natural complexity class However we found that a prop erty is expressible by a sequence of rst order sentences where each has a b ounded n numb er of quantiers if and only if this prop erty is recognized byasimilar sequence of p olynomial size b o olean circuits of b ounded depth It follows that the results of Furst Saxe and Sipser FSS and Sipser Si translate precisely into a pro of that certain prop erties are not expressible in anyrst order language In this pap er wealsointro duce a reduction b etween problems that is new to complexitytheory First order translations as the name implies are xed rst order sentences which translate one kind of structure into another This is a very natural way to get a reduction and at the same time it is very restrictive It seems plausible to prove that such reductions do not exist b etween certain prob lems We present problems which are complete for logspace nondeterministic logspace p olynomial time etc via rst order translations This pap er is organized as follows Section intro duces the complexity classes we will b e considering Section discusses rst order logic Sections and intro duce the languages under consideration Section considers the relationship b etween rst order logic and p olynomial size b ounded depth cir cuits The present lack of knowledge concerning the separation of our various languages is discussed Complexity Classes and Complete Problems In this section we dene those complexity classess whichwe will capture with languages in the following sections We list complete problems for some of the classes In later sections wewillshowhow to express these complete problems in the appropriate languages and we will also show that the problems are complete via rst order translations More information ab out complexity classes maybe found in AHU Consider the following well known sequence of containments L NL L P NP P Here L is deterministic logspace and NL is nondeterministic logspace L S S L is the logspace hierarchyP P is the p olynomial time k k k k hierarchy Most knowledgable p eople susp ect that all of the classes in the ab ove containment are distinct but it is not known that they are not all equal We b egin our list of complete problems with the graph accessibility problem GAP fGj apathinG from v to v g n Theorem Sa GAP is logspacecomplete for NL We will see later that GAP is complete for NL in a much stronger sense The GAP problem maybeweakened to a deterministic logspace problem by only considering those graphs whichhave at most one edge leaving anyvertex GAP fGjG has outdegree and a path in G from v to v g n Theorem HIM GAP is oneway logspacecomplete for L A problem whichliesbetween GAP and GAP in complexityis UGAP fGjG undirected and a path in G from v to v g n Let BPL b ounded probability logspace b e the set of problems S suchthat there exists a logspace coinipping machine M and if w S then ProbM accepts w while if w S then ProbM accepts w It follows from the next theorem that UGAP is in BPL Thus UGAP is probably easier than GAP Theorem AKLL If r is a random walk of length jE jjV j in an undirectedconnectedgraph G then the probability that r includes al l vertices in G is greater than or equal to one half Lewis and Papadimitriou LP dene symmetric machines to b e nondeter ministic turing machines whose next move relation on instantaneous descriptions is symmetric That is if a symmetric machine can move from conguration A to conguration B then it is also allowed to move from B to A Let SymL b e the class of problems accepted by symmetric logspace machines Theorem LP UGAP is logspacecomplete for SymL John Reif Re extended the notion of symmetric machines to allow al ternation Essentially an alternating symmetric machine has a symmetric next move relation except where it alternates b etween existential and universal states S Let SymL SymL b e the symmetric logspace hierarchy Reif k k showed that several interesting prop erties including planarity for graphs of b ounded valence are in the symmetric logspace hierarchy It follows that they are also in BPL O Reif also showed that BPL is contained in O log ntimeandn pro cessors on a probabilistic hardware mo dication machine HMM A a b c Figure An alternating graph with two universal no des a c Theorem Re O SymL BP L ProbHMMTIMElog nPROCn One may also consider harder versions of the GAP problem Let an alternat ing graph G V E A b e a directed graph whose vertices are lab elled universal or existential A V is the set of universal vertices Alternating graphs have a dierent notion of accessibility Let APATHxy b e the smallest relation on vertices of G such that APATHxx If x is existential and for some edge hx z i APATHzy holds then APATHxy If x is universal there is at least one edge leaving x and for all edges hx z i APATHzy holds then APATHxy See Figure where APATHab holds but APATHcb do es not Let AGAP fGjAP AT H v v g G n It is not hard to see that AGAP is the alternating version of GAPandthus is complete for ASPACElogn Recalling that this class is equal to P CKS wehave Theorem Im AGAPislogspacecomplete for P Note that the AGAP problem is easily seen to b e equivalent to the monotone circuit value problem shown to b e complete for P in Go First Order Logic In this section weintro duce the necessary notions from logic The reader is refered to En for more background material A nite structure with vocabulary hR R c c i is a tuple S k r hf n gR R c c i consisting of a universe U f n g k r and relations R R on U corresp onding to the relation symbols R R of k k and constants c c from U corresp onding to the constantsymbols c c from r r For example if hE i consists of a single binary relation symb ol then a structure G hfn gEi with vo cabulary is a graph on n vertices Similarly if hM i consists of a single monadic relation symbol then a structure S hfn gMi with vo cabulary is a binary string of length n If is a vo cabulary let ST RUC T fGjG is a structure with vo cabulary g We will think of a problem as a set of structures of some vo cabulary Of course it suces to only consider problems on binary strings but it is more interesting to b e able to talk ab out other vo cabularies eg graph problems as well Let the relation sxy b e the successor relation on the naturals ie sx y holds i x y Throughout this pap er we will assume that s is a logical relation symb ol denoting the successor relation We will also assume that al l structures under consideration have at least two elements Furthermore the constantsymb ols and m will always refer to the rst and last elements of the universe resp ectively Wenow dene the rst order language L to b e the set of formulas built up from the relation and constantsymbols of and the logical relation symb ols and constant symbols sm using logical connectives variables x y z and quantiers If L let MOD b e the set of nite mo dels of MOD fG ST RUC T jG j g Let FO b e the set of all rst order expressible problems FO