COUNTING HIERARCHIES POLYNOMIAL TIME AND CONSTANT DEPTH CIRCUITS ERIC W ALLENDER Department of Computer Science Rutgers University New Brunswick NJ USA and KLAUS W WAGNER Institut fur Informatik Universitat Wurzbur g D Wurzbur g FRG Abstract In the spring of SeinosukeTo da of the University of ElectroCommu nications in Tokyo Japan proved that the p olynomial hierarchyiscontained PP in P To In this Structural Complexity Column we will briey review To das result and explore how it relates to other topics of interest in computer science In particular wewillintro duce the reader to The Counting Hierarchy a hierarchy of complexity classes contained in PSPACE and containing the Polynomial Hierarchy Threshold Circuits circuits constructed of MAJORITY gates this notion of circuit is b eing studied not only by complexity theoreticians but also by researchers in an active subeld of AI studying neural networks Along the waywell review the imp ortant notion of an operator on a com plexity class The Counting Hierarchy and Op erators on Complexity Classes The counting hierarchywas dened in Wa and indep endently byParb erry and Schnitger in PS The motivation for Wa was the desire to classify precisely the complexity of certain combinatorial problems on graphs with succinct descriptions Parb erry and Schnitger were studying threshold Turing machines in connection with parallel computation One way to dene the counting hierarchyis to take the usual denition of the p olynomial hierarchy Supp orted in part by National Science Foundation Grants CCR and CCR p P p p k NP for k k and replace NP with PP Recall that PP is probabilis tic p olynomial time as A dened by Gill Gi and Simon Si L is in PP if there is a p olynomialti me nondeterministic oracle Turing machine M suchthat x L i more than half of the computation paths of M on input x with oracle A are accepting That gives us the denition p C P p p C k for k C PP k p p PP PP and so on PP C Thus C This characterization of the counting hierarchy is due to Jacob o Toran who has studied the counting hierarchy in depth Tor Tor Although this is p erhaps the simplest way to dene the counting hierarchy it is not the original denition and the pro of that this characterization in terms of oracle Turing machines is equivalent to the one given b elow in terms of generalized quantiers is not obvious Recall that an alternativecharacterization of the p olynomial hierarchy is given by applying b ounded existential and universal quantiers to p olynomialtime predicates One way to formalize this is to dene operators acting on classes of sets That is Denition Let C b e a class of languages Then dene C to b e the set of all languages L such that there is some p olynomial p and some set A C such that x L y jy jpjxjandhx y i A In a similar way one can dene C p P and so on It It is a familiar fact St Wr that NP P has turned out to b e useful to consider other op erators in addition to and In particular the op erators C BP and will b e imp ortant for this survey Denition Let C b e a class of languages Dene CC to b e the set of all languages L such that there is some p olynomial p and some set A C suchthat x L pjxj kfy jy j pjxj and hx y iAgk BP C to b e the set of all languages L such that there is some p olynomial p and some set A C suchthat pjxj kfy jy j pjxjandhx y i A x Lgk Cto b e the set of all languages L such that there is some p olynomial p and some set A C suchthat x L kfy jy j pjxj and hx y iAgk is o dd The original denition of Wa Waa actually used a more general denition of the op erator Chowever it turns out that the complexity classes dened in this way are quite robust to small changes to the denitions and for any reasonable complexity class C the denition given ab ove is equivalent to the denition given in Wa The original denition of the counting hierarchyisthus p C P p p C CC k k Using the op erators Cand BP in dierent combinations one obtains other interesting complexity classes This list of op erators is by no means exclusive For example Toran has shown that an exact counting version of the C op erator which he denotes by C isvery useful to ol for proving results ab out other classes in the counting hierarchyTor Tor The BP op erator is obviously a way to generalize the notion of probabilistic computation to other complexity classes For example BP P is the complexity class AM BMSc The op erator is a way to generalize the complexity class ParityP of PZGP We should mention that Stathis Zachos has an alternative logicbased formulation for dening complexity classes of this sort His approach has also b een quite useful in proving results ab out these classes see HZ Za ZH BHZ ZF Wewould now like to briey catalogue some of the known relationships among the complexity classes dened using these op erators Some of these inclusions are kno wn to hold for al l classes of sets C but some of the other inclusions only hold if C is suciently nice For the sake of simplicitywe will assume here that C is a complexity class dened by applying the op erators from the set f BP Cg to the class P All such classes satisfy the appropriate niceness conditions Note however that one must b e careful what one assumes ab out the classes C Some sub classes of the counting hierarchy such as PP seem not to b e closed under union or intersection while other classes such as NP seem not to b e closed under complementation Information ab out the Theorem Acatalog of facts about subclasses of the Counting Hierarchy C C BP C CC This is a generalization of Gills result Gi that NP and coNP are contained in PP and also of the trivial inclusion BPP PP CC C P C C NP C CC PP Tor C BP C BPP C C P NP BP P This follows from the pro of of VV showing that SATis reducible via probabilistic reductions to the unique satisabili ty problem C C p P P P PZ Thus P is closed under T BPP PP PP KSTT If the underlying class C is closed under p ositive reducibility see Sc then the usual techniques of amplication can b e used to exp onentially reduce the error probability for sets in the class BP CThus for such classes C the following inclusions It should b e noted that classes dened using the C op erator and equalities hold are not known to b e closed under this sort of p ositive reducibility and thus the following inclusions may not hold for such classes Theorem For classes C closed under p ositive reducibility BP C C P BP P BP BP C BP C closure prop erties of classes in the Counting Hierarchymay b e found in Tor In the brief p erio d that has elapsed b etween the time when this article was rst written for the EATCS Bulletin January and the time of the compilation of the presentvolume Septemb er much has b een learned ab out the counting hierarchyItwas shown in BRS that PP is closed under union and intersection This b eautiful and surprising result has b een generalized to the entire counting hierarchy Gu BP C C C Sc This is a generalization of the result of Sia La showing that BPP is in the p olynomial hierarchy BP C BP C This is the op erator swapping lemma of To Some of the inclusions and equalities listed ab ove are quite easy to prove and others are not at all obvious It is imp ortant to note that all of the facts listed ab ove relativize so that the same relationships hold relativetoany oracle Armed with this list of facts we can nowgive a short pro of of the rst part of To das pro of Theorem To The p olynomial hierarchyiscontained in BP P p BP PThus assume inductively Pro of Wehave already observed that p BP P and note that that k p p k NP k BP P BP P P BP BP P BP BP P BP BP P BP P Having nowshown that the p olynomial hierarchyiscontained in BP P CPTo das theorem follows from the following inclusion the pro of of which involves a detailed manipulation of the computation tree of a machine whichwedo not have sucient space to present here PP Theorem To CP P PP To da in fact shows that P contains a seemingly larger class Let PH denote the p olynomial hierarchy PH PP Theorem To PP P Pro of PH BP P PP PP P BPP PP P PP C P PP P An obvious op en question is whether or not PH is contained in PP itself It is NP not even known if P is contained in PP The largest sub class of PH known to b e NPlog contained in PP is the class P the class of languages that can b e recognized with O log n queries to SAT BHW Circuits and Neural Nets To das theorem has some very interesting consequences for circuit complexity Before we can present these consequences it will b e instructive to review some basic notions from circuit complexity A circuit family is a set fC n Ng of circuits where each circuit C takes n n binary inputs of length n and pro duces a single output Each circuit family thus denes a language A circuit family is uniform if the function n C is easily n computable in some sense For the very small circuit complexity classes we discuss here a very strong notion of uniformity is appropriate This is discussed in detail in BIS we will not give detailed denitions concerning uniformity here Two imp ortant circuit complexity classes are NC and AC NC is the class of languages accepted by uniform circuits of AND and OR gates of fanin of p olynomial size and logarithmic depth AC is the class of languages accepted by uniform circuits of AND and OR gates of unb ounded fanin of p olynomial size and O depth AC denotes the languages in AC accepted by circuits of depth k k ClearlyAC NC and b oth classes are contained in P In some