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A Taxonomy of Proof Systems*

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ATaxonomy of Pro of Systems Oded Goldreich Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot Israel September Abstract Several alternative formulations of the concept of an ecient pro of system are nowadays co existing in our eld These systems include the classical formulation of NP interactive proof systems giving rise to the class IP computational lysound proof systemsand probabilistical ly checkable proofs PCP which are closely related to multiprover interactive proofs MI P Although these notions are sometimes intro duced using the same generic phrases they are actually very dierent in motivation applications and expressivepower The main ob jectiveof this essay is to try to clarify these dierences This is a revised version of a survey which has app eared in Complexity Theory Retrospective II LA Hemaspaan dra and A Selman eds Intro duction In recentyears alternativeformulations of the concept of an ecient pro of system have received much attention Not only have talks and pap ers concerning these systems o o ded the eld of theoretical computer science but also some of these developments havereached the nontheory community and a few were even rep orted in nonscientic forums suchastheNew York Times Thus I am quite sure that the reader has heard of phrases suchasinteractive pro ofs and results suchasIP PSPACE By no means am I suggesting that the interest in the various formulations of ecientproof systems has gone out of prop ortion Certainly the notion of an ecient pro of system is central to the eld of computer science and I nd it hard to conceive of circumstances in which one mightsay that it was receiving to o muchattention Furthermore the research area established by these notions has b een one of the most successful and rewarding enterprises in which the theoretical computer science community has b een involved For example zeroknowledge pro ofs haverevolutionized the design of cryptographic proto cols and the characterization of NP in terms of probabilistically checkable pro ofs has contributed to and in fact revived the attempts to classify the complexity of approximation problems Except for NP all pro of systems reviewed b elow are probabilistic and furthermore ha ve a non zero error probabilityHowever the error probability is explicitly b ounded and can b e reduced by successive applications of the pro of system In all cases nonzero error probability is essential to the interesting prop erties and consequences that these probabilistic pro of systems have Referencing convention Ihave decided to pro ceed in a somewhat unconventional way and have decoupled the technical exp osition from the story b ehind its evolution These parts app ear in separate sections whichcan b e read indep endently of each other When reading the technical part the reader should b ear in mind that the references in this part are minimal and denitely substandard with the sole objective of referring the reader to the b est source for more details I wish to stress that the b est source for more details is not necessarily the source that deserves the most credit The situation is reversed in the story part whichcontains only credits or more accurately my evaluation of the contribution of the various works to the development of the eld Addendum The currentversion is augmented by an op en problems section ATechnical Exp osition The notion of a pro of is one of the more fundamental notions of our culture In particular it is central to science and sp ecically to mathematics and computer science Yet although p eople always talk of proofs the fundamental issue is the verication pro cedure Pro of systems are dened by their verication pro cedure and it is the verication pro cedure that gives them their value The notion of a verication pro cedure assumes the notion of computation and furthermore ecient computation This implicit assumption is made explicit in the denition of NP It is the asso ciation of ecient computation with deterministic p olynomialtime algorithms that yields the asso ciation of ecient pro of systems with the class NPNamely to prove the validity of some statement one supplies a relatively short pro of and the verication pro cedure consists of running a p olynomialtime algorithm on input Technical Remarks All complexity measures mentioned in the subsequent exp osition are assumed to b e constructible in p olynomial time We denote by poly the set of all p olynomials and by log the set of all logarithmic functions ie integer functions b ounded by O log n We adopt the def def poly poly standard notations EX P DTIME andNEXP NTIME Interactive Pro of Systems In light of the growing acceptability of randomized and interactive computations it is only natural e p olynomialtime to asso ciate the notion of ecient computation with probabilistic and interactiv computations This leads naturally to the notion of an interactive pro of system in which the veri cation pro cedure is interactive and randomized rather than b eing noninteractive and deterministic as in NP A sketch of the formal denition is given in Item b elow We stress that no com putational restrictions are placed on the prover Items and intro duce additional complexity measures that can b e ignored in a rst reading Denition Interactive Pro ofs IP An interactive pro of system ips for a language L is a pair P V of interactive machines so that V is a probabilistic polynomialtime machine satisfying CompletenessFor every x L the verier V always accepts after interacting with the prover P on common input x SoundnessFor every x L and every potential prover P the verier V rejects with probability at least after interacting with P on common input x Let m and r beinteger functions The complexity class IP mr consists of languages having an interactive proof system in which on common input x the verier uses at most r jxj coin tosses and the total number of messages exchangedbetween the parties is bounded by mjxj Let M and R be sets of integer functions Then def IP M R IP mr mMrR def def IPm poly and IP IP poly Final ly IP m Weavoid the denition of interacting machines This denition can b e found in In Item wehave followed the common convention of sp ecifying b oth the verier and the prover An alternative presentation only sp ecies the verier while rephrasing the completeness condition as follows There exists a machine P a prover so that for every x L the verier V always accepts after interacting with P on common input x The soundness condition allows for errors that is executions in which the verier accepts x L In general one may consider the error probability in Yet the error is explicitly b ounded by the soundness condition as another parameter It is not hard to see that the error probabilityin interactive pro ofs can b e reduced by indep endent sequential andor parallel rep etitions Actually this holds even for somewhat dep endent parallel rep etitions see which is instructive also for the simpler case of indep endent parallel rep etitions On the other hand requiring zero soundness error in interactive pro of systems restricts their existence to languages in NP We stress that although wehave relaxed the requirements from the verication pro cedure by allowing it to interact toss coins and err with b ounded probabilitywe did not restrict the validity of the assertions by assumptions concerning the p otential prover This should b e contrasted with later notions of pro of systems such as computationallysound ones and multiprover ones in which the validity of the soundness condition dep ends on assumptions concerning the external proving entity Known results Clearly IP poly equals coRP whereas IP equals NPFurthermore IP poly con tains BP P see or Hence IP IP polycontains BP P N P whereas we currently do not know whether NP contains BP P It is also easy to see that IP log collapses to IP P whereas IPpoly log collapses to IP NP The main result concerning interactive pro of systems is that they exist for any language recognizable in p olynomial space Namely Theorem IP PSPACE Theorem was established using algebraic metho ds In particular the following approach unprecedented in complexity theory was employed In order to demonstrate that a particular language is in a particular class an arithmetic generalization of the Bo olean problem is presented and elementary algebraic metho ds are applied to show that the arithmetic problem is solvable within the class Interestingly this technique do es not relativize and furthermore yields results eg IP PSPACE that are false relative to most oracles providing a dramatic refutation of the Random Oracle Hyp othesis see Concerning the ner structure of the IPhierarchy the following is known For every integer function f so that f n for all n the class IP O f collapses to the class IP f and in particular IP O collapses to IP The class IP contains languages not known to b e in NP eg Graph NonIsomorphism See for the general technique and for its applicati on yielding the quoted result The class IP is contained in NPp oly ie nonuniformNP analogously to BP P P poly If coNP IP then the p olynomialtime hierarchy collapses It is conjectured that coNP is not contained in IP and consequently that interactive pro ofs with an unb ounded numb er of message exchanges are more p owerful than interactive pro ofs in which only a b ounded ie constant numb er of messages are exchanged The IPhierarchy ie IP equals an analogous hierarchy in whichtheverier
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