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Probabilistic Proof Systems Basic Research in Computer Science BRICS BRICS RS-94-28 O. Goldreich: Probabilistic Proof Systems Basic Research in Computer Science Probabilistic Proof Systems Oded Goldreich BRICS Report Series RS-94-28 ISSN 0909-0878 September 1994 Copyright c 1994, BRICS, Department of Computer Science University of Aarhus. All rights reserved. Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS Department of Computer Science University of Aarhus Ny Munkegade, building 540 DK - 8000 Aarhus C Denmark Telephone:+45 8942 3360 Telefax: +45 8942 3255 Internet: [email protected] Probabilisti c Pro of Systems Oded Goldreich Department of Applied Mathematics and Computer Science Weizmann Institute of Science Rehovot Israel September Abstract Various typ es of probabilistic pro of systems haveplayed a central role in the de velopment of computer science in the last decade In this exp osition we concentrate on three such pro of systems interactive proofs zeroknow ledge proofsand proba bilistic checkable proofs stressing the essential role of randomness in each of them This exp osition is an expanded version of a survey written for the pro ceedings of the International Congress of Mathematicians ICM held in Zurichin It is hop e that this exp osition may b e accessible to a broad audience of computer scientists and mathematians Partially supp orted by grant No from the United States Israel Binational Science Founda tion BSF Jerusalem Israel Revised and expanded while visiting BRICS Basic Research in Computer Science Center of the Danish National ResearchFoundation Intro duction The glory given to the creativity required to nd pro ofs makes us forget that it is the less gloried pro cedure of verication whichgives pro ofs their value Philosophically sp eaking pro ofs are secondary to the verication pro cedure whereas technically sp eaking pro of systems are dened in terms of their verication pro cedures The notion of a verication pro cedure assumes the notion of computation and fur thermore the notion of ecient computation This implicit assumption is made explicit in the denition of NPinwhich ecient computation is asso ciated with deterministic p olynomialtime algorithms Denition NPpro of systems Let S f g and f g f g f g bea function so that x S if and only if there exists a w f g such that x w If is computable in time boundedbyapolynomial in the length of its rst argument then we say that S is an NPset and denes an NPpro of system For example in prop ositional calculus a pro of is a sequence of assertions each b eing a form of an axiom or is obtained by applying an inference rule on previous assertions Thus the verication pro cedure consists of checking the justication of each assertion in the sequence Clearly this pro cedure can b e implemented bya very ecient algorithm In contrast it is widely b elieved that there exists no ecient algorithm for nding pro ofs to given assertions in prop ositional calculus since the task is NPHard see b elow Traditionally NP is dened as the class of NPsets cf Yet eachsuch NPset can b e viewed as a pro of system For example consider the set of satisable Bo olean formulae Clearly a satisfying assignment for a formula constitutes an NPpro of for the assertion is satisable the verication pro cedure consists of substituting the variables of by the values assigned by and computing the value of the resulting Bo olean expression The formulation of NPpro ofs restricts the eective length of pro ofs to b e p olyno mial in length of the corresp onding assertions since the runningtime of the verication pro cedure is restricted to b e p olynomial in the length of the assertion However longer pro ofs may b e allowed by padding the assertion with suciently many blank symb ols So it seems that NP gives a satisfactory formulation of pro of systems with ecientver ication pro cedures This is indeed the case if one asso ciates ecient pro cedures with deterministic p olynomialtime algorithms However we can gain a lot if we are willing to take a somewhat nontraditional step and allow probabilistic verication pro cedures In particular Randomized and interactivev erication pro cedures giving rize to interactive proof systemsseemmuch more p owerful ie expressive than their deterministic coun terparts Such randomized pro cedures allowtheintro duction of zeroknow ledge proofs which are of great theoretical and practical interest NPpro ofs can b e eciently transformed into a redundant form which oers a trade o b etween the numb er of lo cations examined in the NPpro of and the condence in its validitysee probabilistical ly checkable proofs In all ab ovementioned typ es of probabilistic pro of systems explicit b ounds are imp osed on the computational complexityoftheverication pro cedure which in turn is p ersonied by the notion of a verier Furthermore in all these pro of systems the verier is allowed to toss coins and rule by statistical evidence Thus all these pro of systems carry a probability of error yet this probability is explicitly b ounded and furthermore can b e reduced by successive application of the pro of system Basic background from computational complexity The following are standard complexity classes Pdenotes the class of sets in which memb ership can b e decided in deterministic p olynomialtime Namely for every S P there exists a deterministic p olynomial time algorithm A so that x S i Ax for all x f g Note that P is a subset of NP consiting of these NPsets for which pro ofs of memb ership ie NPpro ofs can b e eciently found rather than merely exist RP resp BP P denotes the class of sets in whichmemb ership can b e decided wosided error probability in probabilistic p olynomialtime with onesided resp t Sp ecically for every S RP there exists a probabilistic p olynomialtime algorithm A so that for every x S Prob Ax whereas for every x S Prob Ax where the probabilityistaken uniformly over all p ossible outcomes of the inter nal coin tosses of algorithm A for every S BPP there exists a probabilistic p olynomialtime algorithm A so that for every x S Prob Ax whereas for every x S Prob Ax In b oth cases the nontrivial probabilityboundsmaybechange in various ways preserving the complexity class NPdenotes the class of NPsets and coNP denotes the class of their complements def S NP where S f g S ie S coNP i A set S is polynomialtime reducible to a set T if there exists a p olynomialtime computable function f so that x S i f x T for every xAsetisNPhard if every NPset is p olynomialtime reducible to it A set is NPcomplete if it is b oth NPhard and in NP PSPACE denotes the class of sets in whichmemb ership can b e decided in p olynomial space ie the workspace taken by the decider is p olynomial in length of the input Obviously P RP BPP PSPACE It is not hard to see that RP N P and that N P PSPACE It is widely b elieved that P NP and NP PSPACE Furthermore it is also b elieved that NP coNP NPhard sets or tasks are assumed to b e infeasible since if an NPhard set is in P then NP P by virtue of the reductions of all NPsets to each NPhard set Conventions When presenting a pro of system we state all complexity b ounds in terms of the length of the assertion to b e proven whichisviewed as an input to the verier Namely p olynomial time means time p olynomial in the length of this assertion Note that this convention is consistent with our denition of NPpro ofs Denote by poly the set of all integer functions b ounded by a p olynomial and by log the set of all integer functions b ounded by a logarithmic function ie f log i f n O log n Basic Background from combinatorics A simple graph GisapairV EwhereE is a set of subsets of V ie for every e E it holds je V j The elements of V are called vertices and the elements of E are called edges In this exp osition we consider only simple nite graphs Two graphs G V E and G V E are called isomorphic if there exists a and onto mapping from the v ertex set V to the vertex set V so that fu v g E if and only if fv ug E The edge preserving mapping if existing is called an isomorphism between the graphs A graph G V Eissaidtobe colorable if there exists a function V f g so that v uforevery fu v g E Such a function is called a coloring of the graph Interactive Pro of Systems In light of the growing acceptability of randomized and distributed computations it is only natural to asso ciate the notion of ecient computation with probabilistic and interactive p olynomialtime computations This leads naturally to the notion of interactive pro of systems in whichtheverication pro cedure is interactive and randomized rather than b eing noninteractive and deterministicThus a pro of in this context is not a xed and static ob ject but rather a randomized dynamic pro cess in which the verier interacts with the prover Intuitivelyonemay think of this interaction as consisting of tricky questions asked by the verier to which the prover has to reply convincingly The ab ove discussion as well as the following denition makes explicit reference to a prover whereas a prover is only implicit in the traditional denitions of pro of systems eg NPpro ofs Denition Lo osely sp eaking an interactive pro of is a game b etween a computationally b ounded ver ier and a computationally unb ounded prover whose goal is to convince the verier of the validity of
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