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Probabilistic Proof Systems { Lecture Notes Probabilistic Pro of Systems Lecture Notes Oded Goldreich Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot Israel September Abstract Various typ es of probabilistic pro of systems haveplayed a central role in the development of computer science in the last decade In these notes we concentrate on three such pro of systems interactive proofs zeroknow ledge proofsand probabilistic checkable proofs Remark These are lecture notes in the strict sense of the word Surveys of mine on this sub ject can b e obtained from URL httptheorylcsmitedu odedppshtml These notes were prepared for a series of lectures given in the Theory Student Seminar of the CS Department of UCBerkeley Visiting Miller Professor Aug Sept EECS Dept UCBerkeley Intro duction The glory given to the creativity required to nd pro ofs makes us forget that it is the less glori ed 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 furthermore the notion of ecient computation This implicit assumption is made explicit in the denition of NP in which ecient computation is asso ciated with deterministic p olynomialtime algorithms Traditionally NP is dened as the class of NPsets Yet each such 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 olynomial in length of the corresp onding assertions 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 ecientverication pro cedures This is indeed the case if one asso ciates ecient pro cedures with deterministic p olynomialtime algorithms However wecangainalotifweare willing to take a somewhat nontraditional step and allow probabilistic verication pro cedures In particular interactive proof systems Randomized and interactiveverication pro cedures giving rise to seem much more p owerful ie expressive than their deterministic counterparts Such randomized pro cedures allow the intro 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 tradeo between the numb er of lo cations examined in the NPpro of and the condence in its validity see probabilistical ly checkable proofs In all ab ovementioned typ es of probabilistic pro of systems explicit b ounds are imp osed on the computational complexity of the verication pro cedure which in turn is p ersonied by the notion ofaverier 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 Interactive Pro of Systems In lightofthegrowing 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 which the verica tion pro cedure is interactive and randomized rather than b eing noninteractive and deterministic Thus 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 Intuitively one may think of this interaction as consisting of tricky questions asked by the verier to which the prover has to reply convinc ingly The ab ove discussion as well as the actual denition makes explicit reference to a prover whereas a prover is only implicit in the traditional denitions of pro of systems eg NPpro ofs The Denition Interaction Going b eyond the unidirectional interaction of the NPpro of system If the verier do es not toss coins then interaction can b e collapsed to a single message computationall y unb ounded Prover As in NPwe start by not considering the complexity of proving probabilistic p olynomialtime Verier We maintain the paradigm that verication ought to b e easyalaswe allow random choices in our notion of easiness Completeness and Soundness We relax the traditional soundness condition by allowing small probability of b eing fo oled by false pro ofs The probabilityistaken over the veriers random choices We still require p erfect completeness that is that correct statements are proven with probability Error probability b eing a parameter can b e further reduced by successive rep eti tions Variations Relaxing the p erfect completeness requirement yields a twosided error variantof IP ie error probability allowed also in the completeness condition Restricting the verier to send only random ie uniformly chosen messages yields the restricted ArthurMerlin interactive pro ofs aka publiccoins interactive pro ofs Alas b oth variants are essentially as p owerful as the one ab ove An Example interactive pro of of Graph NonIsomorphism The problem not known to b e in NP Proving that two graphs are isomorphic can b e done by presenting an isomorphism but howdoyou prove that no such isomorphism exists The construction the two dierent ob ject proto col if you claim that two ob jects are dierent then you should b e able to tell which is which when I presentthemtoyou in random order In the context of the Graph NonIsomorphism interactive pro of two supp osedly dierentobjects are dened by taking random isomorphic copies of each of the input graphs If these graphs are indeed nonisomorphic then the ob jects are dierent the distributions have distinct supp ort else the ob jects are identical Interactive pro of of NonSatisability Arithmetization of Bo olean CNF formulae Observe that the arithmetic expression is a low degree p olynomial Observe that in any case the value of the arithmetic expression is b ounded Moving to a Finite Field Whenever wecheck equalitybetween twointegers in M it suces to check equalitymodq where qM The b enet is that the arithmetic is now in a nite eld mo d q and so certain things are nicer eg uniformly selecting a value Thus proving that a CNF formula is not satisable reduces to proving equality of the following form X X x x modq n x x n where is a lowdegreemultivariant p olynomial The construction stripping summations in iterations In each iteration the prover is supp osed to supply the p olynomial describing the expression in one currently stripp ed variable By the ab ove observation this is a low degree p olynomial and so has a short description The verier checks that the p olynomial is of low degree and that it corresp onds to the currentvalue b eing claimed ie p p v Next the verier randomly instantiates the variable yielding a new r for uniformly chosen r GFq value to b e claimed for the resulting expression ie v p The verier sends the uniformly chosen instantiation to the prover At the end of the last iteration the verier has a fully sp ecied expression and can easily check it against the claimed value Completeness of the ab ove When the claim holds the prover has no problem supplying the correct p olynomials and this will lead the verier to always accept Soundness of the ab ove It suces to b ound the probability that for a particular iteration the initial claim is false whereas the ending claim is correct Both claims refer to the current summation expression b eing equal to the currentvalue where current means either at the b eginning of the iteration or at its end Let T b e the actual p olynomial representing the expression when stripping the currentvariable and let pbeany p otential answer by the prover Wemay assume that p p v and that p is of lowdegree as otherwise the verier will reject Using our hyp othesis that the initial claim is false we know that T T v Thus p and T are dierentlowdegree p olynomials and so they may agree on very few p oints In case the verier instantiation do es not happ en to b e one of these few p oints the ending claim is false to o Op en Problem alternative pro of of coNP IP Polynomials play a fundamental role in the above construction and this trend has even deepened in subsequent works on PCP It does not seem possible to abstract that role which seems to be very annoying I consider it important to obtain an alternative proof of coNP IPa proof in which al l the underlying ideas can bepresentedat an abstract level The Power of Interactive Pro ofs IP PSPACE Interactive Pro ofs for PSPACE Recall that PSPACE languages can b e expressed by Quan tied Bo olean Formulae The numb er of quantiers is p olynomial in the input but there are b oth existential and universal quantiers and furthermore these quantiers may alternate Con sidering the arithmetization of these formulae wefacetwo problems Firstly the value of the formulae is only b ounded by a double exp onential function in the length of the input and sec ondly when stripping out summations the expression may b e a p olynomial of high degree due to the universal
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