Lance Fortnow

Non-deterministic exponential time has two-prover interactive protocols

We determine the exact power of two-prover interactive proof systems introduced by Ben-Or, Goldwasser, Kilian, and Wigderson (1988). In this system, two all-powerful noncommunicating provers convince a randomizing polynomial time verifier in polynomial time that the inputx belongs to the languageL. We show that the class of languages having tow-prover interactive proof systems is nondeterministic exponential time. We also show that to prove membership in languages inEXP, the honest provers need the power ofEXP only. The first part of the proof of the main result extends recent techniques of polynomial extrapolation used in the single prover case by Lund, Fortnow, Karloff, Nisan, and Shamir. The second part is averification scheme for multilinearity of a function in several variables held by an oracle and can be viewed as an independent result onprogram verification. Its proof rests on combinatorial techniques employing a simple isoperimetric inequality for certain graphs:

Algebraic methods for interactive proof systems

A new algebraic technique for the construction of interactive proof systems is presented. Our technique is used to prove that every language in the polynomial-time hierarchy has an interactive proof system. This technique played a pivotal role in the recent proofs that IP = PSPACE [28] and that MIP = NEXP [4].

Checking computations in polylogarithmic time

Motivated by Manuel Blum’s concept of inst ante checking, we consider new, very fast and generic mechanisms of checking computations. Our results exploit recent advances in interactive proof protocols [LFKN], [Sh], and especially the MIP = NEXP protocol from [BFL]. WJe show that every nondeterministic computational task S(Z, y), defined as a polynomial time relation between the instance x, representing the input and output combined, and the witness y can be modified to a task S such that: (i) the same instances remain accepted; (ii) each instance/witness pair becomes checkable in poly!ogariihmic Monte Carlo time; and (iii) a witness satisfying S’ can be computed in polynomial time from a witness satisfying S. Here the instance and the description of S have to be provided in error-correcting code (since the checker will not notice slight changes). A modification of the MIP proof was required to achieve polynomial time in (iii); the earlier technique yields N“(*OglOg’1 time only. This result becomes significant if software and hardware reliability are regarded aa a considerable cost factor. The polylogarithmic checker is the only part of the system that needs to be trusted; it can be hard wired. (We use just one Checker for all problems!) The checker is tiny and so presumably can be optimized and checked off-line at a modest cost. In this setup, a single reliable PC can monitor the operation of a herd of supercomputers working with possibly extremely powerful but unreliable software and untested hardware. 1Research partially supported by NSF Grant CCR-871OO78. Email: laci@cs.uchicago. edu 2Rese~& partiay supported by NSF Grant CCR-SMXI

BPP has subexponential time simulations unless EXPTIME has publishable proofs

XEK5. E-mail: fortnow@cs.uc&cago. edu 3Supported by NSF grant CCR-SJC115276. E-mail: Lnd@cs.bu.edu 4111 cummington St., Boston MA 02215. 5E-mail: mario@cs.uchicago .edu ~1100 E 58th St, Chicago IL 60637. Permission to copy without fee all or part of this msterisl is granted provided that tie copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee snd/or specific permission. @ 1991 ACM 089791-397-31911000410021

The complexity of perfect zero-knowledge

1.50 Leonid A. Levin 3

Testing that distributions are close

AbstractWe show thatBPP can be simulated in subexponential time for infinitely many input lengths unless exponential timeℴ collapses to the second level of the polynomial-time hierarchy.ℴ has polynomial-size circuits andℴ has publishable proofs (EXPTIME=MA). We also show thatBPP is contained in subexponential time unless exponential time has publishable proofs for infinitely many input lengths. In addition, we showBPP can be simulated in subexponential time for infinitely many input lengths unless there exist unary languages inMA-P.The proofs are based on the recent characterization of the power of multiprover interactive protocols and on random self-reducibility via low-degree polynomials. They exhibit an interplay between Boolean circuit simulation, interactive proofs and classical complexity classes. An important feature of this proof is that it does not relativize.One of the ingredients of our proof is a lemma that states that ifEXPTIME has polynomial size circuits thenEXPTIME=MA. This extends previous work by Albert Meyer.

On the power of multi-prover interactive protocols

A Perfect Zero-Knowledge interactive proof system convinces a verifier that a string is in a language without revealing any additional knowledge in an information-theoretic sense. We show that for any language that has a perfect zero-knowledge proof system, its complement has a short interactive protocol. This result implies that there are not any perfect zero-knowledge protocols for NP-complete languages unless the polynomial time hierarchy collapses. This paper demonstrates that knowledge complexity can be used to show that a language is easy to prove.

The status of the P versus NP problem

Given two distributions over an n element set, we wish to check whether these distributions are statistically close by only sampling. We give a sublinear algorithm which uses O(n/sup 2/3//spl epsiv//sup -4/ log n) independent samples from each distribution, runs in time linear in the sample size, makes no assumptions about the structure of the distributions, and distinguishes the cases when the distance between the distributions is small (less than max(/spl epsiv//sup 2//32/sup 3//spl radic/n,/spl epsiv//4/spl radic/n=)) or large (more than /spl epsiv/) in L/sub 1/-distance. We also give an /spl Omega/(n/sup 2/3//spl epsiv//sup -2/3/) lower bound. Our algorithm has applications to the problem of checking whether a given Markov process is rapidly mixing. We develop sublinear algorithms for this problem as well.

Nondeterministic exponential time has two-prover interactive protocols

Abstract We look at complexity issues of interactive proof systems with multiple provers separated from each other. This model, developed by Ben-Or et al. (1988) allows the verifier to play the provers off each other. We show this model equivalent to an alternative interactive proof system model using oracles as provers. We also show that every language accepted by these models lies in nondeterministic exponential time. We exhibit a relativized world where a co-NP language does not have multiple prover interactive proofs. Finally, we show a simple example that one cannot parallelize multiple prover protocols as easily as the single prover model.

Testing random variables for independence and identity

Its one of the fundamental mathematical problems of our time, and its importance grows with the rise of powerful computers.