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37
NonDeterministic Exponential Time has TwoProver Interactive Protocols
"... We determine the exact power of twoprover interactive proof systems introduced by BenOr, Goldwasser, Kilian, and Wigderson (1988). In this system, two allpowerful noncommunicating provers convince a randomizing polynomial time verifier in polynomial time that the input z belongs to the language ..."
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Cited by 406 (40 self)
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We determine the exact power of twoprover interactive proof systems introduced by BenOr, Goldwasser, Kilian, and Wigderson (1988). In this system, two allpowerful noncommunicating provers convince a randomizing polynomial time verifier in polynomial time that the input z belongs to the language L. It was previously suspected (and proved in a relativized sense) that coNPcomplete languages do not admit such proof systems. In sharp contrast, we show that the class of languages having twoprover interactive proof systems is nondeterministic exponential time. After the recent results that all languages in PSPACE have single prover interactive proofs (Lund, Fortnow, Karloff, Nisan, and Shamir), this represents a further step demonstrating the unexpectedly immense power of randomization and interaction in efficient provability. Indeed, it follows that multiple provers with coins are strictly stronger than without, since NEXP # NP. In particular, for the first time, provably polynomial time intractable languages turn out to admit “efficient proof systems’’ since NEXP # P. We show that to prove membership in languages in EXP, the honest provers need the power of EXP only. A consequence, linking more standard concepts of structural complexity, states that if EX P has polynomial size circuits then EXP = Cg = MA. The first part of the proof of the main result extends recent techniques of polynomial extrapolation of truth values used in the single prover case. The second part is a verification scheme for multilinearity of an nvariable function held by an oracle and can be viewed as an independent result on program verification. Its proof rests on combinatorial techniques including the estimation of the expansion rate of a graph.
BPP has Subexponential Time Simulations unless EXPTIME has Publishable Proofs (Extended Abstract)
, 1993
"... ) L'aszl'o Babai Noam Nisan y Lance Fortnow z Avi Wigderson University of Chicago Hebrew University Abstract We show that BPP can be simulated in subexponential time for infinitely many input lengths unless exponential time ffl collapses to the second level of the polynomialtime hierarchy, ..."
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Cited by 114 (9 self)
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) L'aszl'o Babai Noam Nisan y Lance Fortnow z Avi Wigderson University of Chicago Hebrew University Abstract We show that BPP can be simulated in subexponential time for infinitely many input lengths unless exponential time ffl collapses to the second level of the polynomialtime hierarchy, ffl has polynomialsize circuits and ffl has publishable proofs (EXPTIME=MA). We also show that BPP is contained in subexponential time unless exponential time has publishable proofs for infinitely many input lengths. In addition, we show BPP can be simulated in subexponential time for infinitely many input lengths unless there exist unary languages in MA n P . The proofs are based on the recent characterization of the power of multiprover interactive protocols and on random selfreducibility 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 ...
Linear time solvable optimization problems on graphs of bounded cliquewidth
, 2000
"... Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decompositions are the best known ones. On graphs of treewidth at most k, i.e., that have tree decompositions of width at most k, where k is fixed, every dec ..."
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Cited by 112 (22 self)
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Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decompositions are the best known ones. On graphs of treewidth at most k, i.e., that have tree decompositions of width at most k, where k is fixed, every decision or optimization problem expressible in monadic secondorder logic has a linear algorithm. We prove that this is also the case for graphs of cliquewidth at most k, where this complexity measure is associated with hierarchical decompositions of another type, and where logical formulas are no longer allowed to use edge set quantifications. We develop applications to several classes of graphs that include cographs and are, like cographs, defined by forbidding subgraphs with “too many” induced paths with four vertices.
In Search of an Easy Witness: Exponential Time vs. Probabilistic Polynomial Time
"... Restricting the search space f0; 1g to the set of truth tables of "easy" Boolean functions on log n variables, as well as using some known hardnessrandomness tradeoffs, we establish a number of results relating the complexity of exponentialtime and probabilistic polynomialtime complexity cla ..."
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Cited by 58 (6 self)
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Restricting the search space f0; 1g to the set of truth tables of "easy" Boolean functions on log n variables, as well as using some known hardnessrandomness tradeoffs, we establish a number of results relating the complexity of exponentialtime and probabilistic polynomialtime complexity classes. In particular, we show that NEXP ae P=poly , NEXP = MA; this can be interpreted as saying that no derandomization of MA (and, hence, of promiseBPP) is possible unless NEXP contains a hard Boolean function. We also prove several downward closure results for ZPP, RP, BPP, and MA; e.g., we show EXP = BPP , EE = BPE, where EE is the doubleexponential time class and BPE is the exponentialtime analogue of BPP.
Instance Complexity
, 1994
"... We introduce a measure for the computational complexity of individual instances of a decision problem and study some of its properties. The instance complexity of a string x with respect to a set A and time bound t, ic t (x : A), is defined as the size of the smallest specialcase program for A that ..."
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Cited by 30 (1 self)
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We introduce a measure for the computational complexity of individual instances of a decision problem and study some of its properties. The instance complexity of a string x with respect to a set A and time bound t, ic t (x : A), is defined as the size of the smallest specialcase program for A that runs in time t, decides x correctly, and makes no mistakes on other strings ("don't know" answers are permitted). We prove that a set A is in P if and only if there exist a polynomial t and a constant c such that ic t (x : A) c for all x
Pselective Selfreducible sets: A New Characterization of P
 In Proceedings of the 8th Structure in Complexity Theory Conference
, 1996
"... We show that any pselective and selfreducible set is in P . As the converse is also true, we obtain a new characterization of the class P . A generalization and several consequences of this theorem are discussed. Among other consequences, we show that under reasonable assumptions autoreducibi ..."
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Cited by 27 (6 self)
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We show that any pselective and selfreducible set is in P . As the converse is also true, we obtain a new characterization of the class P . A generalization and several consequences of this theorem are discussed. Among other consequences, we show that under reasonable assumptions autoreducibility and selfreducibility differ on NP , and that there are nonpT mitotic sets in NP . 1 Introduction Separating complexity classes is a very popular, but rarely won game in complexity theory. Frustrated by misfortune, computer scientists have often turned to attempts of characterizing complexity classes in a different way. The hopes are, that the new characterization of the complexity class may provide new insights and a `handle' to force the separation where earlier attempts have failed. Wellknown examples of this are the many ways to define the class of sets for which there exist small circuits [Pip79], and the identification of various forms of interactive proof systems with stan...
A Downward Collapse Within The Polynomial Hierarchy
, 1998
"... . Downward collapse (also known as upward separation) refers to cases where the equality of two larger classes implies the equality of two smaller classes. We provide an unqualified downward collapse result completely within the polynomial hierarchy. In particular, we prove that, for
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Cited by 23 (9 self)
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.<F3.803e+05> Downward collapse (also known as upward separation) refers to cases where the equality of two larger classes implies the equality of two smaller classes. We provide an unqualified downward collapse result completely within the polynomial hierarchy. In particular, we prove that, for<F3.319e+05> k ><F3.803e+05> 2, if P<F2.821e+05> #<F2.795e+05> p k<F2.821e+05> [1]<F3.803e+05> = P<F2.821e+05> #<F2.795e+05> p k<F2.821e+05> [2]<F3.803e+05> then #<F2.562e+05> p k<F3.803e+05> = #<F2.562e+05> p k<F3.803e+05> = PH. We extend this to obtain a more general downward collapse result.<F4.005e+05> Key words.<F3.803e+05> computational complexity theory, easyhard arguments, downward collapse, polynomial hierarchy<F4.005e+05> AMS subject classifications.<F3.803e+05> 68Q15, 68Q10, 03D15, 03D10<F4.005e+05> PII.<F3.803e+05> S0097539796306474<F5.353e+05> 1. Introduction.<F4.529e+05> The theory of NPcompleteness does not resolve the issue of whether P and NP are equal. However, it do...
Disjoint NPPairs
, 2003
"... We study the question of whether the class DisNP of disjoint pairs (A, B) of NPsets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NPsets that is N ..."
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Cited by 23 (7 self)
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We study the question of whether the class DisNP of disjoint pairs (A, B) of NPsets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NPsets that is NPhard. We show under reasonable hypotheses that nonsymmetric disjoint NPpairs exist, which provides additional evidence for the existence of Pinseparable disjoint NPpairs. We construct
FiniteModel Theory  A Personal Perspective
 Theoretical Computer Science
, 1993
"... Finitemodel theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finitemodel theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph ..."
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Cited by 20 (0 self)
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Finitemodel theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finitemodel theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph.D. thesis. Among the topics discussed are:
Email and the unexpected power of interaction
 Structure in Complexity theory
, 1988
"... This is a true fable about Merlin, the infinitely intelligent but never trusted magician; and Arthur, the reasonable but impatient sovereign with an occasional unorthodox request; about the concept of an efficient proof; about polynomials and interpolation, electronic mail, coin flipping, and the in ..."
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Cited by 18 (3 self)
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This is a true fable about Merlin, the infinitely intelligent but never trusted magician; and Arthur, the reasonable but impatient sovereign with an occasional unorthodox request; about the concept of an efficient proof; about polynomials and interpolation, electronic mail, coin flipping, and the incredible power of interaction. About MIP, IP, #P, P SP ACE, NEXP T IME, and new techniques that do not relativize. About fast progress, fierce competition, and email ethics. 1 How did Merlin end up in the cave? In the court of King Arthur1 there lived 150 knights and 150 ladies. “Why not 150 married couples, ” the King contemplated one rainy afternoon, and action followed the thought. He asked the Royal Secret Agent (RSA) to draw up a diagram with all the 300 names, indicating bonds of mutual interest between lady and knight by a red line; and the lack thereof, by a blue line. The diagram, with its 1502 = 22, 500 colored lines, looked somewhat confusing, yet it should not confuse Merlin, the court magician, to whom it was subsequently presented by Arthur with the express order to find a perfect matching consisting exclusively of red lines. Merlin walked away, looked at the diagram, and, with his unlimited intellectual ability, immediately recognized that none of the 150! possibilities gave an allred perfect matching. He quickly completed the 150! diagrams, highlighting the wrong blue line in