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27
Designing Programs That Check Their Work
, 1989
"... A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It d ..."
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Cited by 313 (17 self)
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A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It designs program checkers for a few specific and carefully chosen problems in the class FP of functions computable in polynomial time. Problems in FP for which checkers are presented in this paper include Sorting, Matrix Rank and GCD. It also applies methods of modern cryptography, especially the idea of a probabilistic interactive proof, to the design of program checkers for group theoretic computations. Two strucural theorems are proven here. One is a characterization of problems that can be checked. The other theorem establishes equivalence classes of problems such that whenever one problem in a class is checkable, all problems in the class are checkable.
PSelective Sets, and Reducing Search to Decision vs. SelfReducibility
, 1993
"... We obtain several results that distinguish selfreducibility of a language L with the question of whether search reduces to decision for L. These include: (i) If NE 6= E, then there exists a set L in NP \Gamma P such that search reduces to decision for L, search does not nonadaptively reduces to de ..."
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Cited by 39 (9 self)
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We obtain several results that distinguish selfreducibility of a language L with the question of whether search reduces to decision for L. These include: (i) If NE 6= E, then there exists a set L in NP \Gamma P such that search reduces to decision for L, search does not nonadaptively reduces to decision for L, and L is not selfreducible. Funding for this research was provided by the National Science Foundation under grant CCR9002292. y Department of Computer Science, State University of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260 z Department of Computer Science, State University of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260 x Research performed while visiting the Department of Computer Science, State University of New York at Buffalo, Jan. 1992Dec. 1992. Current address: Department of Computer Science, University of ElectroCommunications, Chofushi, Tokyo 182, Japan.  Department of Computer Science, State University of New York at Buffalo, 226...
Lower Bounds for the Low Hierarchy
"... this paper. The low hierarchy, as defined in [Sc83], can only be used to classify the complexity of sets in NP. In order to talk about related sets that are not in NP, the extended low hierarchy was introduced in [BBS86]. (The levels of this hierarchy are labeled EL 2 ,EL 2 , A preliminary ver ..."
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Cited by 32 (4 self)
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this paper. The low hierarchy, as defined in [Sc83], can only be used to classify the complexity of sets in NP. In order to talk about related sets that are not in NP, the extended low hierarchy was introduced in [BBS86]. (The levels of this hierarchy are labeled EL 2 ,EL 2 , A preliminary version of this paper was presented at the 16th International Colloquium on Automata, Languages, and Programming [AH89a]
NPhard Sets are PSuperterse Unless R = NP
, 1992
"... A set A is pterse (psuperterse) if, for all q, it is not possible to answer q queries to A by making only q \Gamma 1 queries to A (any set X). Formally, let PF A qtt be the class of functions reducible to A via a polynomialtime truthtable reduction of norm q, and let PF A qT be the class of ..."
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Cited by 28 (5 self)
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A set A is pterse (psuperterse) if, for all q, it is not possible to answer q queries to A by making only q \Gamma 1 queries to A (any set X). Formally, let PF A qtt be the class of functions reducible to A via a polynomialtime truthtable reduction of norm q, and let PF A qT be the class of functions reducible to A via a polynomialtime Turing reduction that makes at most q queries. A set A is pterse if PF A qtt 6` PF A (q\Gamma1)T for all constants q. A is psuperterse if PF A qtt 6` PF X qT for all constants q and sets X . We show that all NPhard sets (under p tt reductions) are psuperterse, unless it is possible to distinguish uniquely satisfiable formulas from satisfiable formulas in polynomial time. Consequently, all NPcomplete sets are psuperterse unless P = UP (oneway functions fail to exist), R = NP (there exist randomized polynomialtime algorithms for all problems in NP), and the polynomialtime hierarchy collapses. This mostly solves the main open...
On Coherence, RandomSelfReducibility, and SelfCorrection
 In Proc. 11th Conference on Computational Complexity
, 1997
"... . We study three types of selfreducibility that are motivated by the theory of program verification. A set A is randomselfreducible if one can determine whether an input x is in A by making random queries to an Aoracle. The distribution of each query may depend only on the length of x. A set B i ..."
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Cited by 12 (6 self)
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. We study three types of selfreducibility that are motivated by the theory of program verification. A set A is randomselfreducible if one can determine whether an input x is in A by making random queries to an Aoracle. The distribution of each query may depend only on the length of x. A set B is selfcorrectable over a distribution D if one can convert a program that is correct on most of the probability mass of D to a probabilistic program that is correct everywhere. A set C is coherent if one can determine whether an input x is in C by asking questions to an oracle for C \Gamma fxg. We first show that adaptive coherence is more powerful than nonadaptive coherence, even if the nonadaptive querier is nonuniform. Blum et al. [Blum, Luby and Rubinfeld, Journal of Computer and System Sciences, 59:549595, 1993] showed that every randomselfreducible function is selfcorrectable. It is unknown, however, whether selfcorrectability implies randomselfreducibility. We show, under ...
Biimmunity Results for Cheatable Sets
 Theoretical Computer Science
, 1995
"... An oracle A is kcheatable if there is a polynomialtime algorithm to determine the answers to 2 k parallel queries to A from the answers to only k queries to some other oracle B. It is known that 1cheatable sets cannot be biimmune for P. In contrast, we construct 2cheatable sets that are biim ..."
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Cited by 10 (6 self)
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An oracle A is kcheatable if there is a polynomialtime algorithm to determine the answers to 2 k parallel queries to A from the answers to only k queries to some other oracle B. It is known that 1cheatable sets cannot be biimmune for P. In contrast, we construct 2cheatable sets that are biimmune for arbitrary time complexity classes. In addition, for each k, we construct a set that is (k + 1)cheatable, but not kcheatable; we show that this separation does not hold with biimmunity. We show that if a recursive set A is biimmune for P then there exists an infinite 1cheatable set that is polynomialtime mreducible to A. Consequently if NP contains a set that is biimmune for P then NP contains a set that is not polynomialtime Turingequivalent to a selfreducible set. 1. Introduction Complexity theory deals with how hard problems are. Time, space, and alternation have served as measures of difficulty. Recently, researchers have Research supported by a Fannie and John Hertz ...
Polynomial time quantum computation with advice
 Inform. Proc. Lett., 90:195–204, 2003. ECCC
"... Abstract. Advice is supplementary information that enhances the computational power of an underlying computation. This paper focuses on advice that is given in the form of a pure quantum state. The notion of advised quantum computation has a direct connection to nonuniform quantum circuits and tall ..."
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Abstract. Advice is supplementary information that enhances the computational power of an underlying computation. This paper focuses on advice that is given in the form of a pure quantum state. The notion of advised quantum computation has a direct connection to nonuniform quantum circuits and tally languages. The paper examines the influence of such advice on the behaviors of an underlying polynomialtime quantum computation with boundederror probability and shows a power and a limitation of advice. Key Words: computational complexity, quantum circuit, advice function 1
On the limitations of locally robust positive reductions
 International Journal of Foundations of Computer Science
, 1991
"... Polynomialtime positive reductions, as introduced by Selman, are by definition globally robust — they are positive with respect to all oracles. This paper studies the extent to which the theory of positive reductions remains intact when their global robustness assumption is removed. We note that tw ..."
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Cited by 9 (1 self)
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Polynomialtime positive reductions, as introduced by Selman, are by definition globally robust — they are positive with respect to all oracles. This paper studies the extent to which the theory of positive reductions remains intact when their global robustness assumption is removed. We note that twosided locally robust positive reductions — reductions that are positive with respect to the oracle to which the reduction is made — are sufficient to retain all crucial properties of globally robust positive reductions. In contrast, we prove absolute and relativized results showing that onesided local robustness fails to preserve fundamental properties of positive reductions, such as the downward closure of NP. Keywords: Structural complexity theory; Polynomialtime reductions; Complexity classes.
On helping and interactive proof systems
 International Journal of Foundations of Computer Science
, 1995
"... We investigate the complexity of honest provers in interactive proof systems. This corresponds precisely to the complexity of oracles helping the computation of robust probabilistic oracle machines. We obtain upper bounds for languages in FewEXP and for sparse sets in NP. Further, interactive protoc ..."
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Cited by 9 (3 self)
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We investigate the complexity of honest provers in interactive proof systems. This corresponds precisely to the complexity of oracles helping the computation of robust probabilistic oracle machines. We obtain upper bounds for languages in FewEXP and for sparse sets in NP. Further, interactive protocols with provers that are reducible to sets of low information content are considered. Specifically, if the verifier communicates only with provers in P=poly, then the accepted language is low for \Sigma p 2. In the case that the provers are polynomialtime reducible to logsparse sets or to sets in strongP/log then the protocol can be simulated by the verifier even without the help of provers. As a consequence we obtain new collapse results under the assumption that intractable sets reduce to sets with low information content. 1 Introduction and overview of results Two extensions of the concept of NP (as the class of languages with efficient proofs of
Characterizations of Logarithmic Advice Complexity Classes
"... The complexity classes P=log and FullP=log, corresponding to the two standard forms of logarithmic advice for polynomial time, are studied. The novel proof technique of "doubly exponential skip" is introduced, and characterizations for these classes are found in terms of several other con ..."
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Cited by 7 (3 self)
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The complexity classes P=log and FullP=log, corresponding to the two standard forms of logarithmic advice for polynomial time, are studied. The novel proof technique of "doubly exponential skip" is introduced, and characterizations for these classes are found in terms of several other concepts, among them easytodescribe boolean circuits and reduction classes of tally sets with high regularity. Similar results hold for many other complexity classes. In this extended abstract most of the proofs are deferred to the appendix, where they are provided for the interested reader but are intended to be read discretionarily. 1. Introduction The study of nonuniform complexity classes stems from the comparison between uniform models of computation, in which a program is valid for arbitrarily long inputs, and nonuniform models in which each program is valid only for inputs of a fixed length. There are many wellknown models for both. Typical examples of uniform models are the Turing machine and...