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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 307 (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...
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 10 (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
Unambiguous Computations and Locally Definable Acceptance Types
 Theoretical Computer Science
, 1998
"... Hertrampf's locally definable acceptance types show that many complexity classes can be defined in terms of polynomial time bounded NTM's with simple local conditions on the nodes of its computation tree, rather than global concepts like number of accepting paths etc. We introduce a modification of ..."
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Cited by 9 (0 self)
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Hertrampf's locally definable acceptance types show that many complexity classes can be defined in terms of polynomial time bounded NTM's with simple local conditions on the nodes of its computation tree, rather than global concepts like number of accepting paths etc. We introduce a modification of Hertrampf's locally definable acceptance types which allows to get a larger number of characterizable complexity classes. Among others the newly characterizable classes are UP and ModZ k P . It is shown how different types of oracle access, e.g., guarded access, can be characterized by this model. This sheds new light on the discussion on how to access unambiguous computation. We present simple functions that describe precisely objects of current research as the unambiguous oracle, alternation, and promise hierarchies. We exhibit the new class UAP which seems to be an unambiguous analogue of Wagner's rP . UAP (and thus rP) contains Few and is currently the smallest class known with this pr...
On Quasilinear Time Complexity Theory
, 1994
"... This paper furthers the study of quasilinear time complexity initiated by Schnorr and Gurevich and Shelah. We show that the fundamental properties of the polynomialtime hierarchy carry over to the quasilineartime hierarchy. ..."
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Cited by 3 (0 self)
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This paper furthers the study of quasilinear time complexity initiated by Schnorr and Gurevich and Shelah. We show that the fundamental properties of the polynomialtime hierarchy carry over to the quasilineartime hierarchy.
Persistent Computations
, 1998
"... We study computational effects of persistent Turing machines, independently introduced by Goldin and Wegner [GW98], and Kosub [Kos98]. Persistence is a mode of interaction which makes it possible to consider the computational behavior of a Turing machine as an infinite sequence of autonomous computa ..."
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Cited by 1 (0 self)
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We study computational effects of persistent Turing machines, independently introduced by Goldin and Wegner [GW98], and Kosub [Kos98]. Persistence is a mode of interaction which makes it possible to consider the computational behavior of a Turing machine as an infinite sequence of autonomous computations. We investigate different computability concepts such as conditional and essential computability. Furthermore, we give a characterization of all nonimmune sets in terms of persistent computations. 1 Introduction Interaction extends the algorithmic view on computation. By the facility of computational devices to communicate with the world outside, in particular to receive informations from there, moments of noncontrolability and of nonpredictability come inherent in computations. The more powerful interaction can be, the less dominating is the algorithm for the behavior the device is actually showing. Not least for that reason, a paradigm shift from algorithmic to interactive view o...
On Helping by Paritylike Languages
 Information Processing Letters
, 1994
"... Introduction A deterministic oracle Turing machine M is robust [Sch85] if for every oracle X, M X and M ; accept the same language. A language H is said to Phelp [Sch85] a robust oracle machine M if there exists a polynomial p such that for every x, M H on input x runs for at most p(jxj) ste ..."
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Cited by 1 (0 self)
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Introduction A deterministic oracle Turing machine M is robust [Sch85] if for every oracle X, M X and M ; accept the same language. A language H is said to Phelp [Sch85] a robust oracle machine M if there exists a polynomial p such that for every x, M H on input x runs for at most p(jxj) steps. Such H is called a Phelper for M . For a language H, let P help (H) denote the class of languages recognized by robust machines for which H behaves as a Phelper and, for a language class C, let P help (C) denote S H2CP help<F
Querymonotonic Turing Reductions
"... ... A for which any set that Turing reduces to A will also reduceto A via both queryincreasing and querydecreasing Turing reductions. In particular, this holds for all tight paddable sets, where a set is said to be tight paddable exactly if it is paddable via a function whose output length is bou ..."
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Cited by 1 (1 self)
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... A for which any set that Turing reduces to A will also reduceto A via both queryincreasing and querydecreasing Turing reductions. In particular, this holds for all tight paddable sets, where a set is said to be tight paddable exactly if it is paddable via a function whose output length is bounded tightlyboth from above and from below in the length of the input. We prove that many natural NPcomplete problems such as satisfiability, clique, and vertex cover aretight paddable.
Constructive Complexity
, 1990
"... Powerful and widely applicable, yet inherently nonconstructive, tools have recently become available for classifying decision problems as solvable in polynomial time, as a result of the work of Robertson and Seymour. These developments challenge the established view that equates tractability with p ..."
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Powerful and widely applicable, yet inherently nonconstructive, tools have recently become available for classifying decision problems as solvable in polynomial time, as a result of the work of Robertson and Seymour. These developments challenge the established view that equates tractability with polynomialtime solvability, since the existence of an inaccessible algorithm is of very little help in solving a problem. In this paper, we attempt to provide the foundations for a constructive theory of complexity, in which membership of a problem in some complexity class indeed implies that we can find out how to solve that problem within the stated bounds. Our approach is based on relations, rather than on sets; we make much use of selfreducibility and oracle machines, both conventional and "blind," to derive a series of results which establish a structure similar to that of classical complexity theory, but in which we are in fact able to prove results which remain conjectural within the ...