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176
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 305 (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.
Nondeterministic Space is Closed Under Complementation
, 1988
"... this paper we show that nondeterministic space s(n) is closed under complementation, for s(n) greater than or equal to log n. It immediately follows that the contextsensitive languages are closed under complementation, thus settling a question raised by Kuroda in 1964 [9]. See Hartmanis and Hunt [4 ..."
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Cited by 238 (15 self)
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this paper we show that nondeterministic space s(n) is closed under complementation, for s(n) greater than or equal to log n. It immediately follows that the contextsensitive languages are closed under complementation, thus settling a question raised by Kuroda in 1964 [9]. See Hartmanis and Hunt [4] for a discussion of the history and importance of this problem, and Hopcroft and Ullman [5] for all relevant background material and definitions. The history behind the proof is as follows. In 1981 we showed that the set of firstorder inductive definitions over finite structures is closed under complementation [6]. This holds with or without an ordering relation on the structure. If an ordering is present the resulting class is P. Many people expected that the result was false in the absence of an ordering. In 1983 we studied firstorder logic, with ordering, with a transitive closure operator. We showed that NSPACE[log n] is equal to (FO + pos TC), i.e. firstorder logic with ordering, plus a transitive closure operation, in which the transitive closure operator does not appear within any negation symbols [7]. Now we have returned to the issue of complementation in the light of recent results on the collapse of the log space hierarchies [10, 2, 14]. We have shown that the class (FO + pos TC) is closed under complementation. Our
Boundedwidth polynomialsize branching programs recognize exactly those languages
 in NC’, in “Proceedings, 18th ACM STOC
, 1986
"... We show that any language recognized by an NC ’ circuit (fanin 2, depth O(log n)) can be recognized by a width5 polynomialsize branching program. As any boundedwidth polynomialsize branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such prog ..."
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Cited by 213 (13 self)
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We show that any language recognized by an NC ’ circuit (fanin 2, depth O(log n)) can be recognized by a width5 polynomialsize branching program. As any boundedwidth polynomialsize branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC’. Further, following
Matching is as Easy as Matrix Inversion
, 1987
"... A new algorithm for finding a maximum matching in a general graph is presented; its special feature being that the only computationally nontrivial step required in its execution is the inversion of a single integer matrix. Since this step can be parallelized, we get a simple parallel (RNC2) algorit ..."
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Cited by 167 (5 self)
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A new algorithm for finding a maximum matching in a general graph is presented; its special feature being that the only computationally nontrivial step required in its execution is the inversion of a single integer matrix. Since this step can be parallelized, we get a simple parallel (RNC2) algorithm. At the heart of our algorithm lies a probabilistic lemma, the isolating lemma. We show applications of this lemma to parallel computation and randomized reductions.
On Uniformity within NC¹
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1990
"... In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defined by ..."
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Cited by 127 (19 self)
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In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defined by firstorder formulas [Im87a,Im87b] and a uniformity corresponding to Buss' deterministic logtime reductions [Bu87]. We show that these two notions are equivalent, leading to a natural notion of uniformity for lowlevel circuit complexity classes. We show that recent results on the structure of NC¹ [Ba89] still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of subclasses of the regular languages based on their logical expressibility, extending recent work of Straubing, Th'erien, and Thomas [STT88]. A preliminary version of this work appeared as [BIS88].
Identifying the minimal transversals of a hypergraph and related problems
 SIAM Journal on Computing
, 1995
"... The paper considers two decision problems on hypergraphs, hypergraph saturation and recognition of the transversal hypergraph, and discusses their significance for several search problems in applied computer science. Hypergraph saturation, i.e., given a hypergraph H, decide if every subset of vertic ..."
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Cited by 125 (7 self)
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The paper considers two decision problems on hypergraphs, hypergraph saturation and recognition of the transversal hypergraph, and discusses their significance for several search problems in applied computer science. Hypergraph saturation, i.e., given a hypergraph H, decide if every subset of vertices is contained in or contains some edge of H, is shown to be coNPcomplete. A certain subproblem of hypergraph saturation, the saturation of simple hypergraphs, is shown to be computationally equivalent to transversal hypergraph recognition, i.e., given two hypergraphs H 1; H 2, decide if the sets in H 2 are all the minimal transversals of H 1. The complexity of the search problem related to the recognition of the transversal hypergraph, the computation of the transversal hypergraph, is an open problem. This task needs time exponential in the input size, but it is unknown whether an outputpolynomial algorithm exists for this problem. For several important subcases, for instance if an upper or lower bound is imposed on the edge size or for acyclic hypergraphs, we present outputpolynomial algorithms. Computing or recognizing the minimal transversals of a hypergraph is a frequent problem in practice, which is pointed out by identifying important applications in database theory, Boolean switching theory, logic, and AI, particularly in modelbased diagnosis.
On The Rapid Computation of Various Polylogarithmic Constants”, manuscript
, 1996
"... We give algorithms for the computation of the dth digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the d ..."
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Cited by 101 (30 self)
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We give algorithms for the computation of the dth digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of log (2) or on a modest work station in a few hours run time. We demonstrate this technique by computing the ten billionth hexadecimal digit of, the billionth hexadecimal digits of 2 2 log(2) and log (2), and the ten billionth decimal digit of log(9=10). These calculations rest on the observation that very special types of identities exist for certain numbers like, 2,log(2) and log 2 (2). These are essentially polylogarithmic ladders in an integer base. A number of these identities that we deriveinthiswork appear to be new, for example the critical identity for:
The Power of Reconfiguration
, 1998
"... This paper concerns the computational aspects of the reconfigurable network model. The computational power of the model is investigated under several network topologies and assuming several variants of the model. In particular, it is shown that there are reconfigurable machines based on simple netwo ..."
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Cited by 85 (7 self)
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This paper concerns the computational aspects of the reconfigurable network model. The computational power of the model is investigated under several network topologies and assuming several variants of the model. In particular, it is shown that there are reconfigurable machines based on simple network topologies, that are capable of solving large classes of problems in constant time. These classes depend on the kinds of switches assumed for the network nodes. Reconfigurable networks are also compared with various other models of parallel computation, like PRAM's and Branching Programs. Part of this work is to be presented at the 18th International Colloquium on Automata, Languages, and Programming (ICALP), July 1991, Madrid. y Department of Computer Science, The Hebrew University, Jerusalem 91904, Israel. Email: yosi@humus.huji.ac.il, Supported by Eshcol Fellowship. z Department of Applied Mathematics and Computer Science, The Weizmann Institute, Rehovot 76100, Israel. Email: p...
The complexity of acyclic conjunctive queries
 Journal of the ACM
, 1998
"... This paper deals with the evaluation of acyclic Boolean conjunctive queries in relational databases. By wellknown results of Yannakakis [1981], this problem is solvable in polynomial time; its precise complexity, however, has not been pinpointed so far. We show that the problem of evaluating acyc ..."
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Cited by 74 (14 self)
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This paper deals with the evaluation of acyclic Boolean conjunctive queries in relational databases. By wellknown results of Yannakakis [1981], this problem is solvable in polynomial time; its precise complexity, however, has not been pinpointed so far. We show that the problem of evaluating acyclic Boolean conjunctive queries is complete for LOGCFL, the class of decision problems that are logspacereducible to a contextfree language. Since LOGCFL is contained in AC 1 and NC 2, the evaluation problem of acyclic Boolean conjunctive queries is highly parallelizable. We present a parallel database algorithm solving this problem with a logarithmic number of parallel join operations. The algorithm is generalized to computing the output of relevant classes of nonBoolean queries. We also show that the acyclic versions of the following wellknown database and AI problems are all LOGCFLcomplete: The Query Output Tuple problem for conjunctive queries, Conjunctive Query Containment, Clause Subsumption, and Constraint Satisfaction. The LOGCFLcompleteness result is extended to the class of queries of bounded treewidth and to other relevant query classes which are more general than the acyclic queries.
Fast parallel circuits for the quantum Fourier transform
 PROCEEDINGS 41ST ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS’00)
, 2000
"... We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our ..."
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Cited by 51 (2 self)
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We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our circuits have depth O(log n). The best previous depth bound was O(n), even for approximations with constant error. Moreover, our circuits have size O(n log(n/ε)). We also give an upper bound of O(n(log n) 2 log log n) on the circuit size of the exact QFT modulo 2 n, for which the best previous bound was O(n 2). As an application of the above depth bound, we show that Shor’s factoring algorithm may be based on quantum circuits with depth only O(log n) and polynomialsize, in combination with classical polynomialtime pre and postprocessing. In the language of computational complexity, this implies that factoring is in the complexity class ZPP BQNC, where BQNC is the class of problems computable with boundederror probability by quantum circuits with polylogarithmic depth and polynomial size. Finally, we prove an Ω(log n) lower bound on the depth complexity of approximations of the