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141
Satisfiability Coding Lemma
 In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, FOCS’97
, 1997
"... We present and analyze two simple algorithms for finding satisfying assignments of kCNFs (Boolean formulae in conjunctive normal form with at most k literals per clause). The first is a randomized algorithm which, with probability approaching 1, finds a satisfying assignment of a satisfiable kCNF ..."
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Cited by 64 (6 self)
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We present and analyze two simple algorithms for finding satisfying assignments of kCNFs (Boolean formulae in conjunctive normal form with at most k literals per clause). The first is a randomized algorithm which, with probability approaching 1, finds a satisfying assignment of a satisfiable kCNF formula F in time O(n 2 jF j2 n\Gamman=k ). The second algorithm is deterministic, and its running time approaches 2 n\Gamman=2k for large n and k. The randomized algorithm is the best known algorithm for k ? 3; the deterministic algorithm is the best known deterministic algorithm for k ? 4. We also show an \Omega\Gamma n 1=4 2 p n ) lower bound on the size of depth 3 circuits of AND and OR gates computing the parity function. This bound is tight up to a constant factor. The key idea used in these upper and lower bounds is what we call the Satisfiability Coding Lemma. This basic lemma shows how to encode satisfying solutions of a kCNF succinctly. 1 Introduction The problem of ...
Lower Bounds for Deterministic and Nondeterministic Branching Programs
 in Proceedings of the FCT'91, Lecture Notes in Computer Science
, 1991
"... We survey lower bounds established for the complexity of computing explicitly given Boolean functions by switchingandrectifier networks, branching programs and switching networks. We first consider the unrestricted case and then proceed to various restricted models. Among these are monotone networ ..."
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Cited by 57 (4 self)
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We survey lower bounds established for the complexity of computing explicitly given Boolean functions by switchingandrectifier networks, branching programs and switching networks. We first consider the unrestricted case and then proceed to various restricted models. Among these are monotone networks, boundedwidth devices , oblivious devices and readk times only devices. 1 Introduction The main goal of the Boolean complexity theory is to prove lower bounds on the complexity of computing "explicitly given" Boolean functions in interesting computational models. By "explicitly given" researchers usually mean "belonging to the class NP ". This is a very plausible interpretation since on the one hand this class contains the overwhelming majority of interesting Boolean functions and on the other hand it is small enough to prevent us from the necessity to take into account counting arguments. To illustrate the second point, let me remind the reader that already the class \Delta p 2 ,...
Models of Computation  Exploring the Power of Computing
"... Theoretical computer science treats any computational subject for which a good model can be created. Research on formal models of computation was initiated in the 1930s and 1940s by Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages, language translators, and oper ..."
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Cited by 57 (7 self)
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Theoretical computer science treats any computational subject for which a good model can be created. Research on formal models of computation was initiated in the 1930s and 1940s by Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages, language translators, and operating systems were under development and therefore became both the subject and basis for a great deal of theoretical work. The power of computers of this period was limited by slow processors and small amounts of memory, and thus theories (models, algorithms, and analysis) were developed to explore the efficient use of computers as well as the inherent complexity of problems. The former subject is known today as algorithms and data structures, the latter computational complexity. The focus of theoretical computer scientists in the 1960s on languages is reflected in the first textbook on the subject, Formal Languages and Their Relation to Automata by John Hopcroft and Jeffrey Ullman. This influential book led to the creation of many languagecentered theoretical computer science courses; many introductory theory courses today continue to reflect the content of this book and the interests of theoreticians of the 1960s and early 1970s. Although
Efficient Inference of Object Types
, 1995
"... Abadi and Cardelli have recently investigated a calculus of objects [2]. The calculus supports a key feature of objectoriented languages: an object can be emulated by another object that has more refined methods. Abadi and Cardelli presented four firstorder type systems for the calculus. The simpl ..."
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Cited by 55 (6 self)
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Abadi and Cardelli have recently investigated a calculus of objects [2]. The calculus supports a key feature of objectoriented languages: an object can be emulated by another object that has more refined methods. Abadi and Cardelli presented four firstorder type systems for the calculus. The simplest one is based on finite types and no subtyping, and the most powerful one has both recursive types and subtyping. Open until now is the question of type inference, and in the presence of subtyping "the absence of minimum typings poses practical problems for type inference" [2]. In this paper...
Unprovability of Lower Bounds on the Circuit Size in Certain Fragments of Bounded Arithmetic
 in Izvestiya of the Russian Academy of Science, mathematics
, 1995
"... To appear in Izvestiya of the RAN We show that if strong pseudorandom generators exist then the statement “α encodes a circuit of size n (log ∗ n) for SATISFIABILITY ” is not refutable in S2 2 (α). For refutation in S1 2 (α), this is proven under the weaker assumption of the existence of generators ..."
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Cited by 54 (6 self)
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To appear in Izvestiya of the RAN We show that if strong pseudorandom generators exist then the statement “α encodes a circuit of size n (log ∗ n) for SATISFIABILITY ” is not refutable in S2 2 (α). For refutation in S1 2 (α), this is proven under the weaker assumption of the existence of generators secure against the attack by small depth circuits, and for another system which is strong enough to prove exponential lower bounds for constantdepth circuits, this is shown without using any unproven hardness assumptions. These results can be also viewed as direct corollaries of interpolationlike theorems for certain “split versions ” of classical systems of Bounded Arithmetic introduced in this paper.
PartitionBased Logical Reasoning for FirstOrder and Propositional Theories
 Artificial Intelligence
, 2000
"... In this paper we provide algorithms for reasoning with partitions of related logical axioms in propositional and firstorder logic (FOL). We also provide a greedy algorithm that automatically decomposes a set of logical axioms into partitions. Our motivation is twofold. First, we are concerned with ..."
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Cited by 52 (8 self)
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In this paper we provide algorithms for reasoning with partitions of related logical axioms in propositional and firstorder logic (FOL). We also provide a greedy algorithm that automatically decomposes a set of logical axioms into partitions. Our motivation is twofold. First, we are concerned with how to reason e#ectively with multiple knowledge bases that have overlap in content. Second, we are concerned with improving the e#ciency of reasoning over a set of logical axioms by partitioning the set with respect to some detectable structure, and reasoning over individual partitions. Many of the reasoning procedures we present are based on the idea of passing messages between partitions. We present algorithms for reasoning using forward messagepassing and using backward messagepassing with partitions of logical axioms. Associated with each partition is a reasoning procedure. We characterize a class of reasoning procedures that ensures completeness and soundness of our messagepassing ...
The History and Status of the P versus NP Question
, 1992
"... this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the re ..."
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Cited by 50 (0 self)
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this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the realization that certain problems are algorithmically unsolvable. At around this time, forerunners of the programmable computing machine were beginning to appear. As mathematicians contemplated the practical capabilities and limitations of such devices, computational complexity theory emerged from the theory of algorithmic unsolvability. Early on, a particular type of computational task became evident, where one is seeking an object which lies
Forming Concepts for Fast Inference
 In Proceedings of the Tenth National Conference on Artificial Intelligence (AAAI92
, 1992
"... Knowledge compilation speeds inference by creating tractable approximations of a knowledge base, but this advantage is lost if the approximations are too large. We show how learning concept generalizations can allow for a more compact representation of the tractable theory. We also give a general in ..."
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Cited by 49 (2 self)
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Knowledge compilation speeds inference by creating tractable approximations of a knowledge base, but this advantage is lost if the approximations are too large. We show how learning concept generalizations can allow for a more compact representation of the tractable theory. We also give a general induction rule for generating such concept generalizations. Finally, we prove that unless NP ` nonuniform P, not all theories have small Horn least upperbound approximations. 1 Introduction Work in machine learning has traditionally been divided into two main camps: concept learning (e.g. [ Kearns, 1990 ] ) and speedup learning (e.g. [ Minton, 1988 ] ). The work reported in this paper bridges these two areas by showing how concept learning can be used to speed up inference by allowing a more compact and efficient representation of a knowledge base. We have been studying techniques for boosting the performance of knowledge representation systems by compiling expressive but intractable repre...
An O(n^(log log n)) Learning Algorithm for DNF under the Uniform Distribution
 Journal of Computer and System Sciences
, 1998
"... We show that a DNF with terms of size at most d can be approximated by a function with at most d O(d log 1=") non zero Fourier coefficients such that the expected error squared, with respect to the uniform distribution, is at most ". This property is used to derive a learning algorithm for DNF, unde ..."
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Cited by 48 (1 self)
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We show that a DNF with terms of size at most d can be approximated by a function with at most d O(d log 1=") non zero Fourier coefficients such that the expected error squared, with respect to the uniform distribution, is at most ". This property is used to derive a learning algorithm for DNF, under the uniform distribution. The learning algorithm uses queries and learns, with respect to the uniform distribution, a DNF with terms of size at most d in time polynomial in n and d O(d log 1=") . The interesting implications are for the case when " is constant. In this case our algorithm learns a DNF with a polynomial number of terms in time n O(log log n) , and a DNF with terms of size at most O(log n= log log n) in polynomial time.
Boolean Expression Diagrams
, 1997
"... This paper presents a new data structure called Boolean Expression Diagrams (BEDs) for representing and manipulating Boolean functions. BEDs are a generalization of Binary Decision Diagrams (BDDs) which can represent any Boolean circuit in linear space and still maintain many of the desirable proper ..."
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Cited by 46 (5 self)
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This paper presents a new data structure called Boolean Expression Diagrams (BEDs) for representing and manipulating Boolean functions. BEDs are a generalization of Binary Decision Diagrams (BDDs) which can represent any Boolean circuit in linear space and still maintain many of the desirable properties of BDDs. Two algorithms are described for transforming a BED into a reduced ordered BDD. One is a generalized version of the BDD applyoperator while the other can exploit the structural information of the Boolean expression. This ability is demonstrated by verifying that two di erent circuit implementations of a 16bit multiplier implement the same Boolean function. Using BEDs, this veri cation problem is solved in less than a second, while using standard BDD techniques this problem is infeasible. Generally, BEDs are useful in applications, for example tautology checking, where the endresult as a reduced ordered BDD is small.