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A Continuous Approach to Inductive Inference
 Mathematical Programming
, 1992
"... In this paper we describe an interior point mathematical programming approach to inductive inference. We list several versions of this problem and study in detail the formulation based on hidden Boolean logic. We consider the problem of identifying a hidden Boolean function F : f0; 1g n ! f0; 1g ..."
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Cited by 38 (2 self)
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In this paper we describe an interior point mathematical programming approach to inductive inference. We list several versions of this problem and study in detail the formulation based on hidden Boolean logic. We consider the problem of identifying a hidden Boolean function F : f0; 1g n ! f0; 1g using outputs obtained by applying a limited number of random inputs to the hidden function. Given this inputoutput sample, we give a method to synthesize a Boolean function that describes the sample. We pose the Boolean Function Synthesis Problem as a particular type of Satisfiability Problem. The Satisfiability Problem is translated into an integer programming feasibility problem, that is solved with an interior point algorithm for integer programming. A similar integer programming implementation has been used in a previous study to solve randomly generated instances of the Satisfiability Problem. In this paper we introduce a new variant of this algorithm, where the Riemannian metric used...
Con Compression for the Xilinx XC6200 FPGA
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
, 1999
"... ..."
An Interior Point Approach to Boolean Vector Function Synthesis
 In Proceedings of the 36th MSCAS
, 1993
"... The Boolean vector function synthesis problem can be stated as follows: Given a truth table with n input variables and m output variables, synthesize a Boolean vector function that describes the table. In this paper we describe a new formulation of the Boolean vector function synthesis problem as a ..."
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Cited by 13 (1 self)
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The Boolean vector function synthesis problem can be stated as follows: Given a truth table with n input variables and m output variables, synthesize a Boolean vector function that describes the table. In this paper we describe a new formulation of the Boolean vector function synthesis problem as a particular type of Satisfiability Problem. The Satisfiability Problem is translated into an integer programming feasibility problem, that is solved with an interior point algorithm for integer programming. Preliminary computational results are presented. Introduction The Boolean Vector Function Synthesis Problem has applications in logic, artificial intelligence, machine learning, and digital integrated circuit design. In this paper, we describe a Satisfiability Problem formulation of the Boolean Vector Function Synthesis Problem. This formulation can be approached with a wide range of algorithms. In this paper, preliminary computational results are presented using an interior point algorit...
Hardness of Approximate Twolevel Logic Minimization and PAC Learning with Membership Queries
 in Proceedings of the 38th Annual ACM Symposium on the Theory of Computing
, 2006
"... Producing a small DNF expression consistent with given data is a classical problem in computer science that occurs in a number of forms and has numerous applications. We consider two standard variants of this problem. The first one is twolevel logic minimization or finding a minimum DNF formula con ..."
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Cited by 11 (0 self)
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Producing a small DNF expression consistent with given data is a classical problem in computer science that occurs in a number of forms and has numerous applications. We consider two standard variants of this problem. The first one is twolevel logic minimization or finding a minimum DNF formula consistent with a given complete truth table (TTMinDNF). This problem was formulated by Quine in 1952 and has been since one of the key problems in logic design. It was proved NPcomplete by Masek in 1979. The best known polynomial approximation algorithm is based on a reduction to the SETCOVER problem and produces a DNF formula of size O(d · OPT), where d is the number of variables. We prove that TTMinDNF is NPhard to approximate within d γ for some constant γ> 0, establishing the first inapproximability result for the problem. The other DNF minimization problem we consider is PAC learning of DNF expressions when the learning algorithm must output a DNF expression as its hypothesis (referred to as proper learning). We prove that DNF expressions are NPhard to PAC learn properly even when the learner has access to membership queries, thereby answering a longstanding open question due to Valiant [40]. Finally, we provide a concrete connection between these variants of DNF minimization problem. Specifically, we prove that inapproximability of TTMinDNF implies hardness results for restricted proper learning of DNF expressions with membership queries even when learning with respect to the uniform distribution only.
Complexity of twolevel logic minimization
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
"... Abstract—The complexity of twolevel logic minimization is a topic of interest to both computeraided design (CAD) specialists and computer science theoreticians. In the logic synthesis community, twolevel logic minimization forms the foundation for more complex optimization procedures that have si ..."
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Cited by 9 (0 self)
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Abstract—The complexity of twolevel logic minimization is a topic of interest to both computeraided design (CAD) specialists and computer science theoreticians. In the logic synthesis community, twolevel logic minimization forms the foundation for more complex optimization procedures that have significant realworld impact. At the same time, the computational complexity of twolevel logic minimization has posed challenges since the beginning of the field in the 1960s; indeed, some central questions have been resolved only within the last few years, and others remain open. This recent activity has classified some logic optimization problems of high practical relevance, such as finding the minimal sumofproducts (SOP) form and maximal term expansion and reduction. This paper surveys progress in the field with selfcontained expositions of fundamental early results, an account of the recent advances, and some new classifications. It includes an introduction to the relevant concepts and terminology from computational complexity, as well a discussion of the major remaining open problems in the complexity of logic minimization. Index Terms—Computational complexity, logic design, logic minimization, twolevel logic. I.
Fundamental CAD algorithms
 IEEE Trans. on ComputerAided Design of Integrated Circuits and Systems
, 2000
"... Abstract—Computeraided design (CAD) tools are now making it possible to automate many aspects of the design process. This has mainly been made possible by the use of effective and efficient algorithms and corresponding software structures. The very large scale integration (VLSI) design process is e ..."
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Cited by 1 (1 self)
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Abstract—Computeraided design (CAD) tools are now making it possible to automate many aspects of the design process. This has mainly been made possible by the use of effective and efficient algorithms and corresponding software structures. The very large scale integration (VLSI) design process is extremely complex, and even after breaking the entire process into several conceptually easier steps, it has been shown that each step is still computationally hard. To researchers, the goal of understanding the fundamental structure of the problem is often as important as producing a solution of immediate applicability. Despite this emphasis, it turns out that results that might first appear to be only of theoretical value are sometimes of profound relevance to practical problems. VLSI CAD is a dynamic area where problem definitions are continually changing due to complexity, technology and design methodology. In this paper, we focus on several of the fundamental CAD abstractions, models, concepts and algorithms that have had a significant impact on this field. This material should be of great value to researchers interested in entering these areas of research, since it will allow them to quickly focus on much of the key material in our field. We emphasize algorithms in the area of test, physical design, logic synthesis, and formal verification. These algorithms are responsible for the effectiveness and efficiency of a variety of CAD tools. Furthermore, a number of these algorithms have found applications in many other domains. Index Terms—Algorithms, computeraided design, computational complexity, formal verification, logic synthesis, physical design, test. I.
i To Elham ii Acknowledgements
, 2001
"... This thesis has been composed by myself, it has not been accepted in any previous application for a degree, the work of which it is a record has been done by myself and all quotations have been distinguished by quotation marks and the sources of information have specifically acknowledged. ..."
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This thesis has been composed by myself, it has not been accepted in any previous application for a degree, the work of which it is a record has been done by myself and all quotations have been distinguished by quotation marks and the sources of information have specifically acknowledged.
The Complexity of Minimizing Disjunctive Normal Form Formulas
, 1999
"... Contents 1 Introduction 3 2 Preliminaries 6 3 Computing a Minimum DNF 8 4 NP is Enough 11 5 Minimum Term DNF 13 5.2 The hA; Biversion . . . . . . . . . . . . . . . . . . . . . . . . 17 5.3 Masek's Result . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6 Minimum Length DNF 32 6.1 Length ..."
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Contents 1 Introduction 3 2 Preliminaries 6 3 Computing a Minimum DNF 8 4 NP is Enough 11 5 Minimum Term DNF 13 5.2 The hA; Biversion . . . . . . . . . . . . . . . . . . . . . . . . 17 5.3 Masek's Result . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6 Minimum Length DNF 32 6.1 Length vs. Terms . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.2 The Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.3 The hA; Biversion . . . . . . . . . . . . . . . . . . . . . . . . 35 6.4 The full truthtable version . . . . . . . . . . . . . . . . . . . 36 7 Minimum depth DNF 38 7.1 f is a Total Function . . . . . . . . . . . . . . . . . . . . . . . 40 7.2 f is a Partial Function . . . . . . . . . . . . . . . . . . . . . . 42 8 Approximation Hardness 42 8.1 Preserved Solution Values . . . . . . . . . . . . . . . . . . . . 43 8.2 Masek's Reduction . . . . . . . . . . . . . . . . . . . . . . . . 44 8.3 Reductions from X3C . . . . . . . . . . . . . . . . . . . . . . . 4