Topic

# Product term

About: Product term is a research topic. Over the lifetime, 1249 publications have been published within this topic receiving 23062 citations.

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TL;DR: A systematic procedure is presented for writing a Boolean function as a minimum sum of products and specific attention is given to terms which can be included in the function solely for the designer's convenience.

Abstract: A systematic procedure is presented for writing a Boolean function as a minimum sum of products This procedure is a simplification and extension of the method presented by W V Quine Specific attention is given to terms which can be included in the function solely for the designer's convenience

1,103 citations

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TL;DR: A simple constructive algorithm for the evaluation of formulas having two literals per clause, which runs in linear time on a random access machine.

Abstract: Let F = Qlxr Qzxz l ** Qnx, C be a quantified Boolean formula with no free variables, where each Qi is either 3 or t, and C is in conjunctive normal form. That is, C is a conjunction of clauses, each clause is a disjunction of literals, and each literal is either a variable, xi, or the negation of a variable, Zi (1 < i f n). We shall use Ui to denote a literal equal to either Xi or Fi. The evaluation problem for quantified Boolean formulas is to determine whether such a formula F is true. The evaluation problem is complete in polynomial space [6], even if C is restricted to contain at most three literals per clause. The satisfiability problem, the special case in which all quantifiers are existential, is NP-complete [ 1,2,4] for formulas with three literals per clause. However, the satisfiability problem for formulas with only two literals per clause is solvable in polynomial time [ 1,2,4] ; Even, Itai, and Shamir [3] outline a linear-time algorithm. Schaefer [5] claims a polynomial time bound for the evaluation problem with two literals per clause, although he gives no proof. In this note we present a simple constructive algorithm for the evaluation of formulas having two literals per clause, which runs in linear time on a random access machine.

970 citations

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08 Apr 2011TL;DR: A new matrix product, called semi-tensor product of matrices, is used, which can covert the Boolean networks into discrete-time linear dynamic systems and the controllability of Boolean control networks is considered in the paper as an application.

Abstract: A Boolean network is a logical dynamic system, which has been used to describe cellular networks. Using a new matrix product, called semi-tensor product of matrices, a logical function can be expressed as an algebraic function. This expression can covert the Boolean networks into discrete-time linear dynamic systems. Similarly, the Boolean control networks can also be converted into discrete time bilinear dynamic systems. Under these forms the standard matrix analysis can be used to consider the structure and the control problems of Boolean (control) networks. After the detailed description of this new approach, the controllability of Boolean control networks is considered in the paper as an application.

834 citations

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TL;DR: This paper introduces a new representation for Boolean functions, called decision lists, and shows that they are efficiently learnable from examples, and strictly increases the set of functions known to be polynomially learnable, in the sense of Valiant (1984).

Abstract: This paper introduces a new representation for Boolean functions, called decision lists, and shows that they are efficiently learnable from examples. More precisely, this result is established for k-;DL – the set of decision lists with conjunctive clauses of size k at each decision. Since k-DL properly includes other well-known techniques for representing Boolean functions such as k-CNF (formulae in conjunctive normal form with at most k literals per clause), k-DNF (formulae in disjunctive normal form with at most k literals per term), and decision trees of depth k, our result strictly increases the set of functions that are known to be polynomially learnable, in the sense of Valiant (1984). Our proof is constructive: we present an algorithm that can efficiently construct an element of k-DL consistent with a given set of examples, if one exists.

833 citations

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TL;DR: The author describes the Boolean satisfiability method for generating test patterns for single stuck-at faults in combinational circuits, which allows for the addition of heuristics used by structural search methods, and has produced excellent results on popular test pattern generation benchmarks.

Abstract: The author describes the Boolean satisfiability method for generating test patterns for single stuck-at faults in combinational circuits. This new method generates test patterns in two steps: first, it constructs a formula expressing the Boolean difference between the unfaulted and faulted circuits, and second, it applies a Boolean satisfiability algorithm to the resulting formula. This approach differs from previous methods now in use, which search the circuit structure directly instead of constructing a formula from it. The new method is general and effective. It allows for the addition of heuristics used by structural search methods, and it has produced excellent results on popular test pattern generation benchmarks. >

704 citations