This paper presents the syntax and semantics of a small language for describing and using abstract concepts in formal software development and hardware design. The language provides definition, abbreviation, extension, and lemma constructs, which have general mathematical descriptive power, plus a computation-specific realization construct. The semantics, which is denotational, includes specification of the requirements ("legality conditions ") that must be met when using each construct. The syntax and semantics are such that a corresponding proof theory requires only first order and inductive proof methods, rather than general higher order techniques as required in some frameworks. The language and some of the main proof issues are illustrated with an extended example of a behavioral and structural description of a carry-lookahead adder circuit, with the circuit realization given in terms of a generic parallel-prefix circuit. Partially supported by NSF Grant Number CCR-8906678. A pr...

Introduction Interest in the theory and applications of rewriting has been growing rapidly, as evidenced in part by four conference proceedings #including this one# #15, 26, 41,66#; three workshop proceedings #33, 47, 77#; #ve special journal issues #5,88, 24, 40, 67#; more than ten surveys #2,7,27, 28, 44, 56,57,76, 82, 81#; one edited collection of papers #1#; four monographs #3, 12,55,65#; and seven books #four of them still in progress# #8,9, 35, 54, 60,75, 84#. To encourage and stimulate continued progress in this area, wehave collected #with the help of colleagues# a number of problems that appear to us to be of interest and regarding whichwe do not know the answer. Questions on rewriting and other equational paradigms have been included; manyhave not aged su#ciently to be accorded the appellation #open problem". Wehave limited ourselves to theoretical questions, though there are certainly many additional interesting questions relating to applications and implementation

We present an extended completion procedure with builtin theories defined by a collection of associativity and commutativity axioms and additional ground equations, and reformulate Buchberger's algorithm for constructing Gröbner bases for polynomial ideals in this formalism. The presentation of completion is at an abstract level, by transition rules, with a suitable notion of fairness used to characterize a wide class of correct completion procedures, among them Buchberger's original algorithm for polynomial rings over a field.

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www.cs.unm.edu/~kapur Abstract. A method for generating polynomial invariants of imperative programs is presented using the abstract interpretation framework. It is shown that for programs with polynomial assignments, an invariant consisting of a conjunction of polynomial equalities can be automatically generated for each program point. The proposed approach takes into account tests in conditional statements as well as in loops, insofar as they can be abstracted to be polynomial equalities and disequalities. The semantics of each statement is given as a transformation on polynomial ideals. Merging of paths in a program is defined as the intersection of the polynomial ideals associated with each path. For a loop junction, a widening operator based on selecting polynomials up to a certain degree is proposed. The algorithm for finding invariants using this widening operator is shown to terminate in finitely many steps. The proposed approach has been implemented and successfully tried on many programs. A table providing details about the programs is given. 1

. Boolean ring is an algebraic structure which uses exclusive \Gamma or instead of the usual or. It yields a unique normal form for every Boolean function. In this paper we present several fundamental properties concerning Boolean rings. We present a simple method for deriving the Boolean ring normal form directly from a truth table. We also describe a notion of normal form of a Boolean function with a don't-care condition, and show an algorithm for generating such a normal form. We then discuss two Boolean ring based theorem proving methods for propositional logic. Finally we give some arguments on why the Boolean ring representation had not been used more extensively, and how it can be used in computing. 1. Introduction Boolean ring is an algebraic structure which is equivalent to Boolean algebra. The major representational differences are that Boolean ring uses exclusive-or (+) instead of or () to represent Boolean functions, and that there is no need for negation in Boolean ring....

Introduction The Boolean ring or first-order polynomial based theorem proving began with the work of Hsiang (1982, 1985). Hsiang extended the idea of using Boolean polynomials to represent propositional formulae to the case of first-order predicate calculus. Based on the completion procedure of Knuth and Bendix (1970), the N-strategy was proposed. Later on, by imitating the framework of Buchberger 's algorithm to compute the Grobner bases of polynomial ideals (Buchberger 1985), Kapur and Narendran (1985) developed another approach which is also referred to as the Grobner basis method. One obvious advantage of using Boolean polynomials is that every propositional formula has a unique representation, and sometimes it is easy to be generalized to some non-classical logic systems (Chazarain et al. 1991; Wu and Tan 1994). Stimulated by them, some approaches and results have been reported (Bachmair and Dershowitz 1987; Dietrich 1986; Kapur and Zhang 1989; Wu and Liu 1998; Zhang 198

. In this article techniques borrowed from Computer Algebra (Grobner Bases) are applied to deal with Medical Appropriateness Criteria including uncertainty. The knowledge was provided in the format of a table. A previous translation of the table into the format of a "Rule Based System" (denoted RBS) based on a three-valued logic is required beforehand to apply these techniques. Once the RBS has been obtained, we apply a Computer Algebra based inference engine, both to detect anomalies and to infer new knowledge. A specific set of criteria for coronary artery surgery (originally presented in the form of a table) is analyzed in detail. Keywords. Verification. Inference Engines. RBSs in Medicine. Grobner Bases. Top i cs : Integration of Logical Reasoning and Computer Algebra. Symbolic Computation for Expert Systems and Machine Learning. 1 Introduction "Appropriateness criteria" are ratings of the appropriateness for a given diagnostic or therapeutic procedure. Whereas other policies such...

Abstract Recently algorithms for solving propositional satisfiability problem, or SAT, have aroused great interest, and more attention is paid to transformation problem solving. The commonly used transformation is representation transform, but since its intermediate computing procedure is a black box from the viewpoint of the original problem, this approach has many limitations. In this paper, a new approach called algorithm transform is proposed and applied to solving SAT by Wu’s method, a general algorithm for solving polynomial equations. By establishing the correspondence between the primitive operation in Wu’s method and clause resolution in SAT, it is shown that Wu’s method, when used for solving SAT, is primarily a restricted clause resolution procedure. While Wu’s method introduces entirely new concepts, e.g., characteristic set of clauses, to resolution procedure, the complexity result of resolution procedure suggests an exponential lower bound to Wu’s method for solving general polynomial equations. Moreover, this algorithm transform can help achieve a more efficient implementation of Wu’s method since it can avoid the complex manipulation of polynomials and can make the best use of domain specific knowledge. Keywords algorithm design, satisfiability problem, Wu’s method, automated reasoning 1

Abstract. The problem of deciding the probability model equivalence of two PRISM programs is addressed. In the finite case this problem can be solved (albeit slowly) using techniques from algebraic statistics, specifically the computation of elimination ideals and Gröbner bases. A very brief introduction to algebraic statistics is given. Consideration is given to cases where shortcuts to proving/disproving model equivalence are available.