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Inductionless Induction
, 1994
"... Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 A few words explaining the title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Some examples of the problem we are considering . . . . . . . . . . . . . . . . . . . ..."
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Cited by 21 (0 self)
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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 A few words explaining the title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Some examples of the problem we are considering . . . . . . . . . . . . . . . . . . . 3 1.3 Outline of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Formal background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Terms and clauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Equational deduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Inductive theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Constructors and sufficient completeness . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Term Rewriting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.6 Standar
A reduction approach to decision procedures
, 2005
"... Abstract. We present an approach for designing decision procedures based on the reduction of complex theories to simpler ones. Specifically, we define reduction functions as a tool for reducing the satisfiability problem of a complex theory to the satisfiability problem of a simpler one. Reduction f ..."
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Abstract. We present an approach for designing decision procedures based on the reduction of complex theories to simpler ones. Specifically, we define reduction functions as a tool for reducing the satisfiability problem of a complex theory to the satisfiability problem of a simpler one. Reduction functions allow us to reduce the theory of lists to the theory of constructors, the theory of arrays to the theory of equality, the theory of sets to the theory of equality, and the theory of multisets to the theory of integers. Finally, we provide a method for combining reduction functions. This method allows us to reduce the satisfiability problem of a combination of complex theories to the combination of simpler ones. 1
Rewriting systems for Coxeter groups
 J. Pure Appl. Algebra
, 1994
"... A finite complete rewriting system for a group is a finite presentation which gives a solution to the word problem and a regular language of normal forms for the group. In this paper it is shown that the fundamental group of an orientable closed surface of genus g has a finite complete rewriting sys ..."
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Cited by 7 (2 self)
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A finite complete rewriting system for a group is a finite presentation which gives a solution to the word problem and a regular language of normal forms for the group. In this paper it is shown that the fundamental group of an orientable closed surface of genus g has a finite complete rewriting system, using the usual generators a1,.., ag, b1,.., bg along with generators representing their inverses. Constructions of finite complete rewriting systems are also given for any Coxeter group G satisfying one of the following hypotheses. 1) G has three or fewer generators. 2) G does not contain a special subgroup of the form
Dependency Pairs for Rewriting with NonFree Constructors
"... Abstract. A method based on dependency pairs for showing termination of functional programs on data structures generated by constructors with relations is proposed. A functional program is specified as an equational rewrite system, where the rewrite system specifies the program and the equations exp ..."
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Abstract. A method based on dependency pairs for showing termination of functional programs on data structures generated by constructors with relations is proposed. A functional program is specified as an equational rewrite system, where the rewrite system specifies the program and the equations express the relations on the constructors that generate the data structures. Unlike previous approaches, relations on constructors can be collapsing, including idempotency and identity relations. Relations among constructors may be partitioned into two parts: (i) equations that cannot be oriented into terminating rewrite rules, and (ii) equations that can be oriented as terminating rewrite rules, in which case an equivalent convergent system for them is generated. The dependency pair method is extended to normalized rewriting, where constructorterms in the redex are normalized first. The method has been applied to several examples, including the Calculus of Communicating Systems and the Propositional Sequent Calculus. Various refinements, such as dependency graphs, narrowing, etc., which increase the power of the dependency pair method, are presented for normalized rewriting. 1
Noncommutative Gröbner Bases For The Commutator Ideal
, 2000
"... In this article we consider the free associative algebra Kha 1 ; :::; am i on finitely many generators over an arbitrary eld K. We show that the only finite Gröbner bases for the commutator ideal in this algebra are the bases induced by the shortlex ordering, and hence there are exactly m! dist ..."
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In this article we consider the free associative algebra Kha 1 ; :::; am i on finitely many generators over an arbitrary eld K. We show that the only finite Gröbner bases for the commutator ideal in this algebra are the bases induced by the shortlex ordering, and hence there are exactly m! distinct finite reduced Gröbner bases for this ideal. We also give a complete description of the universal Gröbner basis for the commutator ideal in Kha; b; ci. As a part of this construction, we show that there are uncountably many distinct reduced Gröbner bases for the commutator ideal with respect to total division orderings (this has also been independently shown by U. Martin [8], and is related to the work of Hermiller, Kramer and Laubenbacher [5]). This article also includes a combinatorial description of the subword closed normal forms for the free commutative monoid on three generators, leading toward a possible classification of all of the infinite Gröbner bases for the co...
COGNITIVE 2013: The Fifth International Conference on Advanced Cognitive Technologies and Applications Cartesian Intuitionism for Program Synthesis
"... Abstract—This paper presents two possible approaches to scientific discovery, the Newtonian and the Cartesian one. The paper explains the main difference between these two styles and underlines the importance of each of them. Its specific scientific contribution is presenting the less known Cartesia ..."
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Abstract—This paper presents two possible approaches to scientific discovery, the Newtonian and the Cartesian one. The paper explains the main difference between these two styles and underlines the importance of each of them. Its specific scientific contribution is presenting the less known Cartesian approach and the main problems that can be solved by it, in the light of our research in Automated Program Synthesis. The paper is thus related to the creative framework of modeling human reasoning mechanisms, cognitive and computational models, as well as modeling brain information processing mechanisms. Keywordscreativity; Program Synthesis; Newtonian style; Cartesian style; Constructive Matching Methodology