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A completeness theorem for strong normalization in minimal deduction modulo
, 2009
"... Abstract. Deduction modulo is an extension of firstorder predicate logic where axioms are replaced by rewrite rules and where many theories, such as arithmetic, simple type theory and some variants of set theory, can be expressed. An important question in deduction modulo is to find a condition of ..."
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Cited by 6 (2 self)
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Abstract. Deduction modulo is an extension of firstorder predicate logic where axioms are replaced by rewrite rules and where many theories, such as arithmetic, simple type theory and some variants of set theory, can be expressed. An important question in deduction modulo is to find a condition of the theories that have the strong normalization property. Dowek and Werner have given a semantic sufficient condition for a theory to have the strong normalization property: they have proved a ”soundness ” theorem of the form: if a theory has a model (of a particular form) then it has the strong normalization property. In this paper, we refine their notion of model in a way allowing not only to prove soundness, but also completeness: if a theory has the strong normalization property, then it has a model of this form. The key idea of our model construction is a refinement of Girard’s notion of reducibility candidates. By providing a sound and complete semantics for theories having the strong normalization property, this paper contributes to explore the idea
The Discovery Of My Completeness Proofs
 Bulletin of Symbolic Logic
, 1996
"... This paper deals with aspects of my doctoral dissertation 1 ..."
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Cited by 5 (0 self)
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This paper deals with aspects of my doctoral dissertation 1
Proving FirstOrder Equality Theorems with HyperLinking
, 1995
"... Lee and Plaisted recently developed a new automated theorem proving strategy called hyperlinking. As part of his dissertation, Lee developed a roundbyround implementation of the hyperlinking strategy, which competes well with other automated theorem provers on a wide range of theorem proving p ..."
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Cited by 2 (0 self)
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Lee and Plaisted recently developed a new automated theorem proving strategy called hyperlinking. As part of his dissertation, Lee developed a roundbyround implementation of the hyperlinking strategy, which competes well with other automated theorem provers on a wide range of theorem proving problems. However, Lee's roundbyround implementation of hyperlinking is not particularly well suited for the addition of special methods in support of equality. In this dissertation, we describe, as alternative to the roundbyround hyperlinking implementation of Lee, a smallest instance first implementation of hyperlinking which addresses many of the inefficiencies of roundbyround hyperlinking encountered when adding special methods in support of equality. Smallest instance first hyperlinking is based on the formalization of generating smallest clauses first, a heuristic widely used in other automated theorem provers. We prove both the soundness and logical completeness of smallest instance first hyperlinking and show that it always generates smallest clauses first under
REVIEW OF THE BIRTH OF MODEL THEORY BY CALIXTO BADESA
"... What do the theorems of GödelDeligne, ChevalleyTarski, AxGrothendieck, TarskiSeidenberg, and WeilHrushovski have in common? And what do they have to do with the book under review? Each of these theorems was proven by techniques in a particular mathematical area and by model theoretic methods. I ..."
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What do the theorems of GödelDeligne, ChevalleyTarski, AxGrothendieck, TarskiSeidenberg, and WeilHrushovski have in common? And what do they have to do with the book under review? Each of these theorems was proven by techniques in a particular mathematical area and by model theoretic methods. In fact, these model theoretic methods often show a pattern that extends across these areas. What are model theoretic methods? Model theory is the activity of a ‘selfconscious’ mathematician. This mathematician distinguishes an object language (syntax) and a class of structures for this language and ‘definable ’ subsets of those structures (semantics). Semantics provides an interpretation of inscriptions in the formal language in the appropriate structures. At its most basic level this allows the recognition that syntactic transformations can clarify the description of the same set of numbers. Thus, x2 − 3x < −6 is rewritten as x < −2 or x> 3; both formulas define the same set of points if they are interpreted in the real numbers. After clarifying these fundamental notions, we give an anachronistic survey of three themes of 20th century model theory: the study of a) properties of first order
A Review of: The birth of model theory: Löwenheim’s theory in the frame of the theory of relatives, . . .
"... What do the theorems of Gödel–Deligne, Chevalley–Tarski, Ax–Grothendieck, Tarski–Seidenberg, and Weil–Hrushovski have in common? And what do they have to do with the book under review? Each of these theorems was proven by techniques in a particular mathematical area and by model theoretic methods. I ..."
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What do the theorems of Gödel–Deligne, Chevalley–Tarski, Ax–Grothendieck, Tarski–Seidenberg, and Weil–Hrushovski have in common? And what do they have to do with the book under review? Each of these theorems was proven by techniques in a particular mathematical area and by model theoretic methods. In fact, these model theoretic methods often show a pattern that extends across these areas. What are model theoretic methods? Model theory is the activity of a “selfconscious” mathematician. This mathematician distinguishes an object language (syntax) and a class of structures for this language and “definable ” subsets of those structures (semantics). Semantics provides an interpretation of inscriptions in the formal language in the appropriate structures. At its most basic level this allows the recognition that syntactic transformations can clarify the description of the same set of numbers. Thus, x2 − 3x <−6 is rewritten as x<−2 orx>3; both formulas define the same set of points if they are interpreted in the real numbers. After clarifying these fundamental notions, we give an anachronistic survey of three themes of twentieth century model theory: the study of a) properties of first
Author manuscript, published in "Proof Search in Type Theory (2009)" Complete reducibility candidates
, 2009
"... Abstract. Deduction modulo is an extension of firstorder predicate logic where axioms are replaced by a congruence relation on propositions and where many theories, such as arithmetic, simple type theory and some variants of set theory, can be expressed. An important question in deduction modulo is ..."
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Abstract. Deduction modulo is an extension of firstorder predicate logic where axioms are replaced by a congruence relation on propositions and where many theories, such as arithmetic, simple type theory and some variants of set theory, can be expressed. An important question in deduction modulo is to find a condition of the theories that have the strong normalization property. Dowek and Werner have given a semantic sufficient condition for a theory to have the strong normalization property: they have proved a ”soundness ” theorem of the form: if a theory has a model (of a particular form) then it has the strong normalization property. In this paper, we refine their notion of model in a way allowing not only to prove soundness, but also completeness: if a theory has the strong normalization property, then it has a model of this form. The key idea of our model construction is a refinement of Girard’s notion of reducibility candidates. By providing a sound and complete semantics for theories having the strong normalization property, this paper contributes