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Proof Transformations in HigherOrder Logic
, 1987
"... We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, ..."
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We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, H, a form of the sequent calculus we define, nonanalytic proofs are represented by natural deductions. A nondeterministic translation algorithm between expansion proofs and Hdeductions is presented and its correctness is proven. We also present an algorithm for translation in the other direction and prove its correctness. A cutelimination algorithm for expansion proofs is given and its partial correctness is proven. Strong termination of this algorithm remains a conjecture for the full higherorder system, but is proven for the firstorder fragment. We extend the translations to a nonanalytic proof system which contains a primitive notion of equality, while leaving the notion of expansion proof unaltered. This is possible, since a nonextensional equality is definable in our system of type theory. Next we extend analytic and nonanalytic proof systems and the translations between them to include extensionality. Finally, we show how the methods and notions used so far apply to the problem of translating expansion proofs into natural deductions. Much care is taken to specify this translation in a
Streams and strings of formal proofs
 Theoretical Computer Science
, 2000
"... www.elsevier.com/locate/tcs ..."
CLASSICAL LOGICS FOR ATTRIBUTEVALUE LANGUAGES J iirgcn Wcdckind
"... description) languages which ate used in unification grammar to describe a certain kind of linguistic object commonly called attributevalue structure (or feature structure). The al gorithm which is used for deciding satisfialfility of a feature description is based on a restricted deductive closur ..."
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description) languages which ate used in unification grammar to describe a certain kind of linguistic object commonly called attributevalue structure (or feature structure). The al gorithm which is used for deciding satisfialfility of a feature description is based on a restricted deductive closure constrnc tion for sets of literals (atomi formulas and negated atomic formulas). In contrast to the Kasper/Rounds approach (cf. [Kasper/Rounds 90]), we can handle cyclicity, without the need for the introduction of complexity norms, as in [Johnson and [Beierle/Pletat 88]. The deductive closure construction is the direct prooftheoretic correlate of the congruence closure algorithm (cf. [Nelson/Oppen 80]), if itwere used in attributevalue langnages for testing antiaffability of finite sets of literals.
INTRODUCTION TO THE COMBINATORICS AND COMPLEXITY OF CUT ELIMINATION
"... Abstract. Modus Ponens says that if you know A and you know that A implies B, then you know B. This is a basic rule that we take for granted ..."
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Abstract. Modus Ponens says that if you know A and you know that A implies B, then you know B. This is a basic rule that we take for granted
Graph Rewriting for Natural Deduction and the Proper Treatment of Variables
"... This paper presents a graph implementation of natural deduction for firstorder intuitionistic logic. Studies on proof complexity and rewriting tend to focus solely on the structure of proofs and ignore the formulas that are the conclusions of each proof step. The current approach has the additional ..."
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This paper presents a graph implementation of natural deduction for firstorder intuitionistic logic. Studies on proof complexity and rewriting tend to focus solely on the structure of proofs and ignore the formulas that are the conclusions of each proof step. The current approach has the additional motivation of investigating the behaviour of variables, and to this end the formulas within a proof are preserved in the graph structure. The graph system treats assumptions and variables uniformly and has no need for the explicit constraints on variable occurrences that earned
Streams and Strings in Formal Proofs
"... Streams are acyclic directed subgraphs of the logical flow graph of a proof and represent bundles of paths with the same origin and the same end. Streams can be described with a natural algebraic formalism which allows to explain in algebraic terms the evolution of proofs during cutelimination. ..."
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Streams are acyclic directed subgraphs of the logical flow graph of a proof and represent bundles of paths with the same origin and the same end. Streams can be described with a natural algebraic formalism which allows to explain in algebraic terms the evolution of proofs during cutelimination. In our approach, "logic" is often forgotten and combinatorial properties of graphs are taken into account to explain logical phenomena.