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30
Cutelimination and redundancyelimination by resolution
 Journal of Symbolic Computation
, 2000
"... A new cutelimination method for Gentzen’s LK is defined. First cutelimination is generalized to the problem of redundancyelimination. Then the elimination of redundancy in LKproofs is performed by a resolution method in the following way: A set of clauses C is assigned to an LKproof ψ and it is ..."
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A new cutelimination method for Gentzen’s LK is defined. First cutelimination is generalized to the problem of redundancyelimination. Then the elimination of redundancy in LKproofs is performed by a resolution method in the following way: A set of clauses C is assigned to an LKproof ψ and it is shown that C is always unsatisfiable. A resolution refutation of C then serves as a skeleton of an LKproof ψ ′ with atomic cuts; ψ ′ can be constructed from the resolution proof and ψ by a projection method. In the last step the atomic cuts are eliminated and a cutfree proof is obtained. The complexity of the method is analyzed and it is shown that a nonelementary speedup over Gentzen’s method can be achieved. Finally an application to automated deduction is presented: it is demonstrated how informal proofs (containing pseudocuts) can be transformed into formal ones by the method of redundancyelimination; moreover, the method can even be used to transform incorrect proofs into correct ones. 1.
The Decidability of Simultaneous Rigid EUnification with One Variable
 REWRITING TECHNIQUES AND APPLICATIONS
, 1997
"... We show that simultaneous rigid Eunification, or SREU for short, is decidable and in fact EXPTIMEcomplete in the case of one variable. This result implies that the ... fragment of intuitionistic logic with equality is decidable. Together with a previous result regarding the undecidability of the ..."
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Cited by 10 (10 self)
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We show that simultaneous rigid Eunification, or SREU for short, is decidable and in fact EXPTIMEcomplete in the case of one variable. This result implies that the ... fragment of intuitionistic logic with equality is decidable. Together with a previous result regarding the undecidability of the 99fragment, we obtain a complete classification of decidability of the prenex fragment of intuitionistic logic with equality, in terms of the quantifier prefix. It is also proved that SREU with one variable and a constant bound on the number of rigid equations is Pcomplete.
Methods of CutElimination
 PROJECTION, LECTURE
"... This short report presents the main topics of methods of cutelimination which will be presented in the course at the ESSLLI'99. It gives a short introduction addressing the problem of cutelimination in general. Furthermore we give a brief description of several methods and refer to other papers ad ..."
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Cited by 10 (9 self)
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This short report presents the main topics of methods of cutelimination which will be presented in the course at the ESSLLI'99. It gives a short introduction addressing the problem of cutelimination in general. Furthermore we give a brief description of several methods and refer to other papers added to the course material.
Applying Tree Languages in Proof Theory
 In AdrianHoria Dediu and Carlos MartínVide, editors, Language and Automata Theory and Applications (LATA) 2012, volume 7183 of Lecture Notes in Computer Science
, 2012
"... Abstract. We introduce a new connection between formal language theory and proof theory. One of the most fundamental proof transformations in a class of formal proofs is shown to correspond exactly to the computation of the language of a certain class of tree grammars. Translations in both direction ..."
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Cited by 8 (7 self)
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Abstract. We introduce a new connection between formal language theory and proof theory. One of the most fundamental proof transformations in a class of formal proofs is shown to correspond exactly to the computation of the language of a certain class of tree grammars. Translations in both directions, from proofs to grammars and from grammars to proofs, are provided. This correspondence allows theoretical as well as practical applications. 1
On the complexity of proof deskolemization
 J. Symbolic Logic
"... Abstract. We consider the following problem: Given a proof of the Skolemization of a formula F, what is the length of the shortest proof of F? For the restriction of this question to cutfree proofs we prove corresponding exponential upper and lower bounds. §1. Introduction. The Skolemization of for ..."
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Cited by 5 (4 self)
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Abstract. We consider the following problem: Given a proof of the Skolemization of a formula F, what is the length of the shortest proof of F? For the restriction of this question to cutfree proofs we prove corresponding exponential upper and lower bounds. §1. Introduction. The Skolemization of formulas is a standard technique in logic. It consists of replacing existential quantifiers by new function symbols whose arguments reflect the dependencies of the quantifier. The Skolemization of a formula is satisfiabilityequivalent to the original formula. This transformation has a number of applications, it is for example crucial for automated theorem
On the form of witness terms
 ARCH. MATHEMATICAL LOGIC
, 2010
"... We investigate the development of terms during cutelimination in firstorder logic and Peano arithmetic for proofs of existential formulas. The form of witness terms in cutfree proofs is characterized in terms of structured combinations of basic substitutions. Based on this result, a regular tree ..."
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Cited by 4 (3 self)
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We investigate the development of terms during cutelimination in firstorder logic and Peano arithmetic for proofs of existential formulas. The form of witness terms in cutfree proofs is characterized in terms of structured combinations of basic substitutions. Based on this result, a regular tree grammar computing witness terms is given and a class of proofs is shown to have only elementary cutelimination.
The epsilon calculus and Herbrand Complexity
 STUDIA LOGICA
, 2006
"... Hilbert’s εcalculus is based on an extension of the language of predicate logic by a termforming operator εx. Two fundamental results about the εcalculus, the first and second epsilon theorem, play a rôle similar to that which the cutelimination theorem plays in sequent calculus. In particular ..."
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Cited by 2 (0 self)
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Hilbert’s εcalculus is based on an extension of the language of predicate logic by a termforming operator εx. Two fundamental results about the εcalculus, the first and second epsilon theorem, play a rôle similar to that which the cutelimination theorem plays in sequent calculus. In particular, Herbrand’s Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure.