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21
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 pape ..."
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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
Cycling in proofs and feasibility
 Transactions of the American Mathematical Society
, 1998
"... Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual b ..."
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Cited by 8 (4 self)
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Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh’s lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles. 1.
Turning Cycles into Spirals
, 1999
"... Introduction The structure of LK proofs presents intriguing combinatorial aspects which turn out to be very difficult to study [6,8]. It is wellknown that as soon as one wants to intervene over the structure of a proof to simplify it, the complexity of the proof might increase enormously [16,12,14 ..."
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Cited by 6 (3 self)
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Introduction The structure of LK proofs presents intriguing combinatorial aspects which turn out to be very difficult to study [6,8]. It is wellknown that as soon as one wants to intervene over the structure of a proof to simplify it, the complexity of the proof might increase enormously [16,12,14]. There is a link between the presence of cut formulas with nested quantifiers and the nonelementary expansion needed to prove a theorem without the help of such formulas. If one considers the graph defined by tracing the flow of occurrences of formulas (in the sense of [2]) for proofs allowing a nonelementary compression, one Preprint submitted to Elsevier Preprint 7 November 1997 finds that such graphs contain cycles [5] or almost cyclic structures[6]. These cycles codify in a small space (i.e. a proof with a small number of lines) all the information which is present in the proof once cuts on formulas wit
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 Cost of a Cycle is a Square
, 1999
"... The logical flow graphs of sequent calculus proofs might contain oriented cycles. For the predicate calculus the elimination of cycles might be nonelementary and this was shown in [Car96]. For the propositional calculus, we prove that if a proof of k lines contains n cycles then there exists an ..."
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Cited by 3 (2 self)
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The logical flow graphs of sequent calculus proofs might contain oriented cycles. For the predicate calculus the elimination of cycles might be nonelementary and this was shown in [Car96]. For the propositional calculus, we prove that if a proof of k lines contains n cycles then there exists an acyclic proof with O(k n+1 ) lines. In particular, there is a quadratic time algorithm which eliminates a single cycle from a proof. These results are motivated by the search for general methods on proving lower bounds on proof size and by the design of more efficient heuristic algorithms for proof search.
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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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.