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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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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.
Extracting Herbrand Disjunctions by Functional Interpretation
"... Abstract. Carrying out a suggestion by Kreisel, we adapt Gödel’s functional interpretation to ordinary firstorder predicate logic(PL) and thus devise an algorithm to extract Herbrand terms from PLproofs. The extraction is carried out in an extension of PL to higher types. The algorithm consists of ..."
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Abstract. Carrying out a suggestion by Kreisel, we adapt Gödel’s functional interpretation to ordinary firstorder predicate logic(PL) and thus devise an algorithm to extract Herbrand terms from PLproofs. The extraction is carried out in an extension of PL to higher types. The algorithm consists of two main steps: first we extract a functional realizer, next we compute the βnormalform of the realizer from which the Herbrand terms can be read off. Even though the extraction is carried out in the extended language, the terms are ordinary PLterms. In contrast to approaches to Herbrand’s theorem based on cut elimination or εelimination this extraction technique is, except for the normalization step, of low polynomial complexity, fully modular and furthermore allows an analysis of the structure of the Herbrand terms, in the spirit of Kreisel ([13]), already prior to the normalization step. It is expected that the implementation of functional interpretation in Schwichtenberg’s MINLOG system can be adapted to yield an efficient Herbrandterm extraction tool. 1.
JACQUES HERBRAND: LIFE, LOGIC, AND AUTOMATED DEDUCTION
"... The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1 ..."
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The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.
Notre Dame Journal of Formal Logic The Computational Content of Arithmetical Proofs
"... Abstract For any extension T of IΣ1 having a cutelimination property extending that of IΣ1, the number of different proofs that can be obtained by cutelimination from a single Tproof cannot be bound by a function which is provably total in T. 1 ..."
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Abstract For any extension T of IΣ1 having a cutelimination property extending that of IΣ1, the number of different proofs that can be obtained by cutelimination from a single Tproof cannot be bound by a function which is provably total in T. 1