ensl-00814278
https://ens-lyon.hal.science/ensl-00814278
https://ens-lyon.hal.science/ensl-00814278v2/document
https://ens-lyon.hal.science/ensl-00814278v2/file/main.pdf
doi:10.4230/LIPIcs.CSL.2013.i
[ENS-LYON] École Normale Supérieure de Lyon
[CNRS] CNRS - Centre national de la recherche scientifique
[INRIA] INRIA - Institut National de Recherche en Informatique et en Automatique
[UNIV-LYON1] Université Claude Bernard - Lyon I
[INRIA-AUT] Inria autres
[UDL] UDL
[UNIV-LYON] Université de Lyon
[ANR] ANR
Extracting Herbrand trees in classical realizability using forcing
Rieg, Lionel
[INFO.INFO-LO] Computer Science [cs]/Logic in Computer Science [cs.LO]
COMM
classical realizability
forcing
Curry-Howard correspondence
Herbrand trees
Krivine presented in [Kri10] a methodology to combine Cohen's forcing with the theory of classical realizability and showed that the forcing condition can be seen as a reference that is not subject to backtracks. The underlying classical program transformation was then analyzed by Miquel [Miq11] in a fully typed setting in classical higher-order arithmetic (PAω⁺). As a case study of this methodology, we present a method to extract a Herbrand tree from a classical realizer of inconsistency, following the ideas underlying the compactness theorem and the proof of Herbrand's theorem. Unlike the traditional proof based on König's lemma (using a fixed enumeration of atomic formulas), our method is based on the introduction of a particular Cohen real. It is formalized as a proof in PAω⁺, making explicit the construction of generic sets in this framework in the particular case where the set of forcing conditions is arithmetical. We then analyze the algorithmic content of this proof.
2013-09-02
2013-12-11
en
Computer Science Logic 2013
Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik
Computer Science Logic 2013
Turin, Italy