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Linking Focusing and Resolution with Selection

Author Guillaume Burel

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Guillaume Burel
  • ENSIIE and Samovar, Télécom SudParis and CNRS, Université Paris-Saclay, Évry, France, Inria and LSV, CNRS and ENS Paris-Saclay, Université Paris-Saclay, Cachan, France

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Guillaume Burel. Linking Focusing and Resolution with Selection. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.MFCS.2018.9

Abstract

Focusing and selection are techniques that shrink the proof search space for respectively sequent calculi and resolution. To bring out a link between them, we generalize them both: we introduce a sequent calculus where each occurrence of an atom can have a positive or a negative polarity; and a resolution method where each literal, whatever its sign, can be selected in input clauses. We prove the equivalence between cut-free proofs in this sequent calculus and derivations of the empty clause in that resolution method. Such a generalization is not semi-complete in general, which allows us to consider complete instances that correspond to theories of any logical strength. We present three complete instances: first, our framework allows us to show that ordinary focusing corresponds to hyperresolution and semantic resolution; the second instance is deduction modulo theory; and a new setting, not captured by any existing framework, extends deduction modulo theory with rewriting rules having several left-hand sides, which restricts even more the proof search space.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof theory
  • Theory of computation → Automated reasoning
Keywords
  • logic in computer science
  • automated deduction
  • proof theory
  • sequent calculus
  • refinements of resolution
  • deduction modulo theory
  • polarization

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