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.
@InProceedings{burel:LIPIcs.MFCS.2018.9, author = {Burel, Guillaume}, title = {{Linking Focusing and Resolution with Selection}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {9:1--9:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.9}, URN = {urn:nbn:de:0030-drops-95917}, doi = {10.4230/LIPIcs.MFCS.2018.9}, annote = {Keywords: logic in computer science, automated deduction, proof theory, sequent calculus, refinements of resolution, deduction modulo theory, polarization} }
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