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- W1505349735 abstract "In 1983, Valentini presented a syntactic proof of cut-elimi nation for a sequent calculus GLSV for the provability logic GL where we have added the subscript V for Valentini. The sequents in GLSV were built from sets, as opposed to multisets, thus avoiding an explicit contraction rule. From a syntactic point of view, it is more satisfying and formal to explicitly identify the applications of the contraction rule that are 'hidden' in these set-based proofs of cut-elimination. There is ofte n an underly- ing assumption that the move to a proof of cut-elimination fo r sequents built from multisets is easy. Recently, however, it has been claimed that Valentini's arguments to eliminate cut do not terminate when applied to a multiset formulation of GLSV with an explicit rule of contraction. The claim has led to much confusion and various authors have sought new proofs of cut-elimination for GL in a multiset setting. Here we refute this claim by placing Valentini's arguments in a formal setting and proving cut-elimination for sequents built fro m multisets. The formal setting is particularly important for sequents built from multisets, in order to accurately account for the interplay between the weakening and contraction rules. Furthermore, Valentini's original proof relies on a novel induction parameter called width which is computed 'globally'. It is diffi- cult to verify the correctness of his induction argument based on width. In our formulation however, verification of the induction argument is straight- forward. Finally, the multiset setting also introduces a new complication in the the case of contractions above cut when the cut-formul a is boxed. We deal with this using a new transformation based on Valentini's original arguments. Finally, we show that the algorithm purporting to show the non-termi- nation of Valentini's arguments is not a faithful representation of the original arguments, but is instead a transformation already known to be insufficient." @default.
- W1505349735 created "2016-06-24" @default.
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- W1505349735 date "2008-01-01" @default.
- W1505349735 modified "2023-09-28" @default.
- W1505349735 title "Valentini's cut-elimination for provability logic resolved" @default.
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