IA Scholar Query: Terminating Calculi and Countermodels for Constructive Modal Logics.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSat, 07 May 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440A theorem prover and countermodel constructor for provability logic in HOL Light
https://scholar.archive.org/work/lwrexzurfjhz7dei2xzkaglmma
We introduce our implementation in HOL Light of a prototype of a general theorem prover for normal modal logics. In the present work, we start by considering G\"odel-L\"ob provability logic (GL). The methodology consists of a shallow embedding of a labelled sequent calculus for GL whose validity relies on our formalised proof in HOL Light of modal completeness for GL w.r.t. possible world semantics, that we present in the first part of the present work. Our theorem prover for GL is thus implemented as a tactic of HOL Light that formalises the proof search in the corresponding labelled sequent calculus, and works as a decision algorithm for that logic: if the algorithm positively terminates, the tactic succeeds in producing a HOL Light theorem stating that the input formula is a theorem of GL; if the algorithm negatively terminates, the tactic extracts a model falsifying the input formula. We discuss our code for the formal proof of modal completeness, and the design of our proof search algorithm. Furthermore some examples of both interactive and automated use of the latter are shown.Marco Maggesi, Cosimo Perini Brogiwork_lwrexzurfjhz7dei2xzkaglmmaSat, 07 May 2022 00:00:00 GMTAutomating Reasoning with Standpoint Logic via Nested Sequents
https://scholar.archive.org/work/xsxpdv6ud5b7tmwg5ezrnssfci
Standpoint logic is a recently proposed formalism in the context of knowledge integration, which advocates a multi-perspective approach permitting reasoning with a selection of diverse and possibly conflicting standpoints rather than forcing their unification. In this paper, we introduce nested sequent calculi for propositional standpoint logics--proof systems that manipulate trees whose nodes are multisets of formulae--and show how to automate standpoint reasoning by means of non-deterministic proof-search algorithms. To obtain worst-case complexity-optimal proof-search, we introduce a novel technique in the context of nested sequents, referred to as "coloring," which consists of taking a formula as input, guessing a certain coloring of its subformulae, and then running proof-search in a nested sequent calculus on the colored input. Our technique lets us decide the validity of standpoint formulae in CoNP since proof-search only produces a partial proof relative to each permitted coloring of the input. We show how all partial proofs can be fused together to construct a complete proof when the input is valid, and how certain partial proofs can be transformed into a counter-model when the input is invalid. These "certificates" (i.e. proofs and counter-models) serve as explanations of the (in)validity of the input.Tim S. Lyon, Lucía Gómez Álvarezwork_xsxpdv6ud5b7tmwg5ezrnssfciThu, 05 May 2022 00:00:00 GMTGeometric Logic, Constructivisation, and Automated Theorem Proving (Dagstuhl Seminar 21472)
https://scholar.archive.org/work/gnnydk6kx5adlmegxfiwolipge
At least from a practical and contemporary angle, the time-honoured question about the extent of intuitionistic mathematics rather is to which extent any given proof is effective, which proofs of which theorems can be rendered effective, and whether and how numerical information such as bounds and algorithms can be extracted from proofs. All this is ideally done by manipulating proofs mechanically or by adequate metatheorems, which includes proof translations, automated theorem proving, program extraction from proofs, proof analysis and proof mining. The question should thus be put as: What is the computational content of proofs? Guided by this central question, the present Dagstuhl seminar puts a special focus on coherent and geometric theories and their generalisations. These are not only widespread in mathematics and non-classical logics such as temporal and modal logics, but also a priori amenable for constructivisation, e.g., by Barr's Theorem, and last but not least particularly suited as a basis for automated theorem proving. Specific topics include categorical semantics for geometric theories, complexity issues of and algorithms for geometrisation of theories including speed-up questions, the use of geometric theories in constructive mathematics including finding algorithms, proof-theoretic presentation of sheaf models and higher toposes, and coherent logic for automatically readable proofs.Thierry Coquand, Hajime Ishihara, Sara Negri, Peter M. Schusterwork_gnnydk6kx5adlmegxfiwolipgeMon, 11 Apr 2022 00:00:00 GMTDagstuhl Reports, Volume 11, Issue 10, October 2021, Complete Issue
https://scholar.archive.org/work/3w5nqw2gangnrkuqgfzp32cw4u
Dagstuhl Reports, Volume 11, Issue 10, October 2021, Complete Issuework_3w5nqw2gangnrkuqgfzp32cw4uMon, 11 Apr 2022 00:00:00 GMTAutomating Reasoning with Standpoint Logic via Nested Sequents
https://scholar.archive.org/work/j5tnygcvkzdm5a64w2ude3ku44
Standpoint logic is a recently proposed formalism in the context of knowledge integration, which advocates a multi-perspective approach permitting reasoning with a selection of diverse and possibly conflicting standpoints rather than forcing their unification. In this paper, we introduce nested sequent calculi for propositional standpoint logics---proof systems that manipulate trees whose nodes are multisets of formulae---and show how to automate standpoint reasoning by means of non-deterministic proof-search algorithms. To obtain worst-case complexity-optimal proof-search, we introduce a novel technique in the context of nested sequents, referred to as "coloring," which consists of taking a formula as input, guessing a certain coloring of its subformulae, and then running proof-search in a nested sequent calculus on the colored input. Our technique lets us decide the validity of standpoint formulae in CoNP since proof-search only produces a partial proof relative to each permitted coloring of the input. We show how all partial proofs can be fused together to construct a complete proof when the input is valid, and how certain partial proofs can be transformed into a counter-model when the input is invalid. These "certificates" (i.e. proofs and counter-models) serve as explanations of the (in)validity of the input.Tim S. Lyon, Lucía Gómez Álvarezwork_j5tnygcvkzdm5a64w2ude3ku44