Multi-type display calculus for propositional dynamic logic

Frittella, Sabine; Greco, Giuseppe; Kurz, Alexander; Palmigiano, Alessandra

doi:10.1093/logcom/exu064

Cited by 30 publications

(43 citation statements)

References 20 publications

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“…Section 3.4 [15]) on some of the defining inequalities of its corresponding heterogeneous algebras. 5 For instance, the soundness of the characteristic rule of D.WS on HWSAs follows from the validity of the inequality (xi) in every HWSA (discussed in Section 4) and from the soundness of the following ALBA reduction in every HWSA: Proposition 24. For every A ∈ L, the sequent A τ ⊢ A τ is derivable in all display calculi introduced in Section 5.2.…”

Section: Soundnessmentioning

confidence: 99%

Semi De Morgan Logic Properly Displayed

et al. 2020

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In the present paper, we endow a family of axiomatic extensions of semi De Morgan logic with proper multi-type display calculi which are sound, complete, conservative, and enjoy cut elimination and subformula property. Our proposal builds on an algebraic analysis of semi De Morgan algebras and its subvarieties and applies the guidelines of the multi-type methodology in the design of display calculi.

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Section: Soundnessmentioning

confidence: 99%

Semi De Morgan Logic Properly Displayed

et al. 2020

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“…Display calculi have been successful in giving adequate proof-theoretic semantic accounts of logics-such as certain modal and substructural logics [30], and more recently also Dynamic Epistemic Logic [25] and PDL [24]-which have notoriously been difficult to treat with other approaches. Here we mainly report and elaborate on the work of Belnap [2], Wansing [51], Goré [30,29], and Restall [43].…”

Section: Preliminaries On Display Calculimentioning

confidence: 99%

“…Hence, after this step, no occurrences of +∧ and −∨ will remain in the PIA subterms. 24 Notice that applying this step to (Ω, ε)-inductive terms produces (Ω, ε)-inductive terms each PIA-subterm of which contains at most one ε-critical variable occurrence.…”

Section: Remark 14mentioning

confidence: 99%

Unified correspondence as a proof-theoretic tool

Greco

Palmigiano

et al. 2016

J Logic Computation

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158

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The present paper aims at establishing formal connections between correspondence phenomena, well known from the area of modal logic, and the theory of display calculi, originated by Belnap.These connections have been seminally observed and exploited by Marcus Kracht, in the context of his characterization of the modal axioms (which he calls primitive formulas) which can be effectively transformed into 'analytic' structural rules of display calculi. In this context, a rule is 'analytic' if adding it to a display calculus preserves Belnap's cut-elimination theorem.In recent years, the state-of-the-art in correspondence theory has been uniformly extended from classical modal logic to diverse families of nonclassical logics, ranging from (bi-)intuitionistic (modal) logics, linear, relevant and other substructural logics, to hybrid logics and mu-calculi. This generalization has given rise to a theory called unified correspondence, the most important technical tools of which are the algorithm ALBA, and the syntactic characterization of Sahlqvisttype classes of formulas and inequalities which is uniform in the setting of normal DLE-logics (logics the algebraic semantics of which is based on bounded distributive lattices).We apply unified correspondence theory, with its tools and insights, to extend Kracht's results and prove his claims in the setting of DLE-logics. The results of the present paper characterize the space of properly displayable DLE-logics. Keywords: Display calculi, unified correspondence, distributive lattice expansions, properly displayable logics. Math. Subject Class. 03B45, 06D50, 06D10, 03G10, 06E15. , and mu-calculus [11,12]. The breadth of this work has stimulated many and varied applications. Some are closely related to the core concerns of the theory itself, such as the understanding of the relationship between different methodologies for obtaining canonicity results [40,15], or of the phenomenon of pseudocorrespondence [19]. Other, possibly surprising applications include the dual characterizations of classes of finite lattices [27]. Finally, the insights of unified correspondence theory have made it possible to determine the extent to which the Sahlqvist theory of classes of normal DLEs can be reduced to the Sahlqvist theory of normal Boolean expansions, by means of Gödel-type translations [20]. These and other results have given rise to a theory called unified correspondence [13]. Contents Tools of unified correspondence theory.The most important technical tools in unified correspondence are: (a) a very general syntactic definition of the class of Sahlqvist formulas, which applies uniformly to each logical signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives; (b) the algorithm ALBA, which effectively computes first-order correspondents of input term-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which, like the Sahlqvist class, can be defi...

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“…An indication that this is possible is that the deep rules in our system G are reminiscent of display calculi. The latter have been successfully applied to obtain a complete proof system for PDL [24].…”

Section: B Future Researchmentioning

confidence: 99%

Completeness for Game Logic

Enqvist

Hansen²,

Kupke

et al. 2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Game logic was introduced by Rohit Parikh in the 1980s as a generalisation of propositional dynamic logic (PDL) for reasoning about outcomes that players can force in determined 2-player games. Semantically, the generalisation from programs to games is mirrored by moving from Kripke models to monotone neighbourhood models. Parikh proposed a natural PDL-style Hilbert system which was easily proved to be sound, but its completeness has thus far remained an open problem.In this paper, we introduce a cut-free sequent calculus for game logic, and two cut-free sequent calculi that manipulate annotated formulas, one for game logic and one for the monotone µ-calculus, the variant of the polymodal µ-calculus where the semantics is given by monotone neighbourhood models instead of Kripke structures. We show these systems are sound and complete, and that completeness of Parikh's axiomatization follows. Our approach builds on recent ideas and results by Afshari & Leigh (LICS 2017) in that we obtain completeness via a sequence of proof transformations between the systems. A crucial ingredient is a validity-preserving translation from game logic to the monotone µ-calculus.

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Multi-type display calculus for propositional dynamic logic

Abstract: We introduce a multi-type display calculus for Propositional Dynamic Logic (PDL). This calculus is complete w.r.t. PDL, and enjoys Belnap-style cut-elimination and subformula property.

Cited by 30 publications

References 20 publications

Semi De Morgan Logic Properly Displayed

Semi De Morgan Logic Properly Displayed

Unified correspondence as a proof-theoretic tool

Completeness for Game Logic

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