Algebraic semantics and model completeness for Intuitionistic Public Announcement Logic

Sadrzadeh, Mehrnoosh; Palmigiano, Alessandra; Ma, Minghui

doi:10.1016/j.apal.2013.11.004

Cited by 40 publications

(70 citation statements)

References 8 publications

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“…Let us illustrate the reduction strategy of the proposition above by applying it to the right-primitive Fischer Servi inequality discussed in Examples 30 and 34 (cf. [37,Lemma 27]). …”

Section: If T( P Q) Is Definite Then the Following Are Equivalent Fmentioning

confidence: 99%

“…In the remainder of the proof, we focus on one of them. 37 The formulas LA(φ 1 )(i/u), LA(φ 2 )(i/u), RA(ψ 1 )(m/u), and RA(ψ 2 )(m/u) do not need to be pure, and in general they are not. However, the assumptions and the general theory of ALBA guarantee that they are ε ∂ -uniform and free of the variable the 'minimal valuation' of which they are part of.…”

Section: If −X ≺ +ψ 2 Then the Following Rule Is Derivable In Albamentioning

confidence: 99%

“…all variables. 37 Applying the Ackermann rule repeatedly in the order indicated by Ω yields the following pure quasi-inequality: 36 The applications of splitting rules at this stage give rise to a set of inequalities, each of which can be treated separately. In the remainder of the proof, we focus on one of them.…”

Section: If −X ≺ +ψ 2 Then the Following Rule Is Derivable In Albamentioning

confidence: 99%

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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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“…Let us illustrate the reduction strategy of the proposition above by applying it to the right-primitive Fischer Servi inequality discussed in Examples 30 and 34 (cf. [37,Lemma 27]). …”

Section: If T( P Q) Is Definite Then the Following Are Equivalent Fmentioning

confidence: 99%

Section: If −X ≺ +ψ 2 Then the Following Rule Is Derivable In Albamentioning

confidence: 99%

Section: If −X ≺ +ψ 2 Then the Following Rule Is Derivable In Albamentioning

confidence: 99%

See 1 more Smart Citation

Unified correspondence as a proof-theoretic tool

Greco

Palmigiano

et al. 2016

J Logic Computation

Self Cite

158

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“…Let us informally expand on (a) how general principles in duality theory are applied to the Stone duality setting for the relational models of EAK, and yield an algebraic characterization of epistemic updates (this is the approach introduced in [18] and applied there to epistemic actions of public-announcement type), and on (b) how the results of [18] are extended from public announcements to general epistemic updates in the style of Baltag-Moss-Solecki. In [2], given a relational model M and an action structure α, the product update M α is defined as a certain submodel of a certain intermediate model M × α, the domain of which is the cartesian product of the domains of M and of α.…”

Section: Introductionmentioning

confidence: 99%

“…Sahlqvist correspondence for substructural logics). Together with [18], the present paper is concerned with applying duality theory to a close cognate of modal logic, namely Dynamic Epistemic Logic, and starting to take stock of the results of this application. The dynamic epistemic logic considered in the present paper is the Logic of Epistemic Actions and Knowledge due to Baltag-Moss-Solecki [2], and we refer to it as EAK.…”

Section: Introductionmentioning

confidence: 99%

Epistemic Updates on Algebras

Kurz¹,

Palmigiano²

2013

Logical Methods in Computer Science

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Abstract. We develop the mathematical theory of epistemic updates with the tools of duality theory. We focus on the Logic of Epistemic Actions and Knowledge (EAK), introduced by Baltag-MossSolecki, without the common knowledge operator. We dually characterize the product update construction of EAK as a certain construction transforming the complex algebras associated with the given model into the complex algebra associated with the updated model. This dual characterization naturally generalizes to much wider classes of algebras, which include, but are not limited to, arbitrary BAOs and arbitrary modal expansions of Heyting algebras (HAOs). As an application of this dual characterization, we axiomatize the intuitionistic analogue of the logic of epistemic knowledge and actions, which we refer to as IEAK, prove soundness and completeness of IEAK w.r.t. both algebraic and relational models, and illustrate how IEAK encodes the reasoning of agents in a concrete epistemic scenario.

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Constructive Dynamic Logic of Relation Changers

Hatano

Sano

2020

Lecture Notes in Computer Science

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Algebraic semantics and model completeness for Intuitionistic Public Announcement Logic

Cited by 40 publications

References 8 publications

Unified correspondence as a proof-theoretic tool

Unified correspondence as a proof-theoretic tool

Epistemic Updates on Algebras

Constructive Dynamic Logic of Relation Changers

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