Structural Extensions of Display Calculi: A General Recipe

Ciabattoni, Agata; Ramanayake, Revantha

doi:10.1007/978-3-642-39992-3_10

Cited by 9 publications

(17 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Section 8, we show that for any language L DLE , any properly displayable DLE-logic is specially displayable, which implies that any properly displayable L * DLE -logic can be axiomatized by means of primitive L * DLE -inequalities. This last result generalizes an analogous statement made by Kracht in the setting of properly displayable tense modal logics, which was proven in [8,9] in the same setting. In Section 9, we outline a comparison between the present treatment and that of [8,9].…”

Section: Contributionssupporting

confidence: 86%

“…This last result generalizes an analogous statement made by Kracht in the setting of properly displayable tense modal logics, which was proven in [8,9] in the same setting. In Section 9, we outline a comparison between the present treatment and that of [8,9]. In Section 10 we present our conclusions.…”

supporting

confidence: 86%

“…The display property is a crucial technical ingredient for Belnap's cut elimination metatheorem: for instance, it provides the core mechanism for the satisfaction of the crucial condition C 8 , discussed in the following subsection.…”

Section: Belnap's Display Logicmentioning

confidence: 99%

“…In the setting of calculi with the relativized display property 8 , if a given analytic structural rule ρ can be applied in concrete derivations of the calculus then ρ is interderivable, modulo applications of display postulates, with a rule of the following form:…”

Section: Remarkmentioning

confidence: 99%

“…Notwithstanding their different and mutually independent starting points, once a concrete setting is defined which provides a common ground for the application of the two methodologies, it is not difficult to recognize striking similarities between the algorithm defined in [8,9] for computing analytic structural rules from input analytic inductive inequalities and the ALBA-based procedure illustrated in Section 7.1. In what follows, we are not giving a formal proof establishing systematic connections between the two procedures, and limit ourselves to illustrating them by means of an example.…”

Section: Quasi-special Inductive Inequalities Are Equivalent To Primimentioning

confidence: 99%

See 4 more Smart Citations

Unified correspondence as a proof-theoretic tool

Greco

Palmigiano

et al. 2016

J Logic Computation

158

View full text Add to dashboard Cite

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...

show abstract

Section: Contributionssupporting

confidence: 86%

supporting

confidence: 86%

Section: Belnap's Display Logicmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Quasi-special Inductive Inequalities Are Equivalent To Primimentioning

confidence: 99%

See 3 more Smart Citations

Unified correspondence as a proof-theoretic tool

Greco

Palmigiano

et al. 2016

J Logic Computation

158

View full text Add to dashboard Cite

show abstract

Tools for the Investigation of Substructural and Paraconsistent Logics

Ciabattoni

Spendier

2014

Logics in Artificial Intelligence

View full text Add to dashboard Cite

Linear Time in Hypersequent Framework

Indrzejczak¹

2016

Bull. symb. log

View full text Add to dashboard Cite

Hypersequent calculus (HC), developed by A. Avron, is one of the most interesting proof systems suitable for nonclassical logics. Although HC has rather simple form, it increases significantly the expressive power of standard sequent calculi (SC). In particular, HC proved to be very useful in the field of proof theory of various nonclassical logics. It may seem surprising that it was not applied to temporal logics so far. In what follows, we discuss different approaches to formalization of logics of linear frames and provide a cut-free HC formalization of Kt4.3, the minimal temporal logic of linear frames, and some of its extensions. The novelty of our approach is that hypersequents are defined not as finite (multi)sets but as finite lists of ordinary sequents. Such a solution allows both linearity of time flow, and symmetry of past and future, to be incorporated by means of six temporal rules (three for future-necessity and three dual rules for past-necessity). Extensions of the basic calculus with simple structural rules cover logics of serial and dense frames. Completeness is proved by Schütte/Hintikka-style argument using models built from saturated hypersequents. §1. Introduction. G. Pottinger [41], and independently, A. Avron [1] introduced a generalised form of Gentzen's sequent calculus (SC) called hypersequent calculus (HC). The main feature of HC is the application of hypersequents which are (multi)sets of ordinary sequents. It may be of interest that at the same time Došen [15] proposed even more general framework where one is dealing with a hierarchy of sequents of order n + 1 with arguments being finite sets of sequents of order n. In particular, sequents of order 2 consist of finite sets of ordinary sequents (of order 1) on both sides, where elements of the antecedent are treated conjunctively, and elements of the succedent disjunctively. Thus one can treat hypersequents as sequents of order 2 with empty antecedents. Similar ideas based on embedding of sequents inside other sequents were later exploited by Kashima [28], Stouppa [46], Brünnler [10], Poggiolesi [40], and one may notice that HC may be interpreted as a restricted version of such approaches.HC may be seen also as a special kind of display calculus (DC) introduced by Belnap [9] (see also [14]). While in DC a family of structural connectives of fixed arity is introduced, in HC a separator of sequents may be treated as the only added structural connective of nonfixed arity. One may find results concerning embedding of HC in DC in Wansing [50] and Ramanayake [42].

show abstract

Structural Extensions of Display Calculi: A General Recipe

Cited by 9 publications

References 18 publications

Unified correspondence as a proof-theoretic tool

Unified correspondence as a proof-theoretic tool

Tools for the Investigation of Substructural and Paraconsistent Logics

Linear Time in Hypersequent Framework

Contact Info

Product

Resources

About