A multi-labelled sequent calculus for Topo-Logic

2020 ◽  
Vol 30 (2) ◽  
pp. 663-696
Author(s):  
Ian Shillito
Keyword(s):  
Structural Properties ◽  
Epistemic Logic ◽  
Search Strategy ◽  
Sequent Calculus ◽  
Search Tree ◽  
Topological Spaces ◽  
Label Technique ◽  
Proof Search ◽  
Labelled Sequent Calculus

Abstract We present a labelled sequent calculus for a trimodal epistemic logic exhibitied in Baltag et al. (2017, Logic, Rationality, and Interaction, pp. 330–346), an extension of the so called ‘Topo-Logic’. To the best of our knowledge, our calculus is the first proof-calculus for this logic. This calculus is obtained via an adaptation of the label technique by internalizing a semantics over topological spaces. This internalization leads to the generation of two kinds of labels in our calculus and the labelling of formulae by pairs of labels. These novelties give tools to provide a simple calculus that is intuitively connected to the semantics. We prove that this calculus enjoys many structural properties such as admissibility of cut, admissibility of contraction and invertibility of its rules. Finally, we exhibit a proof search strategy for our calculus that allows us to prove completeness in a direct way by the extraction of a countermodel from a failure of proof. To define this strategy, we design a tool for controlling the generation of labels in the construction of a search tree, although the termination of this strategy is still open.

Cut elimination and complexity bounds for intuitionistic epistemic logic

2020 ◽  
Vol 30 (1) ◽  
pp. 281-294
Author(s):  
Vladimir N Krupski
Keyword(s):  
Epistemic Logic ◽  
Sequent Calculus ◽  
Formal System ◽  
Point Of View ◽  
Cut Elimination ◽  
Polynomial Space ◽  
Complexity Bounds ◽  
Proof Search

Abstract The formal system of intuitionistic epistemic logic (IEL) was proposed by S. Artemov and T. Protopopescu. It provides the formal foundation for the study of knowledge from an intuitionistic point of view based on Brouwer–Heyting–Kolmogorov semantics of intuitionism. We construct a cut-free sequent calculus for IEL and establish that polynomial space is sufficient for the proof search in it. We prove that IEL is PSPACE-complete.

A cut-free labelled sequent calculus for dynamic epistemic logic

2020 ◽  
Vol 30 (1) ◽  
pp. 321-348
Author(s):  
Shoshin Nomura ◽  
Hiroakira Ono ◽  
Katsuhiko Sano
Keyword(s):  
Modal Logic ◽  
Epistemic Logic ◽  
Sequent Calculus ◽  
Dynamic Epistemic Logic ◽  
Kripke Semantics ◽  
Cut Rule ◽  
Public Announcement ◽  
Public Announcement Logic ◽  
Soundness And Completeness ◽  
Labelled Sequent Calculus

Abstract Dynamic epistemic logic is a logic that is aimed at formally expressing how a person’s knowledge changes. We provide a cut-free labelled sequent calculus ($\textbf{GDEL}$) on the background of existing studies of Hilbert-style axiomatization $\textbf{HDEL}$ of dynamic epistemic logic and labelled calculi for public announcement logic. We first show that the $cut$ rule is admissible in $\textbf{GDEL}$ and show that $\textbf{GDEL}$ is sound and complete for Kripke semantics. Moreover, we show that the basis of $\textbf{GDEL}$ is extended from modal logic K to other familiar modal logics including S5 with keeping the admissibility of cut, soundness and completeness.

scholarly journals A labelled sequent calculus for BBI: proof theory and proof search

2015 ◽  
Vol 28 (4) ◽  
pp. 809-872
Author(s):  
Zhé Hóu ◽  
Rajeev Goré ◽  
Alwen Tiu
Keyword(s):  
Proof Theory ◽  
Sequent Calculus ◽  
Proof Search ◽  
Labelled Sequent Calculus

Contraction-free sequent calculi for intuitionistic logic

1992 ◽  
Vol 57 (3) ◽  
pp. 795-807 ◽  
Author(s):  
Roy Dyckhoff
Keyword(s):  
Sequent Calculus ◽  
Main Idea ◽  
Simple Algorithm ◽  
Propositional Calculus ◽  
Search Tree ◽  
Sequent Calculi ◽  
Proof Search ◽  
Intuitionistic Propositional Calculus ◽  
Effective Decision ◽  
Subformula Property

Gentzen's sequent calculus LJ, and its variants such as G3 [21], are (as is well known) convenient as a basis for automating proof search for IPC (intuitionistic propositional calculus). But a problem arises: that of detecting loops, arising from the use (in reverse) of the rule ⊃⇒ for implication introduction on the left. We describe below an equivalent calculus, yet another variant on these systems, where the problem no longer arises: this gives a simple but effective decision procedure for IPC.The underlying method can be traced back forty years to Vorob′ev [33], [34]. It has been rediscovered recently by several authors (the present author in August 1990, Hudelmaier [18], [19], Paulson [27], and Lincoln et al. [23]). Since the main idea is not plainly apparent in Vorob′ev's work, and there are mathematical applications [28], it is desirable to have a simple proof. We present such a proof, exploiting the Dershowtiz-Manna theorem [4] on multiset orderings.Consider the task of constructing proofs in Gentzen's sequent calculus LJ of intuitionistic sequents Γ⇒ G, where Γ is a set of assumption formulae and G is a formula (in the language of zero-order logic, using the nullary constant f [absurdity], the unary constant ¬ [negation, with ¬A =defA ⊃ f] and the binary constants &, ∨, and ⊃ [conjunction, disjunction, and implication respectively]). By the Hauptsatz [15], there is an apparently simple algorithm which breaks up the sequent, growing the proof tree until one reaches axioms (of the form Γ⇒ A where A is in Γ), or can make no further progress and must backtrack or even abandon the search. (Gentzen's argument in fact was to use the subformula property derived from the Hauptsatz to limit the size of the search tree. Došen [5] improves on this argument.)

PROOF ANALYSIS FOR LEWIS COUNTERFACTUALS

2015 ◽  
Vol 9 (1) ◽  
pp. 44-75 ◽  
Author(s):  
SARA NEGRI ◽  
GIORGIO SBARDOLINI
Keyword(s):  
Possible Worlds ◽  
Sequent Calculus ◽  
Deductive System ◽  
Axiomatic System ◽  
Relational Semantics ◽  
Completeness Proof ◽  
Analytic Proof ◽  
Proof Search ◽  
Similarity Relations ◽  
Labelled Sequent Calculus

AbstractA deductive system for Lewis counterfactuals is presented, based directly on the influential generalisation of relational semantics through ternary similarity relations introduced by Lewis. This deductive system builds on a method of enriching the syntax of sequent calculus by labels for possible worlds. The resulting labelled sequent calculus is shown to be equivalent to the axiomatic system VC of Lewis. It is further shown to have the structural properties that are needed for an analytic proof system that supports root-first proof search. Completeness of the calculus is proved in a direct way, such that for any given sequent either a formal derivation or a countermodel is provided; it is also shown how finite countermodels for unprovable sequents can be extracted from failed proof search, by which the completeness proof turns into a proof of decidability.

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