Hypersequent Calculus Research Articles

On a multilattice analogue of a hypersequent S5 calculus

In this paper, we present a logic MMLS5n which is a combination of multilattice logic and modal logic S5. MMLS5n is an extension of Kamide and Shramko’s modal multilattice logic which is a multilattice analogue of S4. We present a cut-free hypersequent calculus for MMLS5n in the spirit of Restall’s one for S5 and develop a Kripke semantics for MMLS5n, following Kamide and Shramko’s approach. Moreover, we prove theorems for embedding MMLS5n into S5 and vice versa. As a result, we obtain completeness, cut elimination, decidability, and interpolation theorems for MMLS5n. Besides, we show the duality principle for MMLS5n. Additionally, we introduce a modification of Kamide and Shramko’s sequent calculus for their multilattice version of S4 which (in contrast to Kamide and Shramko’s original one) proves the interdefinability of necessity and possibility operators. Last, but not least, we present Hilbert-style calculi for all the logics in question as well as for a larger class of modal multilattice logics.

Generalisation of proof simulation procedures for Frege systems by M.L. Bonet and S.R. Buss

ABSTRACTIn this paper, we present a generalisation of proof simulation procedures for Frege systems by Bonet and Buss to some logics for which the deduction theorem does not hold. In particular, we study the case of finite-valued Łukasiewicz logics. To this end, we provide proof systems and which augment Avron's Frege system HŁuk with nested and general versions of the disjunction elimination rule, respectively. For these systems, we provide upper bounds on speed-ups w.r.t. both the number of steps in proofs and the length of proofs. We also consider Tamminga's natural deduction and Avron's hypersequent calculus GŁuk for 3-valued Łukasiewicz logic and generalise our results considering the disjunction elimination rule to all finite-valued Łukasiewicz logics.

Hypersequents and Systems of Rules

We define a bi-directional embedding between hypersequent calculi and a subclass of systems of rules (2-systems). In addition to showing that the two proof frameworks have the same expressive power, the embedding allows for the recovery of the benefits of locality for 2-systems, analyticity results for a large class of such systems, and a rewriting of hypersequent rules as natural deduction rules.

Infinite-Valued First-Order Łukasiewicz Logic: Hypersequent Calculi Without Structural Rules and Proof Search for Sentences in the Prenex Form

The rational first-order Pavelka logic is an expansion of the infinite-valued first-order Łukasiewicz logic Ł∀ by truth constants. For this logic, we introduce a cumulative hypersequent calculus G1Ł∀ and a noncumulative hypersequent calculus G2Ł∀ without structural inference rules. We compare these calculi with the Baaz–Metcalfe hypersequent calculus GŁ∀ with structural rules. In particular, we show that every GŁ∀-provable sentence is G1Ł∀-provable and a Ł∀-sentence in the prenex form is GŁ∀-provable if and only if it is G2Ł∀-provable. For a tableau version of the calculus G2Ł∀, we describe a family of proof search algorithms that allow us to construct a proof of each G2Ł∀-provable sentence in the prenex form.

Intermediate Logics Admitting a Structural Hypersequent Calculus

We characterise the intermediate logics which admit a cut-free hypersequent calculus of the form mathbf {HLJ} + mathscr {R}, where mathbf {HLJ} is the hypersequent counterpart of the sequent calculus mathbf {LJ} for propositional intuitionistic logic, and mathscr {R} is a set of so-called structural hypersequent rules, i.e., rules not involving any logical connectives. The characterisation of this class of intermediate logics is presented both in terms of the algebraic and the relational semantics for intermediate logics. We discuss various—positive as well as negative—consequences of this characterisation.

Cut Elimination Theorem for Non-Commutative Hypersequent Calculus

Hypersequent calculi (HC) can formalize various non-classical logics. In [9] we presented a non-commutative variant of HC for the weakest temporal logic of linear frames Kt4.3 and some its extensions for dense and serial flow of time. The system was proved to be cut-free HC formalization of respective temporal logics by means of Schütte/Hintikka-style semantical argument using models built from saturated hypersequents. In this paper we present a variant of this calculus for Kt4.3 with a constructive syntactical proof of cut elimination.

One-step Heyting Algebras and Hypersequent Calculi with the Bounded Proof Property

We investigate proof-theoretic properties of hypersequent calculi for intermediate logics using algebraic methods. More precisely, we consider a new weakly analytic subformula property (the bounded proof property) of such calculi. Despite being strictly weaker than both cut-elimination and the subformula property, this property is sufficient to ensure decidability of finitely axiomatized calculi. We introduce one-step Heyting algebras and establish a semantic criterion characterizing calculi for intermediate logics with the bounded proof property and the finite model property in terms of one-step Heyting algebras. Finally, we show how this semantic criterion can be applied to a number of calculi for well-known intermediate logics such as LC,KC and BD2.

Density revisited

In this (part survey) paper, we revisit algebraic and proof-theoretic methods developed by Franco Montagna and his co-authors for proving that the chains (totally ordered members) of certain varieties of semilinear residuated lattices embed into dense chains of these varieties, a key step in establishing standard completeness results for fuzzy logics. Such varieties are precisely the varieties that are generated as quasivarieties by their dense chains. By showing that all dense chains satisfy a certain e-cyclicity equation, we give a short proof that the variety of all semilinear residuated lattices is not densifiable (first proved by Wang and Zhao). We then adapt the Jenei---Montagna standard completeness proof for monoidal t-norm logic to show that any variety of integral semilinear residuated lattices axiomatized by additional lattice-ordered monoid equations is densifiable. We also generalize known results to show that certain varieties of cancellative semilinear residuated lattices are densifiable. Finally, we revisit the Metcalfe---Montagna proof-theoretic approach, which establishes densifiability of a variety via the elimination of a density rule for a suitable hypersequent calculus, focussing on the case of commutative semilinear residuated lattices.

Hypersequent rules with restricted contexts for propositional modal logics

As part of a general research programme into the expressive power of different generalisations of the sequent framework we investigate hypersequent calculi given by rules of the newly introduced format of hypersequent rules with context restrictions. The introduced rule format is used to prove uniform syntactic cut elimination, decidability and complexity results. We also introduce transformations between hypersequent rules of this format and Hilbert axioms, entailing a result about the limits of such rules. As case studies, we apply our methods to several modal logics and obtain e.g. a complexity-optimal decision procedure for the logic S5 and new calculi for the logic K4.2 as well as combinations of modal logics in the form of simply dependent bimodal logics.

Algebraic proof theory: Hypersequents and hypercompletions

We continue our program of establishing connections between proof-theoretic and order-algebraic properties in the setting of substructural logics and residuated lattices. Extending our previous work that connects a strong form of cut-admissibility in sequent calculi with closure under MacNeille completions of corresponding varieties, we now consider hypersequent calculi and more general completions; these capture logics/varieties that were not covered by the previous approach and that are characterized by Hilbert axioms (algebraic equations) residing in the level P3 of the substructural hierarchy. We provide algebraic foundations for substructural hypersequent calculi and an algorithm to transform P3 axioms/equations into equivalent structural hypersequent rules. Using residuated hyperframes we link strong analyticity in the resulting calculi with a new algebraic completion, which we call hyper-MacNeille.

Free-Variable Semantic Tableaux for the Logic of Fuzzy Inequalities

We present a free-variable tableau calculus for the logic of fuzzy inequalities F∀, which is an extension of infinite-valued first-order Lukasiewicz logic L∀. The set of all L∀-sentences provable in the hypersequent calculus of Baaz and Metcalfe for L∀ is embedded into the set of all F∀-sentences provable in the given tableau calculus. We prove NPcompleteness of the problem of checking tableau closability and propose an algorithm, which is based on unification, for solving the problem.

Proof theory for lattice-ordered groups

Proof-theoretic methods are developed and exploited to establish properties of the variety of lattice-ordered groups. In particular, a hypersequent calculus with a cut rule is used to provide an alternative syntactic proof of the generation of the variety by the lattice-ordered group of automorphisms of the real number chain. Completeness is also established for an analytic (cut-free) hypersequent calculus using cut elimination and it is proved that the equational theory of the variety is co-NP complete.

(Hyper)sequent Calculi for the ALC(S4) Description Logics

Description logics (DL) form a well-known family of knowledge representation languages. One of its main applications is on the Semantic Web as a reasoning framework in the form of the Ontology Web Language (OWL). In this paper, we propose a cut-free tree hypersequent calculus for terminological reasoning in the Description Logic ALC. We show the calculus is sound and complete. Also, an implementation is provided together with a complexity analysis. In addition, we also describe a cut-free sequent calculus for the description logic ALC with reflexive and transitive roles. Soundness and completeness are proven, and a complexity analysis and an implementation are also provided.

Hypersequent Calculi for S5: The Methods of Cut Elimination

S5 is one of the most important modal logic with nice syntactic, semantic and algebraic properties. In spite of that, a successful (i.e. cut-free) formalization of S5 on the ground of standard sequent calculus (SC) was problematic and led to the invention of numerous nonstandard, generalized forms of SC. One of the most interesting framework which was very often used for this aim is that of hypersequent calculi (HC). The paper is a survey of HC for S5 proposed by Pottinger, Avron, Restall, Poggiolesi, Lahav and Kurokawa. We are particularly interested in examining different methods which were used for proving the eliminability/admissibility of cut in these systems and present our own variant of a system which admits relatively simple proof of cut elimination.

A cut-free calculus for second-order Gödel logic

We prove that the extension of the known hypersequent calculus for standard first-order Gödel logic with usual rules for second-order quantifiers is sound and (cut-free) complete for Henkin-style semantics for second-order Gödel logic. The proof is semantic, and it is similar in nature to Schütte and Tait's proof of Takeuti's conjecture.

Eliminability of cut in hypersequent calculi for some modal logics of linear frames

Hypersequent calculi, introduced independently by Pottinger and Avron, provide a powerful generalization of ordinary sequent calculi. In the paper we present a proof of eliminability of cut in hypersequent calculi for three modal logics of linear frames: K4.3, KD4.3 and S4.3. Our cut-free calculus is based on Avron's HC formalization for Gödel–Dummett's logic. The presented proof of eliminability of cut is purely syntactical and based on Ciabattoni, Metcalfe, Montagna's proof of eliminability of cut for hypersequent calculi for some fuzzy logics with modalities.

Hypersequent and Display Calculi – a Unified Perspective

This paper presents an overview of the methods of hypersequents and display sequents in the proof theory of non-classical logics. In contrast with existing surveys dedicated to hypersequent calculi or to display calculi, our aim is to provide a unified perspective on these two formalisms highlighting their differences and similarities and discussing applications and recent results connecting and comparing them.

A note on standard completeness for some extensions of uninorm logic

We prove standard completeness for uninorm logic extended with knotted axioms. This is done following a proof-theoretical approach, based on the elimination of the density rule in suitable hypersequent calculi.

Semantic investigation of canonical Gödel hypersequent systems

We define a general family of hypersequent systems with well-behaved logical rules, of which the known hypersequent calculus for (propositional) Godel logic, is a particular instance. We present a method to obtain (possibly, non-deterministic) many-valued semantics for every system of this family. The detailed semantic analysis provides simple characterizations of cut-admissibility and axiom-expansion for the systems of this family.

From Single Agent to Multi-Agent via Hypersequents

In this paper we present a sequent calculus for the multi-agent system S5m. First, we introduce a particularly simple alternative Kripke semantics for the system S5m. Then, we construct a hypersequent calculus for S5m that reflects at the syntactic level this alternative interpretation. We prove that this hypersequent calculus is theoremwise equivalent to the Hilbert-style system S5m, that it is contraction-free and cut-free, and finally that it is decidable. All results are proved in a purely syntactic way and the cut-elimination procedure yields an upper bound of ip2 (n, 0) where ip2 is an hyperexponential function of base 2.