Hypersequent Calculus Research Articles

Semiconic idempotent logic I: Structure and local deduction theorems

Semiconic idempotent logic is a common generalization of intuitionistic logic, relevance logic with mingle, and semilinear idempotent logic. It is an algebraizable logic and it admits a cut-free hypersequent calculus. We give a structural decomposition of its characteristic algebraic semantics, conic idempotent residuated lattices, showing that each of these is an ordinal sum of simpler partially ordered structures. This ordinal sum is indexed by a totally ordered residuated lattice, which serves as its skeleton and is both a subalgebra and nuclear image. We equationally characterize the totally ordered residuated lattices appearing as such skeletons. Further, we describe both congruence and subalgebra generation in conic idempotent residuated lattices, proving that every variety generated by these enjoys the congruence extension property. In particular, this holds for semilinear idempotent residuated lattices. Based on this analysis, we obtain a local deduction theorem for semiconic idempotent logic, which also specializes to semilinear idempotent logic.

Linear Abelian Modal Logic

A many-valued modal logic, called linear abelian modal logic \(\rm {\mathbf{LK(A)}}\) is introduced as an extension of the abelian modal logic \(\rm \mathbf{K(A)}\). Abelian modal logic \(\rm \mathbf{K(A)}\) is the minimal modal extension of the logic of lattice-ordered abelian groups. The logic \(\rm \mathbf{LK(A)}\) is axiomatized by extending \(\rm \mathbf{K(A)}\) with the modal axiom schemas \(\Box(\varphi\vee\psi)\rightarrow(\Box\varphi\vee\Box\psi)\) and \((\Box\varphi\wedge\Box\psi)\rightarrow\Box(\varphi\wedge\psi)\). Completeness theorem with respect to algebraic semantics and a hypersequent calculus admitting cut-elimination are established. Finally, the correspondence between hypersequent calculi and axiomatization is investigated.

Fractional-Valued Modal Logic and Soft Bilateralism

In a recent paper, under the auspices of an unorthodox variety of bilateralism, we introduced a new kind of proof-theoretic semantics for the base modal logic \(\mathbf{K}\), whose values lie in the closed interval \([0,1]\) of rational numbers [14]. In this paper, after clarifying our conception of bilateralism – dubbed “soft bilateralism” – we generalize the fractional method to encompass extensions and weakenings of \(\mathbf{K}\). Specifically, we introduce well-behaved hypersequent calculi for the deontic logic \(\mathbf{D}\) and the non-normal modal logics \(\mathbf{E}\) and \(\mathbf{M}\) and thoroughly investigate their structural properties.

Linear logic in a refutational setting

Abstract Sequent-style refutation calculi with non-invertible rules are challenging to design because multiple proof-search strategies need to be simultaneously verified. In this paper, we present a refutation calculus for the multiplicative–additive fragment of linear logic ($\textsf{MALL}$) whose binary rule for the multiplicative conjunction $(\otimes )$ and the unary rule for the additive disjunction $(\oplus )$ fail invertibility. Specifically, we design a cut-free hypersequent calculus $\textsf{HMALL}$, which is equivalent to $\textsf{MALL}$, and obtained by transforming the usual tree-like shape of derivations into a parallel and linear structure. Next, we develop a refutation calculus $\overline{\textsf{HMALL}}$ based on the calculus $\textsf{HMALL}$. As far as we know, this is also the first refutation calculus for a substructural logic. Finally, we offer a fractional semantics for $\textsf{MALL}$—whereby its formulas are interpreted by a rational number in the closed interval [0, 1] —thus extending to the substructural landscape the project of fractional semantics already pursued for classical and modal logics.

Rooted Hypersequent Calculus for Modal Logic S5

Rooted Hypersequent Calculus for Modal Logic S5

Proof theory for the logics of bringing-it-about: Ability, coalitions and means-end relationship

Abstract The logic of bringing-it-about (BIAT) aims to capture a notion of agency in which actions are analysed in terms of their results: ‘An agent does something’ means that the agent brings it about that something takes place. Our starting point is the basic BIAT logic as introduced by Elgesem in the ‘90s: this logic contains only a modal operator to express BIAT statements by single agents. Several extensions have been proposed by Elgesem himself and others, notably with the capability operator, coalitions of agents and means-end BIAT statements (i.e. of the form ‘the agent does B by doing A’). We first propose a variant of the neighbourhood semantics, called bi-neighbourhood semantics, for the basic BIAT logic and the mentioned extensions, in which a world is equipped by a set of pairs or neighbourhoods. Differently from the semantics defined in the literature, this reformulation is well suited for countermodel construction. We then introduce modular hypersequent calculi for all logics considered in this work. Our calculi enjoy the fundamental property of cut admissibility, from which it follows their completeness with respect to the axiomatization. Moreover, our calculi provide at the same time a decision procedure, as well as the first practical countermodel extraction procedure: from a single failed proof it is possible to build directly a finite countermodel of the formula under verification in the bi-neighbourhood semantics. By this last result, we obtain constructive proofs of the semantic completeness of the calculi and consequently of the finite model property for all logics.

A non-distributive logic for semiconcepts and its modal extension with semantics based on Kripke contexts

A non-distributive two-sorted hypersequent calculus PDBL and its modal extension MPDBL are proposed for the classes of pure double Boolean algebras and pure double Boolean algebras with operators respectively. A relational semantics for PDBL is next proposed, where any formula is interpreted as a semiconcept of a context. For MPDBL, the relational semantics is based on Kripke contexts, and a formula is interpreted as a semiconcept of the underlying context. The systems are shown to be sound and complete with respect to the relational semantics. Adding appropriate sequents to MPDBL results in logics with semantics based on reflexive, symmetric or transitive Kripke contexts. In particular, one gets the logics MPDBL4 and MPDBL5 with semantics respectively based on reflexive, transitive Kripke contexts and reflexive, symmetric and transitive Kripke contexts. MPDBL4 also happens to be the logic for topological pure double Boolean algebra. It is demonstrated that, using PDBL, the basic notions and relations of conceptual knowledge can be expressed and inferences involving negations can be obtained. Further, drawing a connection with rough set theory, lower and upper approximations of semiconcepts of a context are defined. It is then shown that, using the formulae and sequents involving modal operators in MPDBL5, these approximation operators and their properties can be captured.

Comparing Calculi for First-Order Infinite-Valued Łukasiewicz Logic and First-Order Rational Pavelka Logic

We consider first-order infinite-valued Łukasiewicz logic and its expansion, first-order rational Pavelka logic RPL∀. From the viewpoint of provability, we compare several Gentzen-type hypersequent calculi for these logics with each other and with Hájek’s Hilbert-type calculi for the same logics. To facilitate comparing previously known calculi for the logics, we define two new analytic calculi for RPL∀ and include them in our comparison. The key part of the comparison is a density elimination proof that introduces no cuts for one of the hypersequent calculi considered.

On infinitary Gödel logics

Abstract We study propositional and first-order Gödel logics over infinitary languages, which are motivated semantically by corresponding interpretations into the unit interval $[0,1]$. We provide infinitary Hilbert-style calculi for the particular (propositional and first-order) cases with con-/disjunctions of countable length and prove corresponding completeness theorems by extending the usual Lindenbaum–Tarski construction to the infinitary case for a respective algebraic semantics via complete linear Heyting algebras. We provide infinitary hypersequent calculi and prove corresponding cut-elimination theorems in the Schütte–Tait style. Initial observations are made regarding truth-value sets other than $[0,1]$.

Proof Theory of Riesz Spaces and Modal Riesz Spaces

We design hypersequent calculus proof systems for the theories of Riesz spaces and modal Riesz spaces and prove the key theorems: soundness, completeness and cut elimination. These are then used to obtain completely syntactic proofs of some interesting results concerning the two theories. Most notably, we prove a novel result: the theory of modal Riesz spaces is decidable. This work has applications in the field of logics of probabilistic programs since modal Riesz spaces provide the algebraic semantics of the Riesz modal logic underlying the probabilistic mu-calculus.

Calculi, countermodel generation and theorem prover for strong logics of counterfactual reasoning

AbstractWe present hypersequent calculi for the strongest logics in Lewis’ family of conditional systems, characterized by uniformity and total reflexivity. We first present a non-standard hypersequent calculus, which allows a syntactic proof of cut elimination. We then introduce standard hypersequent calculi, in which sequents are enriched by additional structures to encode plausibility formulas and diamond formulas. Proof search using these calculi is terminating, and the completeness proof shows how a countermodel can be constructed from a branch of a failed proof search. We then describe tuCLEVER, a theorem prover that implements the standard hypersequent calculi. The prover provides a decision procedure for the logics, and it produces a countermodel in case of proof search failure. The prover tuCLEVER is inspired by the methodology of leanTAP and it is implemented in Prolog. Preliminary experimental results show that the performances of tuCLEVER are promising.1

S5-Style Non-Standard Modalities in a Hypersequent Framework

The aim of the paper is to present some non-standard modalities (such as non-contingency, contingency, essence and accident) based on S5-models in a framework of cut-free hypersequent calculi. We also study negated modalities, i.e. negated necessity and negated possibility, which produce paraconsistent and paracomplete negations respectively. As a basis for our calculi, we use Restall's cut-free hypersequent calculus for S5. We modify its rules for the above-mentioned modalities and prove strong soundness and completeness theorems by a Hintikka-style argument. As a consequence, we obtain a cut admissibility theorem. Finally, we present a constructive syntactic proof of cut elimination theorem.

Second-Order Modal Logic

AbstractThe dissertation introduces new sequent-calculi for free first- and second-order logic, and a hyper-sequent calculus for modal logics K, D, T, B, S4, and S5; to attain the calculi for the stronger modal logics, only external structural rules need to be added to the calculus for K, while operational and internal structural rules remain the same. Completeness and cut-elimination are proved for all calculi presented.Philosophically, the dissertation develops an inferentialist, or proof-theoretic, theory of meaning. It takes as a starting point that the sense of a sentence is determined by the rules governing its use. In particular, there are two features of the use of a sentence that jointly determine its sense, the conditions under which it is coherent to assert that sentence and the conditions under which it is coherent to deny that sentence. The dissertation develops a theory of quantification as marking coherent ways a language can be expanded and modality as the means by which we can reflect on the norms governing the assertion and denial conditions of our language. If the view of quantification that is argued for is correct, then there is no tension between second-order quantification and nominalism. In particular, the ontological commitments one can incur through the use of a quantifier depend wholly on the ontological commitments one can incur through the use of atomic sentences. The dissertation concludes by applying the developed theory of meaning to the metaphysical issue of necessitism and contingentism. Two objections to a logic of contingentism are raised and addressed. The resulting logic is shown to meet all the requirement that the dissertation lays out for a theory of meaning for quantifiers and modal operators.Abstract prepared by Andrew ParisiE-mail: andrew.p.parisi@gmail.comURL: https://opencommons.uconn.edu/dissertations/1480/

BOUNDED-ANALYTIC SEQUENT CALCULI AND EMBEDDINGS FOR HYPERSEQUENT LOGICS

AbstractA sequent calculus with the subformula property has long been recognised as a highly favourable starting point for the proof theoretic investigation of a logic. However, most logics of interest cannot be presented using a sequent calculus with the subformula property. In response, many formalisms more intricate than the sequent calculus have been formulated. In this work we identify an alternative: retain the sequent calculus but generalise the subformula property to permit specific axiom substitutions and their subformulas. Our investigation leads to a classification of generalised subformula properties and is applied to infinitely many substructural, intermediate, and modal logics (specifically: those with a cut-free hypersequent calculus). We also develop a complementary perspective on the generalised subformula properties in terms of logical embeddings. This yields new complexity upper bounds for contractive-mingle substructural logics and situates isolated results on the so-called simple substitution property within a general theory.

Hypersequent calculi for non-normal modal and deontic logics: countermodels and optimal complexity

AbstractWe present some hypersequent calculi for all systems of the classical cube and their extensions with axioms ${T}$, ${P}$ and ${D}$ and for every $n \geq 1$, rule ${RD}_n^+$. The calculi are internal as they only employ the language of the logic, plus additional structural connectives. We show that the calculi are complete with respect to the corresponding axiomatization by a syntactic proof of cut elimination. Then, we define a terminating proof search strategy in the hypersequent calculi and show that it is optimal for coNP-complete logics. Moreover, we show that from every failed proof of a formula or hypersequent it is possible to directly extract a countermodel of it in the bi-neighbourhood semantics of polynomial size for coNP logics, and for regular logics also in the relational semantics. We finish the paper by giving a translation between hypersequent rule applications and derivations in a labelled system for the classical cube.

Repetition-free and infinitary analytic calculi for first-order rational Pavelka logic

We present a hypersequent calculus $\text{G}^3\text{\L}\forall$ for first-order infinite-valued {\L}ukasiewicz logic and for an extension of it, first-order rational Pavelka logic; the calculus is intended for bottom-up proof search. In $\text{G}^3\text{\L}\forall$, there are no structural rules, all the rules are invertible, and designations of multisets of formulas are not repeated in any premise of the rules. The calculus $\text{G}^3\text{\L}\forall$ proves any sentence that is provable in at least one of the previously known hypersequent calculi for the given logics. We study proof-theoretic properties of $\text{G}^3\text{\L}\forall$ and thereby provide foundations for proof search algorithms.

A Hypersequent Solution to the Inferentialist Problem of Modality

The standard inferentialist approaches to modal logic tend to suffer from not being able to uniquely characterize the modal operators, require that introduction and elimination rules be interdefined, or rely on the introduction of possible-world like indexes into the object language itself. In this paper I introduce a hypersequent calculus that is flexible enough to capture many of the standard modal logics and does not suffer from the above problems. It is therefore an ideal candidate to underwrite an inferentialist theory of meaning for modal operators. Here I treat specifically the modal logics K, D, T, S4, B, and S5. I show that the calculi are adequate for each set of models, and show that they meet a large set of criteria that are generally thought necessary for a calculus to underwrite a theory of meaning.

Resource sharing linear logic

AbstractIn this paper, we introduce a new logic that we call ‘resource sharing linear logic (RSLL)’. In linear logic (LL), formulas without modality express some resource-conscious situation (a formula can be used only once); formulas with modality express a situation with unlimited resources. We introduce the logic RSLL in which we have a strengthened modality (S5-modality) that can be understood as expressing not only unlimited resources but also resources shared by different agents. Observing that merely strengthening the modality allows weakening axiom to be derivable in a Hilbert-style formulation of this logic, we reformulate RSLL as a logic similar to affine logic by a hypersequent calculus that has weakening as a primitive rule. We prove the completeness of the hypersequent calculus with respect to phase semantics and the cut-elimination theorem for the system by a syntactical method. We also prove the decidability of RSLL via a computational interpretation of RSLL, which is a parallel version of Kopylov’s computational model for LL. We then introduce an explicit counterpart of RSLL in the style of Artemov’s justication logic (JRSLL). We prove a realization theorem for RSLL via its explicit counterpart.

Game-theoretical interpretation of abelian logic A

In the present paper we introduce a variation of Giles's game that captures the semantics of Slaney and Meyer's Abelian logic. This is a variation of the game earlier proposed for the Łukasiewicz infinitely-valued logic. We discuss two possible interpretations of this game. One of the interpretations involves a reference to different types of agents. We also give a brief description of the Abelian logic which as well corresponds to one of the comparative logics proposed by Casari. By different types of agents, we understand agents with diverse cognitive presumptions and capabilities. This reflects the idea that different agents can be encoded by a game (dialogue) semantics and truth (and validity) can be seen as a product of different types of communications between agents, establishing the relation between various types of moves available to the players and the resulting type of rationality. However, the main focus of the paper is concentrated on the technical result concerning the game proposed in the paper. In a separate section, we prove that this game is adequate to the Abelian logic. The game can be extended to the one allowing for the disjunctive strategies. As immediate future research, we suggest proving that Proponent’s winning strategies for some formula $F$ in the game for Abelian logic $\textbf{A}$ with disjunctive strategies correspond to a derivation of the formula $F$ in the hypersequent calculus $\textbf{GA}$.

CUT ELIMINATION IN HYPERSEQUENT CALCULUS FOR SOME LOGICS OF LINEAR TIME

AbstractThis is a sequel article to [10] where a hypersequent calculus (HC) for some temporal logics of linear frames includingKt4.3and its extensions for dense and serial flow of time was investigated in detail. A distinctive feature of this approach is that hypersequents are noncommutative, i.e., they are finite lists of sequents in contrast to other hypersequent approaches using sets or multisets. Such a system in [10] was proved to be cut-free HC formalization of respective logics by means of semantical argument. In this article we present an equivalent variant of this calculus for which a constructive syntactical proof of cut elimination is provided.