Showing papers on "Normal modal logic published in 2020"

Belnap–dunn modal logics: truth constants vs. truth values

Abstract: We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for ‘neither’ or/and ‘both’ leads to quite unexpected results. To be more precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics.

Coalgebraic Reasoning with Global Assumptions in Arithmetic Modal Logics.

Abstract: We establish a generic upper bound ExpTime for reasoning with global assumptions (also known as TBoxes) in coalgebraic modal logics. Unlike earlier results of this kind, our bound does not require a tractable set of tableau rules for the instance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that potentially avoids building the entire exponential-sized space of candidate states, and thus offers a basis for practical reasoning. This algorithm still involves frequent fixpoint computations; we show how these can be handled efficiently in a concrete algorithm modelled on Liu and Smolka's linear time fixpoint algorithm. Finally, we show that the upper complexity bound is preserved under adding nominals to the logic, i.e. in coalgebraic hybrid logic.

About the Unification Type of Modal Logics Between $$\mathbf {KB}$$ and $$\mathbf {KTB}$$

Abstract: The unification problem in a normal modal logic is to determine, given a formula φ, whether there exists a substitution σ such that σ(φ) is in that logic. In that case, σ is a unifier of φ. We shall say that a set of unifiers of a unifiable formula φ is minimal complete if for all unifiers σ of φ, there exists a unifier τ of φ in that set such that τ is more general than σ and for all σ,τ in that set, σ≠τ, neither σ is more general than τ, nor τ is more general than σ. When a unifiable formula has no minimal complete set of unifiers, the formula is nullary. We usually distinguish between elementary unification and unification with parameters. In elementary unification, all variables are likely to be replaced by formulas when one applies a substitution. In unification with parameters, some variables—called parameters—remain unchanged. In this paper, we prove that normal modal logics KB, KDB and KTB as well as infinitely many normal modal logics between KDB and KTB possess nullary formulas for unification with parameters.

Rosser Provability and Normal Modal Logics

Abstract: In this paper, we investigate Rosser provability predicates whose provability logics are normal modal logics. First, we prove that there exists a Rosser provability predicate whose provability logic is exactly the normal modal logic $$\mathsf{KD}$$. Secondly, we introduce a new normal modal logic $$\mathsf{KDR}$$ which is a proper extension of $$\mathsf{KD}$$, and prove that there exists a Rosser provability predicate whose provability logic includes $$\mathsf{KDR}$$.

A journey in modal proof theory: From minimal normal modal logic to discrete linear temporal logic

Abstract: Extending and generalizing the approach of 2-sequents (Masini, 1992), we present sequent calculi for the classical modal logics in the K, D, T, S4 spectrum. The systems are presented in a uniform way-different logics are obtained by tuning a single parameter, namely a constraint on the applicability of a rule. Cut-elimination is proved only once, since the proof goes through independently from the constraints giving rise to the different systems. A sequent calculus for the discrete linear temporal logic ltl is also given and proved complete. Leitmotiv of the paper is the formal analogy between modality and first-order quantification.

Modal meet-implication logic.

Abstract: We extend the meet-implication fragment of propositional intuitionistic logic with a meet-preserving modality. We give semantics based on semilattices and a duality result with a suitable notion of descriptive frame. As a consequence we obtain completeness and identify a common (modal) fragment of a large class of modal intuitionistic logics. We recognise this logic as a dialgebraic logic, and as a consequence obtain expressivity-somewhere-else. Within the dialgebraic framework, we then investigate the extension of the meet-implication fragment of propositional intuitionistic logic with a monotone modality and prove completeness and expressivity-somewhere-else for it.