Named Models in Coalgebraic Hybrid Logic

Hybrid logic extends modal logic with support for reasoning about individual states, designated by so-called nominals. We study hybrid logic in the broad context of coalgebraic semantics, where Kripke frames are replaced with coalgebras for a given functor, thus covering a wide range of reasoning principles including, e.g., probabilistic, graded, default, or coalitional operators. Specifically, we establish generic criteria for a given coalgebraic hybrid logic to admit named canonical models, with ensuing completeness proofs for pure extensions on the one hand, and for an extended hybrid language with local binding on the other. We instantiate our framework with a number of examples. Notably, we prove completeness of graded hybrid logic with local binding

Named Models in Coalgebraic Hybrid Logic

Hybrid logic extends modal logic with support for reasoning about individual states, designated by so-called nominals. We study hybrid logic in the broad context of coalgebraic semantics, where Kripke frames are replaced with coalgebras for a given functor, thus covering a wide range of reasoning principles including, e.g., probabilistic, graded, default, or coalitional operators. Specifically, we establish generic criteria for a given coalgebraic hybrid logic to admit named canonical models, with ensuing completeness proofs for pure extensions on the one hand, and for an extended hybrid language with local binding on the other. We instantiate our framework with a number of examples. Notably, we prove completeness of graded hybrid logic with local binding

Many-Valued Hybrid Logic

In this paper we define a family of many-valued semantics for hybrid logic, where each semantics is based on a finite Heyting algebra of truth-values. We provide sound and complete tableau systems for these semantics. Moreover, we show how the tableau systems can be made terminating and thereby give rise to decision procedures for the logics in question. Our many-valued hybrid logics turn out to be "intermediate" logics between intuitionistic hybrid logic and classical hybrid logic in a specific sense explained in the paper. Our results show that many-valued hybrid logic is indeed a natural enterprise

Modal Hybrid Logic

This is an extended version of the lectures given during the 12-th Conference on Applications of Logic in Philosophy and in the Foundations of Mathematics in Szklarska Poręba (7–11 May 2007). It contains a survey of modal hybrid logic, one of the branches of contemporary modal logic. In the first part a variety of hybrid languages and logics is presented with a discussion of expressivity matters. The second part is devoted to thorough exposition of proof methods for hybrid logics. The main point is to show that application of hybrid logics may remarkably improve the situation in modal proof theory

Hierarchical hybrid logic

We introduce HHL, a hierarchical variant of hybrid logic. We study first order correspondence results and prove a Hennessy-Milner like theorem relating (hierarchical) bisimulation and modal equivalence for HHL. Combining hierarchical transition structures with the ability to refer to specific states at different levels, this logic seems suitable to express and verify properties of hierarchical transition systems, a pervasive semantic structure in Computer Science.ERDF European Regional Development Fund, through the COMPETE Programme, and by National Funds through FCT - Portuguese Foundation for Science and Technology - within projects POCI-01-0145-FEDER-016692 and UID/MAT/04106/2013, as well by project “SmartEGOV: Harnessing EGOV for Smart Governance (Foundations, Methods, Tools) / NORTE-01-0145-FEDER-000037”, supported by Norte Portugal Regional Operational Programme (NORTE 2020), under the PORTUGAL 2020 Partnership Agreement. A. Madeira and R. Neves are further supported by the FCT individual grants SFRH/BPD/103004/2014 and SFRH/BD/52234/201

Synthetic Completeness for a Terminating Seligman-Style Tableau System

Hybrid logic extends modal logic with nominals that name worlds. Seligman-style tableau systems for hybrid logic divide branches into blocks named by nominals to achieve a local proof style. We present a Seligman-style tableau system with a formalization in the proof assistant Isabelle/HOL. Our system refines an existing system to simplify formalization and we claim termination from this relationship. Existing completeness proofs that account for termination are either analytic or based on translation, but synthetic proofs have been shown to generalize to richer logics and languages. Our main result is the first synthetic completeness proof for a terminating hybrid logic tableau system. It is also the first formalized completeness proof for any hybrid logic proof system

Paraconsistency in hybrid logic

As in standard knowledge bases, hybrid knowledge bases (i.e. sets of information specified by hybrid formulas) may contain inconsistencies arising from different sources, namely from the many mechanisms used to collect relevant information. Being a fact, rather than a queer anomaly, inconsistency also needs to be addressed in the context of hybrid logic applications. This article introduces a paraconsistent version of hybrid logic which is able to accommodate inconsistencies at local points without implying global failure. A main feature of the resulting logic, crucial to our approach, is the fact that every hybrid formula has an equivalent formula in negation normal form. The article also provides a measure to quantify the inconsistency of a hybrid knowledge base, useful as a possible basis for comparing knowledge bases. Finally, the concepts of extrinsic and intrinsic inconsistency of a theory are discussed

Introducing hierarquical hybrid logic

This paper introduces HHL, a hierarchical variant of hybrid logic. First-order correspondence and a Hennessy-Milner like theorem relating (hierarchical) bisimulation and logical equivalence for HHL are presented. Combining hierarchical transition structures with the ability to refer to specific states at any level of description, this logic seems suitable to express and verify properties of hierarchical transition systems, a pervasive semantic structure in Computer Science

A hilbert-style axiomatisation for equational hybrid logic

This paper introduces an axiomatisation for equational hybrid logic based on previous axiomatizations and natural deduction systems for propositional and first-order hybrid logic. Its soundness and completeness is discussed. This work is part of a broader research project on the development a general proof calculus for hybrid logics