We continue our investigation into hybrid polyadic multi-sorted logic with a focus on expresivity related to the operational and axiomatic semantics of rogramming languages, and relations with first-order logic. We identify a fragment of the full logic, for which we prove sound and complete deduction and we show that it is powerful enough to represent both the programs and their semantics in an uniform way. Although weaker than other hybrid systems previously developed, this system is expected to have better computational properties. Finally, we provide a standard translation from full hybrid many-sorted logic to first-order logic.
This paper presents a many-sorted polyadic modal logic that generalizes some of the existing approaches. The algebraic semantics has led us to a many-sorted generalization of boolean algebras with operators, for which we prove the analogue of the Jónsson-Tarski theorem. While the transition from the mono-sorted logic to many-sorted one is a smooth process, we see our system as a step towards deepening the connection between modal logic and program verification, since our system can be seen as the propositional fragment of Matching logic, a first-order logic for specifying and reasoning about p…
In the present paper syntax and semantics will be presented for an expansion of ordinary n-agent QML with constant domain, non-rigid constants, rigid variables and including both functions, relations, and equality. Further, the number of agents will be specified axiomatically thus ensuring maximal flexibility wrt. the cardinality of the set of agents. Domain, variables, and constants will be partitioned in an agent-part and an object-part and the syntax will be expanded to include strings in which indexes of modal operators are quantified over as wff's of the language. This will enhance expres…
Propositional term modal logic is interpreted over Kripke structures with unboundedly many accessibility relations and hence the syntax admits variables indexing modalities and quantification over them. This logic is undecidable, and we consider a variable-free propositional bi-modal logic with implicit quantification. Thus $[\forall] α$ asserts necessity over all accessibility relations and $[\exists] α$ is classical necessity over some accessibility relation. The logic is associated with a natural bisimulation relation over models and we show that the logic is exactly the bisimulation invari…
Forcing and Interpolation in first-order hybrid Logic with rigid symbols
Institute of Mathematics for Industry, Kyushu University（九州大学产业数学研究所）；Joint Graduate School of Mathematics for Innovation, Kyushu University（九州大学数学创新联合研究生院）
Inquisitive modal logic, InqML, is a generalisation of standard Kripke-style modal logic. In its epistemic incarnation, it extends standard epistemic logic to capture not just the information that agents have, but also the questions that they are interested in. Technically, InqML fits within the family of logics based on team semantics. From a model-theoretic perspective, it takes us a step in the direction of monadic second-order logic, as inquisitive modal operators involve quantification over sets of worlds. We introduce and investigate the natural notion of bisimulation equivalence in the …
Building on our previous work on hybrid polyadic modal logic we identify modal logic equivalents for Matching Logic, a logic for program specification and verification. This provides a rigorous way to transfer results between the two approaches, which should benefit both systems.
Relation-changing modal logics are extensions of the basic modal logic that allow changes to the accessibility relation of a model during the evaluation of a formula. In particular, they are equipped with dynamic modalities that are able to delete, add, and swap edges in the model, both locally and globally. We provide translations from these logics to hybrid logic along with an implementation. In general, these logics are undecidable, but we use our translations to identify decidable fragments. We also compare the expressive power of relation-changing modal logics with hybrid logics.
We propose a hybrid-dynamic first-order logic as a formal foundation for specifying and reasoning about reconfigurable systems. As the name suggests, the formalism we develop extends (many-sorted) first-order logic with features that are common to hybrid logics and to dynamic logics. This provides certain key advantages for dealing with reconfigurable systems, such as: (a) a signature of nominals, including operation and relation symbols, that allows references to specific possible worlds / system configurations -- as in the case of hybrid logics; (b) distinguished signatures of rigid and flex…
We characterise non-distributive positive logic as the fragment of a single-sorted first-order language that is preserved by a new notion of simulation called a meet-simulation. Meet-simulations distinguish themselves from simulations because they relate pairs of states from one model to single states from another. En route to this result we use a more traditional notion of simulations and prove a Hennessy-Milner style theorem for it, using an analogue of modal saturation called meet-compactness.
Game semantics and winning strategies offer a potential conceptual bridge between semantics and proof systems of logics. We illustrate this link for hybrid logic -- an extension of modal logic that allows for explicit reference to worlds within the language. The main result is that the systematic search of winning strategies over all models can be finitized and thus reformulated as a proof system.
This paper investigates first-order game logic and first-order modal mu-calculus, which extend their propositional modal logic counterparts with first-order modalities of interpreted effects such as variable assignments. Unlike in the propositional case, both logics are shown to have the same expressive power and their proof calculi to have the same deductive power. Both calculi are also mutually relatively complete. In the presence of differential equations, corollaries obtain usable and complete translations between differential game logic, a logic for the deductive verification of hybrid ga…
Freie Universität Berlin（柏林自由大学）；University of Luxembourg（卢森堡大学）
The automated theorem prover Leo-III for classical higher-order logic with Henkin semantics and choice is presented. Leo-III is based on extensional higher-order paramodulation and accepts every common TPTP dialect (FOF, TFF, THF), including their recent extensions to rank-1 polymorphism (TF1, TH1). In addition, the prover natively supports almost every normal higher-order modal logic. Leo-III cooperates with first-order reasoning tools using translations to many-sorted first-order logic and produces verifiable proof certificates. The prover is evaluated on heterogeneous benchmark sets.
In recent work, comonads and associated structures have been used to analyse a range of important notions in finite model theory, descriptive complexity and combinatorics. We extend this analysis to Hybrid logic, a widely-studied extension of basic modal logic, which corresponds to the bounded fragment of first-order logic. In addition to characterising the various resource-indexed equivalences induced by Hybrid logic and the bounded fragment, and the associated combinatorial decompositions of structures, we also give model-theoretic characterisations of bounded formulas in terms of invariance…
