This study introduces refutation-aware Gentzen-style sequent calculi and Kripke-style semantics for propositional until-free linear-time temporal logic. The sequent calculi and semantics are constructed on the basis of the refutation-aware setting for Nelson’s paraconsistent logic. The cut-elimination and completeness theorems for the proposed sequent calculi and semantics are proven.

We combine linear temporal logic (with both past and future modalities) with a deontic version of justification logic to provide a framework for reasoning about time and epistemic and normative reasons. In addition to temporal modalities, the resulting logic contains two kinds of justification assertions: epistemic justification assertions and deontic justification assertions. The former presents justification for the agent’s knowledge and the latter gives reasons for why a proposition is obligatory. We present two kinds of semantics for the logic: one (...) based on Fitting models and the other based on neighborhood models. The use of neighborhood semantics enables us to define the dual of deontic justification assertions properly, which corresponds to a notion of permission in deontic logic. We then establish the soundness and completeness of an axiom system of the logic with respect to these semantics. Further, we formalize the Protagoras versus Euathlus paradox in this logic and present a precise analysis of the paradox, and also briefly discuss Leibniz’s solution. (shrink)

Dynamic Topological Logic () is a combination of , under its topological interpretation, and the temporal logic interpreted over the natural numbers. is used to reason about properties of dynamical systems based on topological spaces. Semantics are given by dynamic topological models, which are tuples , where is a topological space, f a function on X and V a truth valuation assigning subsets of X to propositional variables. Our main result is that the set of valid formulas of over spaces (...) with continuous functions is recursively enumerable. We show this by defining alternative semantics for . Under standard semantics, is not complete for Kripke frames. However, we introduce the notion of a non-deterministic quasimodel, where the function f is replaced by a binary relation g assigning to each world multiple temporal successors. We place restrictions on the successors so that the logic remains unchanged; under these alternative semantics, becomes Kripke-complete. We then apply model-search techniques to enumerate the set of all valid formulas. (shrink)

We combine linear temporal logic (with both past and future modalities) with a deontic version of justification logic to provide a framework for reasoning about time and epistemic and normative reasons. In addition to temporal modalities, the resulting logic contains two kinds of justification assertions: epistemic justification assertions and deontic justification assertions. The former presents justification for the agent’s knowledge and the latter gives reasons for why a proposition is obligatory. We present two kinds of semantics for the logic: one (...) based on Fitting models and the other based on neighborhood models. The use of neighborhood semantics enables us to define the dual of deontic justification assertions properly, which corresponds to a notion of permission in deontic logic. We then establish the soundness and completeness of an axiom system of the logic with respect to these semantics. Further, we formalize the Protagoras versus Euathlus paradox in this logic and present a precise analysis of the paradox, and also briefly discuss Leibniz’s solution. (shrink)

We study a propositional probabilistic temporal epistemic logic |$\textbf {PTEL}$| with both future and past temporal operators, with non-rigid set of agents and the operators for agents’ knowledge and for common knowledge and with probabilities defined on the sets of runs and on the sets of possible worlds. A semantics is given by a class |${\scriptsize{\rm Mod}}$| of Kripke-like models with possible worlds. We prove decidability of |$\textbf {PTEL}$| by showing that checking satisfiability of a formula in |${\scriptsize{\rm Mod}}$| is (...) equivalent to checking its satisfiability in a finite set of finitely representable structures. The same procedure can be applied to the class of all synchronous |${\scriptsize{\rm Mod}}$|-models. We give an upper complexity bound for the satisfiability problem for |${\scriptsize{\rm Mod}}$|⁠. (shrink)

Temporal justification logic is a new family of temporal logics of knowledge in which the knowledge of agents is modelled using a justification logic. In this paper, we present various temporal justification logics involving both past and future time modalities. We combine Artemov’s logic of proofs with linear temporal logic with past, and we also investigate several principles describing the interaction of justification and time. We present two kinds of semantics for our temporal justification logics, one based on interpreted systems (...) and Fitting models and the other based on Mkrtychev models, and further, we establish soundness and completeness. We show that the internalization property holds in some of the temporal justification logics. We further investigate the two well-known epistemic-temporal notions of no forgetting and no learning in the framework of justification logics. Finally, we present temporal justification logics that avoid the logical omniscience problem. (shrink)

The language of linear temporal logic can be interpreted on the class of dynamic topological systems, giving rise to the intuitionistic temporal logic ${\sf ITL}^{\sf c}_{\Diamond \forall }$, recently shown to be decidable by Fernández-Duque. In this article we axiomatize this logic, some fragments, and prove completeness for several familiar spaces.

The distributed temporal logic (DTL) is a logic for reasoning about temporal properties of distributed systems from the local point of view of the system’s agents, which are assumed to execute sequentially and to interact by means of synchronous event sharing. Different versions of DTL have been provided over the years for a number of different applications, reflecting different perspectives on how non-local information can be accessed by each agent. In this paper, we propose an automata-theoretic model checking algorithm for (...) DTL. To this end, we propose a notion of distributed transition system that will be used to specify the system to be verified. The properties that the system should meet are specified in DTL. In order to capture the models of these properties, we propose the notions of generalized distributed Büchi automaton and of distributed Büchi automaton. With these concepts, we are able to adapt results from automata-theoretic approaches to model checking in LTL to the distributed case. (shrink)

We present a variant of ATL with incomplete information which includes the distributed knowledge operators corresponding to synchronous action and perfect recall. The cooperation modalities assume the use the distributed knowledge of coalitions and accordingly refer to perfect recall incomplete information strategies. We propose a model-checking algorithm for the logic. It is based on techniques for games with imperfect information and partially observable objectives, and involves deciding emptiness for automata on infinite trees. We also propose an axiomatic system and prove (...) its completeness for a rather expressive subset. As for the constructs left outside this completely axiomatised subset, we present axioms by which they can be defined in the subset on the class of models in which every state has finitely many successors and give a complete axiomatisation for a “flat” subset of the logic with these constructs included. (shrink)

The paper offers a formalization of reasoning about distributed multi-agent systems. The presented propositional probabilistic temporal epistemic logic |$\textbf {PTEL}$| is developed in full detail: syntax, semantics, soundness and strong completeness theorems. As an example, we prove consistency of the blockchain protocol with respect to the given set of axioms expressed in the formal language of the logic. We explain how to extend |$\textbf {PTEL}$| to axiomatize the corresponding first-order logic.

We present an overview of linear-time temporal logics with Presburger constraints whose models are sequences of tuples of integers. Such formal specification languages are well-designed to specify and verify systems that can be modelled with counter systems. The paper recalls the general framework of LTL over concrete domains and presents the main decidability and complexity results related to fragments of Presburger LTL. Related formalisms are also briefly presented.