7 research outputs found
G\"odel-Dummett linear temporal logic
We investigate a version of linear temporal logic whose propositional
fragment is G\"odel-Dummett logic (which is well known both as a
superintuitionistic logic and a t-norm fuzzy logic). We define the logic using
two natural semantics: first a real-valued semantics, where statements have a
degree of truth in the real unit interval and second a `bi-relational'
semantics. We then show that these two semantics indeed define one and the same
logic: the statements that are valid for the real-valued semantics are the same
as those that are valid for the bi-relational semantics. This G\"odel temporal
logic does not have any form of the finite model property for these two
semantics: there are non-valid statements that can only be falsified on an
infinite model. However, by using the technical notion of a quasimodel, we show
that every falsifiable statement is falsifiable on a finite quasimodel,
yielding an algorithm for deciding if a statement is valid or not. Later, we
strengthen this decidability result by giving an algorithm that uses only a
polynomial amount of memory, proving that G\"odel temporal logic is
PSPACE-complete. We also provide a deductive calculus for G\"odel temporal
logic, and show this calculus to be sound and complete for the above-mentioned
semantics, so that all (and only) the valid statements can be proved with this
calculus.Comment: arXiv admin note: substantial text overlap with arXiv:2205.00574,
arXiv:2205.0518
The Stable Model Semantics of Datalog with Metric Temporal Operators
We introduce negation under the stable model semantics in DatalogMTL - a
temporal extension of Datalog with metric temporal operators. As a result, we
obtain a rule language which combines the power of answer set programming with
the temporal dimension provided by metric operators. We show that, in this
setting, reasoning becomes undecidable over the rational timeline, and
decidable in EXPSPACE in data complexity over the integer timeline. We also
show that, if we restrict our attention to forward-propagating programs,
reasoning over the integer timeline becomes PSPACE-complete in data complexity,
and hence, no harder than over positive programs; however, reasoning over the
rational timeline in this fragment remains undecidable. Under consideration in
Theory and Practice of Logic Programming (TPLP).Comment: Under consideration in Theory and Practice of Logic Programming
(TPLP
The stable model semantics of datalog with metric temporal operators
We introduce negation under the stable model semantics in DatalogMTL - a temporal extension of Datalog with metric temporal operators. As a result, we obtain a rule language which combines the power of answer set programming with the temporal dimension provided by metric operators. We show that, in this setting, reasoning becomes undecidable over the rational timeline, and decidable in EXPSPACE in data complexity over the integer timeline. We also show that, if we restrict our attention to forward-propagating programs, reasoning over the integer timeline becomes PSPACE-complete in data complexity, and hence, no harder than over positive programs; however, reasoning over the rational timeline in this fragment remains undecidable. Under consideration in Theory and Practice of Logic Programming (TPLP)
Linear-Time Temporal Answer Set Programming
[Abstract]: In this survey, we present an overview on (Modal) Temporal Logic Programming in view of its application to Knowledge Representation and Declarative Problem Solving. The syntax of this extension of logic programs is the result of combining usual rules with temporal modal operators, as in Linear-time Temporal Logic (LTL). In the paper, we focus on the main recent results of the non-monotonic formalism called Temporal Equilibrium Logic (TEL) that is defined for the full syntax of LTL but involves a model selection criterion based on Equilibrium Logic, a well known logical characterization of Answer Set Programming (ASP). As a result, we obtain a proper extension of the stable models semantics for the general case of temporal formulas in the syntax of LTL. We recall the basic definitions for TEL and its monotonic basis, the temporal logic of Here-and-There (THT), and study the differences between finite and infinite trace length. We also provide further useful results, such as the translation into other formalisms like Quantified Equilibrium Logic and Second-order LTL, and some techniques for computing temporal stable models based on automata constructions. In the remainder of the paper, we focus on practical aspects, defining a syntactic fragment called (modal) temporal logic programs closer to ASP, and explaining how this has been exploited in the construction of the solver telingo, a temporal extension of the well-known ASP solver clingo that uses its incremental solving capabilities.Xunta de Galicia; ED431B 2019/03We are thankful to the anonymous reviewers for their thorough work and their useful
suggestions that have helped to improve the paper. A special thanks goes to Mirosaw
Truszczy´nski for his support in improving the quality of our paper. We are especially
grateful to David Pearce, whose help and collaboration on Equilibrium Logic was the
seed for a great part of the current paper. This work was partially supported by MICINN,
Spain, grant PID2020-116201GB-I00, Xunta de Galicia, Spain (GPC ED431B 2019/03),
R´egion Pays de la Loire, France, (projects EL4HC and etoiles montantes CTASP), European
Union COST action CA-17124, and DFG grants SCHA 550/11 and 15, Germany