The complexity of linear-time temporal logic over the class of ordinals

We consider the temporal logic with since and until modalities. This temporal logic is expressively equivalent over the class of ordinals to first-order logic by Kamp's theorem. We show that it has a PSPACE-complete satisfiability problem over the class of ordinals. Among the consequences of our proof, we show that given the code of some countable ordinal alpha and a formula, we can decide in PSPACE whether the formula has a model over alpha. In order to show these results, we introduce a class of simple ordinal automata, as expressive as B\"uchi ordinal automata. The PSPACE upper bound for the satisfiability problem of the temporal logic is obtained through a reduction to the nonemptiness problem for the simple ordinal automata.Comment: Accepted for publication in LMC

Two-Variable Logic over Countable Linear Orderings

We study the class of languages of finitely-labelled countable linear orderings definable in two-variable first-order logic. We give a number of characterisations, in particular an algebraic one in terms of circle monoids, using equations. This generalises the corresponding characterisation, namely variety DA, over finite words to the countable case. A corollary is that the membership in this class is decidable: for instance given an MSO formula it is possible to check if there is an equivalent two-variable logic formula over countable linear orderings. In addition, we prove that the satisfiability problems for two-variable logic over arbitrary, countable, and scattered linear orderings are NEXPTIME-complete

The complexity of clausal fragments of LTL

We introduce and investigate a number of fragments of propositional temporal logic LTL over the flow of time (ℤ, <). The fragments are defined in terms of the available temporal operators and the structure of the clausal normal form of the temporal formulas. We determine the computational complexity of the satisfiability problem for each of the fragments, which ranges from NLogSpace to PTime, NP and PSpace

Decidability of Difference Logic over the Reals with Uninterpreted Unary Predicates

First-order logic fragments mixing quantifiers, arithmetic, and uninterpreted predicates are often undecidable, as is, for instance, Presburger arithmetic extended with a single uninterpreted unary predicate. In the SMT world, difference logic is a quite popular fragment of linear arithmetic which is less expressive than Presburger arithmetic. Difference logic on integers with uninterpreted unary predicates is known to be decidable, even in the presence of quantifiers. We here show that (quantified) difference logic on real numbers with a single uninterpreted unary predicate is undecidable, quite surprisingly. Moreover, we prove that difference logic on integers, together with order on reals, combined with uninterpreted unary predicates, remains decidable.Comment: This is the preprint for the submission published in CADE-29. It also includes an additional detailed proof in the appendix. The Version of Record of this contribution will be published in CADE-2

A Hypersequent Calculus with Clusters for Tense Logic over Ordinals

Prior\u27s tense logic forms the core of linear temporal logic, with both past- and future-looking modalities. We present a sound and complete proof system for tense logic over ordinals. Technically, this is a hypersequent system, enriched with an ordering, clusters, and annotations. The system is designed with proof search algorithms in mind, and yields an optimal coNP complexity for the validity problem. It entails a small model property for tense logic over ordinals: every satisfiable formula has a model of order type at most omega^2. It also allows to answer the validity problem for ordinals below or exactly equal to a given one

A cookbook for temporal conceptual data modelling with description logic

We design temporal description logics suitable for reasoning about temporal conceptual data models and investigate their computational complexity. Our formalisms are based on DL-Lite logics with three types of concept inclusions (ranging from atomic concept inclusions and disjointness to the full Booleans), as well as cardinality constraints and role inclusions. In the temporal dimension, they capture future and past temporal operators on concepts, flexible and rigid roles, the operators `always' and `some time' on roles, data assertions for particular moments of time and global concept inclusions. The logics are interpreted over the Cartesian products of object domains and the flow of time (Z,<), satisfying the constant domain assumption. We prove that the most expressive of our temporal description logics (which can capture lifespan cardinalities and either qualitative or quantitative evolution constraints) turn out to be undecidable. However, by omitting some of the temporal operators on concepts/roles or by restricting the form of concept inclusions we obtain logics whose complexity ranges between PSpace and NLogSpace. These positive results were obtained by reduction to various clausal fragments of propositional temporal logic, which opens a way to employ propositional or first-order temporal provers for reasoning about temporal data models

On the expressiveness and monitoring of metric temporal logic

It is known that Metric Temporal Logic (MTL) is strictly less expressive than the Monadic First-Order Logic of Order and Metric (FO[<, +1]) when interpreted over timed words; this remains true even when the time domain is bounded a priori. In this work, we present an extension of MTL with the same expressive power as FO[<, +1] over bounded timed words (and also, trivially, over time-bounded signals). We then show that expressive completeness also holds in the general (time-unbounded) case if we allow the use of rational constants q ∈ Q in formulas. This extended version of MTL therefore yields a definitive real-time analogue of Kamp’s theorem. As an application, we propose a trace-length independent monitoring procedure for our extension of MTL, the first such procedure in a dense real-time setting

Insights into Modal Slash Logic and Modal Decidability

The present paper has a two-fold task. On the one hand, it aims to provide an overview on Independence friendly modal logic as defined in (Tulenheimo, 2003; Tulenheimo, 2004) and studied in a number of subsequent publications. For systematic reasons to be explained, the logic is here referred to as modal slash logic (MsL). On the other hand, we take a close look at a syntactic fragment of MsL, to be termed MsL0, first formulated in (Tulenheimo and Sevenster, 2006). We push the study of this logic deeper at several points: a model-theoretic criterion is presented which serves to tell when a formula of MsL0 is not truth-equivalent to any formula of basic modal logic (ML); the game-theoretic property of ‘bounded quasi-positionality' of MsL0 is studied in detail; an alternative syntax for MsL0 is discerned and the logic obtained is shown to enjoy the property of quasi-locality (generalizing the notion of locality familiar from ML); and we formulate an asymmetric bisimulation concept and use it to prove that MsL0 is not closed under complementation. Drawing from insights provided by the study of MsL0, we conclude by general observations about claims made on the ‘reasons' why various modal logics are computationally well-behaved

Programmation par contraintes sur les flux de données

We study the generalization of constraint programming on variables finite domains with variable flow. On the one hand, the flow of concepts, infinite sequences and infinite words have been the subject of numerous studies, and a goal is to achieve a state of the art covering language theory, classical and temporal logics as well as many related formalisms. The reconciliation performed with temporal logics is a first step towards unification formalisms on flows and temporal logics being themselves many, we establish a classification of these will allow the extrapolation of contributions to other contexts. The second objective is to identify the elements of these formalisms that allow the processing of satisfaction problems with the techniques of constraint programming on finite domain variables. Compared to the expressiveness of temporal logic, that of our formalism is more limited. This is due to the fact that constraint programming allows only the conjunction of constraints and requires integrating the disjunction in the notion of constraint propagator. Our formalism allows a gain in conciseness and reuse of the concept of propagator. The issue of generalization to more expressive logics is left open.Nous étudions la généralisation de la programmation par contraintes sur les variables à domaines finis aux variables flux. D'une part, les concepts de flux, de séquences infinies et de mots infinis ont fait l'objet de nombreux travaux, et un objectif consiste à réaliser un état de l'art qui couvre la théorie des langages, les logiques classiques et temporelles, ainsi que les nombreux formalismes apparentés. Le rapprochement effectué avec les logiques temporelles est un premier pas vers l'unification des formalismes sur les flux, et les logiques temporelles étant elles-même nombreuses, nous établissons une classification de celles-ci qui permettra l'extrapolation des contributions à d'autres contextes. Le second objectif consiste à identifier les éléments de ces formalismes qui permettent le traitement des problèmes de satisfaction avec les techniques de la programmation par contraintes sur les variables à domaines finis. Comparée à l'expressivité des logiques temporelles, celle de notre formalisme est plus limitée. Ceci est dû au fait que la programmation par contraintes ne permet que la conjonction de contraintes, et impose d'intégrer la disjonction dans la notion de propagateur de contraintes. Notre formalisme permet un gain en concision et la réutilisation de la notion de propagateur. La question de la généralisation à des logiques plus expressives est laissée ouverte