On Decidability and Expressiveness of Propositional Interval Neighborhood Logics (original) (raw)
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Propositional interval neighborhood temporal logics
2003
Logics for time intervals provide a natural framework for dealing with time in various areas of computer science and artificial intelligence, such as planning, natural language processing, temporal databases, and formal specification. In this paper we focus our attention on propositional interval temporal logics with temporal modalities for neighboring intervals over linear orders. We study the class of propositional neighborhood logics (PN L) over two natural semantics, respectively admitting and excluding point-intervals. First, we introduce interval neighborhood frames and we provide representation theorems for them; then, we develop complete axiomatic systems and semantic tableaux for logics in PN L. HS features four basic operators: B (begin) and E (end ), and their transposes B and E . Given a formula ϕ and an interval [d 0 , d 1 ], B ϕ holds at [d 0 , d 1 ] if ϕ holds at [d 0 , d 2 ], for some d 2 < d 1 , and E ϕ holds at [d 0 , d 1 ] if ϕ holds at [d 2 , d 1 ], for some d 2 > d 0 .
Interval Temporal Logics over Strongly Discrete Linear Orders: the Complete Picture
Interval temporal logics provide a general framework for temporal reasoning about interval structures over linearly ordered domains, where intervals are taken as the primitive ontological entities. In this paper, we identify all fragments of Halpern and Shoham's interval temporal logic HS with a decidable satisfiability problem over the class of strongly discrete linear orders. We classify them in terms of both their relative expressive power and their complexity. We show that there are exactly 44 expressively different decidable fragments, whose complexity ranges from NP to EXPSPACE. In addition, we identify some new undecidable fragments (all the remaining HS fragments were already known to be undecidable over strongly discrete linear orders). We conclude the paper by an analysis of the specific case of natural numbers, whose behavior slightly differs from that of the whole class of strongly discrete linear orders. The number of decidable fragments over natural numbers raises ...
Interval temporal logics over strongly discrete linear orders: Expressiveness and complexity
Theoretical Computer Science, 2014
ABSTRACT Interval temporal logics provide a natural framework for temporal reasoning about interval structures over linearly ordered domains, where intervals are taken as the primitive ontological entities. Their computational behavior mainly depends on two parameters: the set of modalities they feature and the linear orders over which they are interpreted. In this paper, we identify all fragments of Halpern and Shoham's interval temporal logic HS with a decidable satisfiability problem over the class of strongly discrete linear orders as well as over its relevant subclasses (the class of finite linear orders, ZZ, NN, and Z−Z−). We classify them in terms of both their relative expressive power and their complexity, which ranges from NP-completeness to non-primitive recursiveness.
Propositional interval neighborhood logics: Expressiveness, decidability, and undecidable extensions
In this paper, we investigate the expressiveness of the variety of propositional interval neighborhood logics (PNL), we establish their decidability on linearly ordered domains and some important subclasses, and we prove the undecidability of a number of extensions of PNL with additional modalities over interval relations. All together, we show that PNL form a quite expressive and nearly maximal decidable fragment of Halpern-Shoham's interval logic HS.
—Interval temporal logics are temporal logics that take time intervals, instead of time instants, as their primitive temporal entities. One of the most studied interval temporal logics is Halpern and Shoham's modal logic of time intervals (HS), which has a distinct modality for each binary relation between intervals over a linear order. As HS turns out to be undecidable over most classes of linear orders, the study of HS fragments, featuring a proper subset of HS modalities, is a major item in the research agenda for interval temporal logics. A characterization of HS fragments in terms of their relative expressive power has been given for the class of all linear orders. Unfortunately, there is no easy way to directly transfer such a result to other meaningful classes of linear orders. In this paper, we provide a complete classification of the expressiveness of HS fragments over the class of (all) dense linear orders.
Lecture Notes in Computer Science, 2008
Interval temporal logics are based on temporal structures where time intervals, rather than time instants, are the primitive ontological entities. They employ modal operators corresponding to various relations between intervals, known as Allen's relations. Technically, validity in interval temporal logics translates to dyadic second-order logic, thus explaining their complex computational behavior. The full modal logic of Allen's relations, called HS, has been proved to be undecidable by Halpern and Shoham under very weak assumptions on the class of interval structures, and this result was discouraging attempts for practical applications and further research in the field. A renewed interest has been recently stimulated by the discovery of interesting decidable fragments of HS. This paper contributes to the characterization of the boundary between decidability and undecidability of HS fragments. It summarizes known positive and negative results, it describes the main techniques applied so far in both directions, and it establishes a number of new undecidability results for relatively small fragments of HS.
Non-finite Axiomatizability and Undecidability of Interval Temporal Logics with C, D, and T
Lecture Notes in Computer Science, 2008
Interval logics are an important area of computer science. Although attention has been mainly focused on unary operators, an early work by introduced an expressively complete interval logic language called CDT, based on binary operators, which has many potential applications and a strong theoretical interest. Many very natural questions about CDT and its fragments, such as (non-)finite axiomatizability and (un-)decidability, are still open (as a matter of fact, only a few undecidability results, including the undecidability of CDT, are known). In this paper, we answer most of these questions, showing that almost all fragments of CDT, containing at least one binary operator, are neither finitely axiomatizable with standard rules nor decidable. A few cases remain open.
The Dark Side of Interval Temporal Logic: Sharpening the Undecidability Border
Unlike the Moon, the dark side of interval temporal logics is the one we usually see: their ubiquitous undesirability. Identifying minimal undecidable interval logics is thus a natural and important issue in the research agenda in the area. The decidability status of a logic often depends on the class of models (in our case, the class of interval structures)in which it is interpreted. In this paper, we have identified several new minimal undecidable logics amongst the fragments of Halpern-Shoham logic HS, including the logic of the overlaps relation, over the classes of all and finite linear orders, as well as the logic of the meet and subinterval relations, over the class of dense linear orders. Together with previous undecid ability results, this work contributes to delineate the border of the dark side of interval temporal logics quite sharply.
The Light Side of Interval Temporal Logic: The Bernays-Schönfinkel's Fragment of CDT
Decidability and complexity of the satisfiability problem for the logics of time intervals have been extensively studied in the last years. Even though most interval logics turnout to be undecidable, meaningful exceptions exist, such as the logics of temporal neighborhood and (some of) the logics of the subinterval relation. In this paper, we explore a different path to decidability: instead of restricting the set of modalities or imposing suitable semantic restrictions, we take the most expressive interval temporal logic studied so far, namely, Venema's CDT, and we suitably limit the nesting degree of modalities. The decidability of the satisfiability problem for the resulting CDT fragment is proved by embedding it into a well-known decidable prefix quantifier class of first-order logic, namely, the Bernays-Schonfinkel's class. In addition, we show that such a fragment is in fact NP-complete (theBernays-Schonfinkel's class is NEXPTIME-complete), and that any natural ext...
The light side of interval temporal logic: the Bernays-Schönfinkel fragment of CDT
Annals of Mathematics and Artificial Intelligence, 2013
Decidability and complexity of the satisfiability problem for the logics of time intervals have been extensively studied in the last years. Even though most interval logics turn out to be undecidable, meaningful exceptions exist, such as the logics of temporal neighborhood and (some of) the logics of the subinterval relation. In this paper, we explore a different path to decidability: instead of restricting the set of modalities or imposing suitable semantic restrictions, we take the most expressive interval temporal logic studied so far, namely, Venema's CDT, and we suitably limit the nesting degree of modalities. The decidability of the satisfiability problem for the resulting CDT fragment is proved by embedding it into a well-known decidable prefix quantifier class of first-order logic, namely, the Bernays-Schönfinkel's class. In addition, we show that such a fragment is in fact NP-complete (the Bernays-Schönfinkel's class is NEXPTIME-complete), and that any natural extension of it is undecidable.