Temporal Logics over Linear Time Domains Are in PSPACE

Rabinovich, Alexander

doi:10.1007/978-3-642-15349-5_3

Cited by 8 publications

(6 citation statements)

References 16 publications

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“…PROOF. The upper bound follows from the reduction · ‡ above and the fact that PT L is PSPACEcomplete over (Z, <) [Rabinovich 2010;Reynolds 2010;Sistla and Clarke 1982]. The lower bound is an immediate consequence of the observation that T FPX DL-Lite N bool can encode formulas of the form θ ∧ ✷ * i (ϕ i → F ψ i ), where θ, the ϕ i and ψ i are conjunctions of propositional variables: satisfiability of such formulas is known to be PSPACE-hard (see e.g., [Gabbay et al 1994]).…”

Section: Complexity Of Tus Dl-lite N Bool and Its Fragmentsmentioning

confidence: 99%

“…We believe, however, that these fragments are of sufficient interest on their own, independently of temporal conceptual modelling and reasoning. Sistla and Clarke [1982] showed that full PT L is PSPACE-complete; see also [Halpern and Reif 1981;Lichtenstein et al 1985;Rabinovich 2010;Reynolds 2010]. Ono and Nakamura [1980] proved that for formulas with only ✷ F and ✸ F the satisfiability problem becomes NP-complete.…”

Section: Clausal Fragments Of Propositional Temporal Logicmentioning

confidence: 99%

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A Cookbook for Temporal Conceptual Data Modelling with Description Logics

Artale

Kontchakov

Ryzhikov

et al. 2014

ACM Trans. Comput. Logic

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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. The logics are interpreted over the Cartesian products of object domains and the flow of time (Z, <), satisfying the constant domain assumption. Concept and role inclusions of the TBox hold at all moments of time (globally) and data assertions of the ABox hold at specified moments of time. To express temporal constraints of conceptual data models, the languages are equipped with flexible and rigid roles, standard future and past temporal operators on concepts and operators 'always' and 'sometime' on roles. The most expressive of our temporal description logics (which can capture lifespan cardinalities and either qualitative or quantitative evolution constraints) turns out to be undecidable. However, by omitting some of the temporal operators on concepts/roles or by restricting the form of concept inclusions we construct logics whose complexity ranges between NLOGSPACE and PSPACE. These positive results are 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.

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Section: Complexity Of Tus Dl-lite N Bool and Its Fragmentsmentioning

confidence: 99%

Section: Clausal Fragments Of Propositional Temporal Logicmentioning

confidence: 99%

A Cookbook for Temporal Conceptual Data Modelling with Description Logics

Artale

Kontchakov

Ryzhikov

et al. 2014

ACM Trans. Comput. Logic

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“…For instance the uniform satisfiability problem is pspace-complete and we obtain alternative proofs for results in [DN07]. Recent results about the polynomial space upper bound for LTL over various classes of linear orderings can be found in [Rab10a,Rab10b] by using the so-called composition technique and the automata-based technique used in this paper.…”

Section: Resultsmentioning

confidence: 88%

“…More precisely, satisfiability for LTL(U, S) augmented with future and past Stavi operators is in 2expspace [Cri09]. Nevertheless, complexity of LTL(U, S) over the class of linear orderings has been recently solved: for any temporal logic with a finite set of modalities definable in the existential fragment of secondorder logic has a pspace satisfiability problem over the class of linear orderings [Rab10a,Rab10b] (see also [Rey10a]). Moreover, observe that LTL(U, S) over the reals has been recently shown in pspace in [Rey10a], which allows us to obtain in a different way that LTL(U, S) over the countable ordinals is in pspace (see the full arguments in [Rab10a, Section 13]).…”

Section: Related Workmentioning

confidence: 99%

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

Demri¹,

Rabinovich²

2010

Logical Methods in Computer Science

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Abstract. 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 α and a formula, we can decide in pspace whether the formula has a model over α. In order to show these results, we introduce a class of simple ordinal automata, as expressive as Büchi 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.

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“…The tight complexity bounds in Table 1 show how the complexity of the satisfiability problem for LTL-formulas depends on the form of clauses and the available temporal operators. The PSpace upper bound for LTL ✷, bool is wellknown [17,25,23,24]; the matching lower bound can be obtained already for LTL ✷, horn without ✷ F and ✷ P by a standard encoding of deterministic Turing machines with polynomial tape [9]. The NP upper bound for LTL ✷ bool is also well-known [21], and the PTime and NLogSpace lower bounds for LTL * ✷ horn and LTL * ✷ core coincide with the complexity of the respective non-temporal languages.…”

Section: Introductionmentioning

confidence: 99%

The Complexity of Clausal Fragments of LTL

Artale

Kontchakov

Ryzhikov

et al. 2013

Logic for Programming, Artificial Intelligence, and Reasoning

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Abstract. 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.

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Temporal Logics over Linear Time Domains Are in PSPACE

Cited by 8 publications

References 16 publications

A Cookbook for Temporal Conceptual Data Modelling with Description Logics

A Cookbook for Temporal Conceptual Data Modelling with Description Logics

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

The Complexity of Clausal Fragments of LTL

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