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Decidable and undecidable fragments of Halpern and Shoham’s interval temporal logic: towards a complete classification
 In Proc. of the 15th Int. Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR), volume 5330 of LNCS
, 2008
"... Abstract. 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 interv ..."
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Abstract. 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 secondorder 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. 1
Maximal decidable fragments of Halpern and Shoham’s modal logic of intervals
, 2010
"... Abstract. In this paper, we focus our attention on the fragment of Halpern and Shoham’s modal logic of intervals (HS) that features four modal operators corresponding to the relations “meets”, “met by”, “begun by”, and “begins ” of Allen’s interval algebra (AĀBB ̄ logic). AĀBB̄ properly extends i ..."
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Abstract. In this paper, we focus our attention on the fragment of Halpern and Shoham’s modal logic of intervals (HS) that features four modal operators corresponding to the relations “meets”, “met by”, “begun by”, and “begins ” of Allen’s interval algebra (AĀBB ̄ logic). AĀBB̄ properly extends interesting interval temporal logics recently investigated in the literature, such as the logic BB ̄ of Allen’s “begun by/begins ” relations and propositional neighborhood logic AĀ, in its many variants (including metric ones). We prove that the satisfiability problem for AĀBB̄, interpreted over finite linear orders, is decidable, but not primitive recursive (as a matter of fact, AĀBB ̄ turns out to be maximal with respect to decidability). Then, we show that it becomes undecidable when AĀBB ̄ is interpreted over classes of linear orders that contains at least one linear order with an infinitely ascending sequence, thus including the natural time flows N, Z, and R. 1
Metric propositional neighborhood logics on natural numbers
 SOFTW SYST MODEL
, 2011
"... Interval logics formalize temporal reasoning on interval structures over linearly (or partially) ordered domains, where time intervals are the primitive ontological entities and truth of formulae is defined relative to time intervals, rather than time points. In this paper, we introduce and study ..."
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Cited by 11 (7 self)
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Interval logics formalize temporal reasoning on interval structures over linearly (or partially) ordered domains, where time intervals are the primitive ontological entities and truth of formulae is defined relative to time intervals, rather than time points. In this paper, we introduce and study Metric Propositional Neighborhood Logic (MPNL) over natural numbers. MPNL features two modalities referring, respectively, to an interval that is “met by” the current one and to an interval that “meets” the current one, plus an infinite set of length constraints, regarded as atomic propositions, to constrain the length of intervals. We
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 undecidability. 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 cl ..."
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Cited by 7 (5 self)
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Unlike the Moon, the dark side of interval temporal logics is the one we usually see: their ubiquitous undecidability. 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 HalpernShoham 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 undecidability results, this work contributes to delineate the border of the dark side of interval temporal logics quite sharply.
Expressiveness of the Interval Logics of Allen’s Relations on the Class of all Linear Orders: Complete Classification
 PROCEEDINGS OF THE TWENTYSECOND INTERNATIONAL JOINT CONFERENCE ON ARTIFICIAL INTELLIGENCE
, 2011
"... We compare the expressiveness of the fragments of Halpern and Shoham’s interval logic (HS), i.e., of all interval logics with modal operators associated with Allen’s relations between intervals in linear orders. We establish a complete set of interdefinability equations between these modal operators ..."
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Cited by 5 (4 self)
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We compare the expressiveness of the fragments of Halpern and Shoham’s interval logic (HS), i.e., of all interval logics with modal operators associated with Allen’s relations between intervals in linear orders. We establish a complete set of interdefinability equations between these modal operators, and thus obtain a complete classification of the family of 2 12 fragments of HS with respect to their expressiveness. Using that result and a computer program, we have found that there are 1347 expressively different such interval logics over the class of all linear orders.
DECIDABILITY OF THE INTERVAL TEMPORAL LOGIC ABB OVER THE NATURAL NUMBERS
, 2010
"... In this paper, we focus our attention on the interval temporal logic of the Allen’s relations “meets”, “begins”, and “begun by” (ABB for short), interpreted over natural numbers. We first introduce the logic and we show that it is expressive enough to model distinctive interval properties, such as a ..."
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Cited by 3 (3 self)
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In this paper, we focus our attention on the interval temporal logic of the Allen’s relations “meets”, “begins”, and “begun by” (ABB for short), interpreted over natural numbers. We first introduce the logic and we show that it is expressive enough to model distinctive interval properties, such as accomplishment conditions, to capture basic modalities of pointbased temporal logic, such as the until operator, and to encode relevant metric constraints. Then, we prove that the satisfiability problem for ABB over natural numbers is decidable by providing a small model theorem based on an original contraction method. Finally, we prove the EXPSPACEcompleteness of the problem.
Decidability of the interval temporal logic AĀBB̄ over the rationals
, 2014
"... Abstract. The classification of the fragments of Halpern and Shoham’s logic with respect to decidability/undecidability of the satisfiability problem is now very close to the end. We settle one of the few remaining questions concerning the fragment AĀBB̄, which comprises Allen’s interval relation ..."
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Abstract. The classification of the fragments of Halpern and Shoham’s logic with respect to decidability/undecidability of the satisfiability problem is now very close to the end. We settle one of the few remaining questions concerning the fragment AĀBB̄, which comprises Allen’s interval relations “meets ” and “begins ” and their symmetric versions. We already proved that AĀBB ̄ is decidable over the class of all finite linear orders and undecidable over ordered domains isomorphic to N. In this paper, we first show that AĀBB ̄ is undecidable over R and over the class of all Dedekindcomplete linear orders. We then prove that the logic is decidable over Q and over the class of all linear orders. 1
Tractable Interval Temporal Propositional and Description Logics
"... We design a tractable Horn fragment of the HalpernShoham temporal logic and extend it to intervalbased temporal description logics, instance checking in which is Pcomplete for both combined and data complexity. ..."
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We design a tractable Horn fragment of the HalpernShoham temporal logic and extend it to intervalbased temporal description logics, instance checking in which is Pcomplete for both combined and data complexity.