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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
Undecidability of interval temporal logics with the Overlap modality
 In Proc. of 16th International Symposium on Temporal Representation and Reasoning (TIME), 88– 95. IEEE Computer Society Press, 2009. 2828 Della Monica D., Goranko V., Montanari A., Sciavicco G.: Crossing
"... We investigate fragments of HalpernShoham’s interval logic HS involving the modal operators for the relations of left or right overlap of intervals. We prove that most of these fragments are undecidable, by employing a nontrivial reduction from the octant tiling problem. 1. ..."
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Cited by 8 (7 self)
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We investigate fragments of HalpernShoham’s interval logic HS involving the modal operators for the relations of left or right overlap of intervals. We prove that most of these fragments are undecidable, by employing a nontrivial reduction from the octant tiling problem. 1.
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.
Undecidability of the logic of Overlap relation over discrete linear orderings
"... The validity/satisfiability problem for most propositional interval temporal logics is (highly) undecidable, under very weak assumptions on the class of interval structures in which they are interpreted. That, in particular, holds for most fragments of Halpern and Shoham’s interval modal logic HS. S ..."
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Cited by 5 (4 self)
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The validity/satisfiability problem for most propositional interval temporal logics is (highly) undecidable, under very weak assumptions on the class of interval structures in which they are interpreted. That, in particular, holds for most fragments of Halpern and Shoham’s interval modal logic HS. Still, decidability is the rule for the fragments of HS with only one modal operator, based on an Allen’s relation. In this paper, we show that the logic O of the Overlap relation, when interpreted over discrete linear orderings, is an exception. The proof is based on a reduction from the undecidable octant tiling problem. This is one of the sharpest undecidability result for fragments of HS.
Right propositional neighborhood logic over natural numbers with integer constraints for interval lengths
 In Proc. of the 7th IEEE Int. Conference on Software Engineering and Formal Methods (SEFM
, 2009
"... Interval temporal logics are based on interval structures over linearly (or partially) ordered domains, where time intervals, rather than time instants, are the primitive ontological entities. In this paper we introduce and study Right Propositional Neighborhood Logic over natural numbers with int ..."
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Cited by 4 (1 self)
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Interval temporal logics are based on interval structures over linearly (or partially) ordered domains, where time intervals, rather than time instants, are the primitive ontological entities. In this paper we introduce and study Right Propositional Neighborhood Logic over natural numbers with integer constraints for interval lengths, which is a propositional interval temporal logic featuring a modality for the ‘right neighborhood ’ relation between intervals and explicit integer constraints for interval lengths. We prove that it has the bounded model property with respect to ultimately periodic models and is therefore decidable. In addition, we provide an EXPSPACE procedure for satisfiability checking and we prove EXPSPACEhardness by a reduction from the exponential corridor tiling problem. 1
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.
Crossing the undecidability border with extensions of propositional neighborhood logic over natural numbers
 Journal of Universal Computer Science
"... Abstract: Propositional Neighborhood Logic (PNL) is an interval temporal logic featuring two modalities corresponding to the relations of right and left neighborhood between two intervals on a linear order (in terms of Allen’s relations, meets and met by). Recently, it has been shown that PNL interp ..."
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Cited by 1 (1 self)
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Abstract: Propositional Neighborhood Logic (PNL) is an interval temporal logic featuring two modalities corresponding to the relations of right and left neighborhood between two intervals on a linear order (in terms of Allen’s relations, meets and met by). Recently, it has been shown that PNL interpreted over several classes of linear orders, including natural numbers, is decidable (NEXPTIMEcomplete) and that some of its natural extensions preserve decidability. Most notably, this is the case with PNL over natural numbers extended with a limited form of metric constraints and with the future fragment of PNL extended with modal operators corresponding to Allen’s relations begins, begun by, and before. This paper aims at demonstrating that PNL and its metric version MPNL, interpreted over natural numbers, are indeed very close to the border with undecidability, and even relatively weak extensions of them become undecidable. In particular, we show that (i) the addition of binders on integer variables ranging over interval lengths makes the resulting hybrid extension of MPNL undecidable, and (ii) a very weak firstorder extension of the future fragment of PNL, obtained by replacing proposition letters by a restricted subclass of firstorder formulae where only one variable is allowed, is undecidable (in contrast with the decidability of similar firstorder extensions of pointbased temporal logics).
I T L: J
"... We discuss a family of modal logics for reasoning about relational structures of intervals over (usually) linear orders, with modal operators associated with the various binary relations between such intervals, known as Allen’s interval relations. The formulae of these logics are evaluated at inte ..."
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We discuss a family of modal logics for reasoning about relational structures of intervals over (usually) linear orders, with modal operators associated with the various binary relations between such intervals, known as Allen’s interval relations. The formulae of these logics are evaluated at intervals rather than points and the main effect of that semantic feature is substantially higher expressiveness and computational complexity of the interval logics as compared to pointbased ones. Without purporting to provide a comprehensive survey of the field, we take the reader to a journey through the main developments in it over the past 10 years and outline some landmark results on expressiveness and (un)decidability of the satisfiability problem for the family of interval logics. 1