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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
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.
Hybrid Metric Propositional Neighborhood Logics with Interval Length Binders
"... We investigate the question of how much hybrid machinery can be added to the interval neighbourhood logic PNL and its metric extension MPNL without losing the decidability of their satisfiability problem in N. In particular, we consider the natural hybrid extension of MPNL obtained by adding binders ..."
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We investigate the question of how much hybrid machinery can be added to the interval neighbourhood logic PNL and its metric extension MPNL without losing the decidability of their satisfiability problem in N. In particular, we consider the natural hybrid extension of MPNL obtained by adding binders on integer variables ranging over lengths of intervals, thus enabling storage of the length of the current interval and further references to it. We show that even a very weak natural fragment of such extensions becomes undecidable, which is somewhat surprising, being in contrast with the decidability of MPNL, which can be seen as a hybrid language with length constraints only involving constants over interval lengths. These results show that MPNL itself is, in this sense, a maximal decidable (weakly) hybrid extension of PNL.