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**11 - 17**of**17**### Interval Temporal Logics:a Journey

"... 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 interv ..."

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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 point-based 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

### Decidability of the Logics of the Reflexive Sub-interval and Super-interval Relations over Finite Linear Orders

"... Abstract—An interval temporal logic is a propositional, multimodal logic interpreted over interval structures of partial orders. The semantics of each modal operator are given in the standard way with respect to one of the natural accessibility relations defined on such interval structures. In this ..."

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Abstract—An interval temporal logic is a propositional, multimodal logic interpreted over interval structures of partial orders. The semantics of each modal operator are given in the standard way with respect to one of the natural accessibility relations defined on such interval structures. In this paper, we consider the modal operators based on the (reflexive) subinterval relation and the (reflexive) super-interval relation. We show that the satisfiability problems for the interval temporal logics featuring either or both of these modalities, interpreted over interval structures of finite linear orders, are all PSPACEcomplete. These results fill a gap in the known complexity results for interval temporal logics. Keywords-interval temporal logic; decidability; computational complexity I.

### I T L: J

"... We discuss a family of modal logics for reasoning about relational struc-tures of intervals over (usually) linear orders, with modal operators asso-ciated 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 struc-tures of intervals over (usually) linear orders, with modal operators asso-ciated 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 in-terval logics as compared to point-based ones. Without purporting to pro-vide 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

### 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.

### The Last Paper on the Halpern–Shoham Interval Temporal Logic Draft (October 22, 2010)

"... The Halpern–Shoham logic is a modal logic of time in-tervals. Some effort has been put in last ten years to clas-sify fragments of this beautiful logic with respect to decid-ability of its satisfiability problem. We contribute to this effort by showing – what we believe is quite an unexpected result ..."

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The Halpern–Shoham logic is a modal logic of time in-tervals. Some effort has been put in last ten years to clas-sify fragments of this beautiful logic with respect to decid-ability of its satisfiability problem. We contribute to this effort by showing – what we believe is quite an unexpected result – that the logic of subintervals, the fragment of the Halpern–Shoham where only the operator “during”, or D, is allowed, is undecidable over discrete structures. This is surprising as this logic is decidable over dense orders [14] and its reflexive variant is known to be decidable over dis-crete structures [13]. Our result subsumes a lot of previ-ous results for the discrete case, like the undecidability for

### DL-Lite and Interval Temporal Logics: a Marriage Proposal (extended version)

, 2014

"... Description logics [10] (DLs) are widely-used logical formalisms for knowl-edge representation, where the domain of interest is structured in concepts whose properties are specified by roles. Complex concepts and role expres-sions are constructed, starting from atomic ones, by applying suitable (log ..."

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Description logics [10] (DLs) are widely-used logical formalisms for knowl-edge representation, where the domain of interest is structured in concepts whose properties are specified by roles. Complex concepts and role expres-sions are constructed, starting from atomic ones, by applying suitable (log-