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**1 - 5**of**5**### 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

### 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

### 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

### 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