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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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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
COMPLEXITY HIERARCHIES BEYOND ELEMENTARY
, 2013
"... We introduce a hierarchy of fastgrowing complexity classes and show its suitability for completeness statements of many non elementary problems. This hierarchy allows the classification of many decision problems with a nonelementary complexity, which occur naturally in logic, combinatorics, formal ..."
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We introduce a hierarchy of fastgrowing complexity classes and show its suitability for completeness statements of many non elementary problems. This hierarchy allows the classification of many decision problems with a nonelementary complexity, which occur naturally in logic, combinatorics, formal languages, verification, etc., with complexities ranging from simple towers of exponentials to Ackermannian and beyond.
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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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 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
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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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).
Decidability of the Logics of the Reflexive Subinterval and Superinterval 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) superinterval 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. Keywordsinterval temporal logic; decidability; computational complexity I.
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.
This work is licensed under the Creative Commons Attribution License. Interval Temporal Logics over Strongly Discrete Linear Orders: the Complete Picture
"... c © D. Bresolin et al. ..."
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Begin, After, and Later: a Maximal Decidable Interval Temporal Logic∗
"... Interval temporal logics (ITLs) are logics for reasoning about temporal statements expressed over intervals, i.e., periods of time. The most famous ITL studied so far is Halpern and Shoham’s HS, which is the logic of the thirteen Allen’s interval relations. Unfortunately, HS and most of its fragmen ..."
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Interval temporal logics (ITLs) are logics for reasoning about temporal statements expressed over intervals, i.e., periods of time. The most famous ITL studied so far is Halpern and Shoham’s HS, which is the logic of the thirteen Allen’s interval relations. Unfortunately, HS and most of its fragments have an undecidable satisfiability problem. This discouraged the research in this area until recently, when a number nontrivial decidable ITLs have been discovered. This paper is a contribution towards the complete classification of all different fragments of HS. We consider different combinations of the interval relations begins (B), after (A), later (L) and their inverses A, B and L. We know from previous works that the combinationABBA is decidable only when finite domains are considered (and undecidable elsewhere), and thatABB is decidable over the natural numbers. We extend these results by showing that decidability of ABB can be further extended to capture the language ABBL, which lies in between ABB and ABBA, and that turns out to be maximal w.r.t decidability over strongly discrete linear orders (e.g. finite orders, the naturals, the integers). We also prove that the proposed decision procedure is optimal with respect to the EXPSPACE complexity class. 1