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23
Twovariable logic on data words
, 2007
"... In a data word each position carries a label from a finite alphabet and a data value from some infinite domain. These models have been already considered in the realm of semistructured data, timed automata and extended temporal logics. It is shown that satisfiability for the twovariable firstorder ..."
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Cited by 35 (4 self)
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In a data word each position carries a label from a finite alphabet and a data value from some infinite domain. These models have been already considered in the realm of semistructured data, timed automata and extended temporal logics. It is shown that satisfiability for the twovariable firstorder logic FO 2 (∼,<,+1) is decidable over finite and over infinite data words, where ∼ is a binary predicate testing the data value equality and +1, < are the usual successor and order predicates. The complexity of the problem is at least as hard as Petri net reachability. Several extensions of the logic are considered, some remain decidable while some are undecidable.
Decidable and undecidable fragments of Halpern and Shoham’s interval temporal logic: towards a complete classification
 In Proc. of the 15th Int. Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR), volume 5330 of LNCS
, 2008
"... Abstract. Interval temporal logics are based on temporal structures where time intervals, rather than time instants, are the primitive ontological entities. They employ modal operators corresponding to various relations between intervals, known as Allen’s relations. Technically, validity in interv ..."
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Cited by 17 (15 self)
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Abstract. Interval temporal logics are based on temporal structures where time intervals, rather than time instants, are the primitive ontological entities. They employ modal operators corresponding to various relations between intervals, known as Allen’s relations. Technically, validity in interval temporal logics translates to dyadic secondorder logic, thus explaining their complex computational behavior. The full modal logic of Allen’s relations, called HS, has been proved to be undecidable by Halpern and Shoham under very weak assumptions on the class of interval structures, and this result was discouraging attempts for practical applications and further research in the field. A renewed interest has been recently stimulated by the discovery of interesting decidable fragments of HS. This paper contributes to the characterization of the boundary between decidability and undecidability of HS fragments. It summarizes known positive and negative results, it describes the main techniques applied so far in both directions, and it establishes a number of new undecidability results for relatively small fragments of HS. 1
Propositional Interval Neighborhood Logics: Expressiveness, Decidability, and Undecidable Extensions
"... In this paper, we investigate the expressiveness of the variety of propositional interval neighborhood logics (PNL), we establish their decidability on linearly ordered domains and some important subclasses, and we prove undecidability of a number of extensions of PNL with additional modalities ov ..."
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Cited by 16 (11 self)
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In this paper, we investigate the expressiveness of the variety of propositional interval neighborhood logics (PNL), we establish their decidability on linearly ordered domains and some important subclasses, and we prove undecidability of a number of extensions of PNL with additional modalities over interval relations. All together, we show that PNL form a quite expressive and nearly maximal decidable fragment of HalpernShoham’s interval logic HS.
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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Cited by 14 (10 self)
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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
Twovariable logic with two order relations
 In CSL, volume 6247 of Lecture Notes in Computer Science
"... The finite satisfiability problem for twovariable logic over structures with unary relations and two order relations is investigated. Firstly, decidability is shown for structures with one total preorder relation and one linear order relation. More specifically, we show that this problem is compl ..."
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Cited by 12 (3 self)
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The finite satisfiability problem for twovariable logic over structures with unary relations and two order relations is investigated. Firstly, decidability is shown for structures with one total preorder relation and one linear order relation. More specifically, we show that this problem is complete for EXPSPACE. As a consequence, the same upper bound applies to the case of two linear orders. Secondly, we prove undecidability for structures with two total preorder relations as well as for structures with one total preorder and two linear order relations. Further, we point out connections to other logics. Decidability is shown for twovariable logic on data words with orders on both positions and data values, but without a successor relation. We also study “partial models ” of compass and interval temporal logic and prove decidability for some of their fragments. 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
On decidability and expressiveness of propositional interval neighborhood logics
 In Proceedings of the International Symposium on Logical Foundations of Computer Science (LFCS), volume 4514 of Lecture Notes in Computer Science
, 2007
"... Abstract. Intervalbased temporal logics are an important research area in computer science and artificial intelligence. In this paper we investigate decidability and expressiveness issues for Propositional Neighborhood Logics (PNLs). We begin by comparing the expressiveness of the different PNLs. ..."
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Cited by 10 (7 self)
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Abstract. Intervalbased temporal logics are an important research area in computer science and artificial intelligence. In this paper we investigate decidability and expressiveness issues for Propositional Neighborhood Logics (PNLs). We begin by comparing the expressiveness of the different PNLs. Then, we focus on the most expressive one, namely, PNLpi+, and we show that it is decidable over various classes of linear orders by reducing its satisfiability problem to that of the twovariable fragment of firstorder logic with binary relations over linearly ordered domains, due to Otto. Next, we prove that PNLpi+ is expressively complete with respect to such a fragment. We conclude the paper by comparing PNLpi+ expressiveness with that of other intervalbased temporal logics.
On the Satisfiability of TwoVariable Logic over Data Words
"... Data trees and data words have been studied extensively in connection with XML reasoning. These are trees or words that, in addition to labels from a finite alphabet, carry labels from an infinite alphabet (data). While in general logics such as MSO or FO are undecidable for such extensions, decida ..."
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Cited by 9 (2 self)
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Data trees and data words have been studied extensively in connection with XML reasoning. These are trees or words that, in addition to labels from a finite alphabet, carry labels from an infinite alphabet (data). While in general logics such as MSO or FO are undecidable for such extensions, decidablity results for their fragments have been obtained recently, most notably for the twovariable fragments of FO and existential MSO. The proofs, however, are very long and nontrivial, and some of them come with no complexity guarantees. Here we give a much simplified proof of the decidability of twovariable logics for data words with the successor and dataequality predicates. In addition, the new proof provides several new fragments of lower complexity. The proof mixes databaseinspired constraints with encodings in Presburger arithmetic.
Optimal Tableaux for Right Propositional Neighborhood Logic over Linear Orders
 In Proc. of the 11th European Conference on Logics in AI, number 5293 in LNAI
"... Abstract. The study of interval temporal logics on linear orders is a meaningful research area in computer science and artificial intelligence. Unfortunately, even when restricted to propositional languages, most interval logics turn out to be undecidable. Decidability has been usually recovered by ..."
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Cited by 7 (6 self)
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Abstract. The study of interval temporal logics on linear orders is a meaningful research area in computer science and artificial intelligence. Unfortunately, even when restricted to propositional languages, most interval logics turn out to be undecidable. Decidability has been usually recovered by imposing severe syntactic and/or semantic restrictions. In the last years, tableaubased decision procedures have been obtained for logics of the temporal neighborhood and logics of the subinterval relation over specific classes of temporal structures. In this paper, we develop an optimal NEXPTIME tableaubased decision procedure for the future fragment of Propositional Neighborhood Logic over the whole class of linearly ordered domains. 1
TwoVariable Logic on 2Dimensional Structures
"... This paper continues the study of the twovariable fragment of firstorder logic (FO 2) over twodimensional structures, more precisely structures with two orders, their induced successor relations and arbitrarily many unary relations. Our main focus is on ordered data words which are finite sequence ..."
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Cited by 4 (2 self)
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This paper continues the study of the twovariable fragment of firstorder logic (FO 2) over twodimensional structures, more precisely structures with two orders, their induced successor relations and arbitrarily many unary relations. Our main focus is on ordered data words which are finite sequences from the set Σ×D where Σ is a finite alphabet and D is an ordered domain. These are naturally represented as labelled finite sets with a linear order ≤l and a total preorder ≤p. We introduce ordered data automata, an automaton model for ordered data words. An ordered data automaton is a composition of a finite state transducer and a finite state automaton over the product Boolean algebra of finite and cofinite subsets of N. We show that ordered data automata are equivalent to the closure of FO 2 (+1l, ≤p, +1p) under existential quantification of unary relations. Using this automaton model we prove that the finite satisfiability problem for this logic is decidable on structures where the ≤pequivalence classes are of bounded size. As a corollary, we obtain that finite satisfiability of FO 2 is decidable (and it is equivalent to the reachability problem of vector addition systems) on structures with two linear order successors and a linear order corresponding to one of the successors. Further we prove undecidability of FO 2 on several other twodimensional structures.