99 research outputs found
An Optimal Decision Procedure for MPNL over the Integers
Interval temporal logics provide a natural framework for qualitative and
quantitative temporal reason- ing over interval structures, where the truth of
formulae is defined over intervals rather than points. In this paper, we study
the complexity of the satisfiability problem for Metric Propositional Neigh-
borhood Logic (MPNL). MPNL features two modalities to access intervals "to the
left" and "to the right" of the current one, respectively, plus an infinite set
of length constraints. MPNL, interpreted over the naturals, has been recently
shown to be decidable by a doubly exponential procedure. We improve such a
result by proving that MPNL is actually EXPSPACE-complete (even when length
constraints are encoded in binary), when interpreted over finite structures,
the naturals, and the in- tegers, by developing an EXPSPACE decision procedure
for MPNL over the integers, which can be easily tailored to finite linear
orders and the naturals (EXPSPACE-hardness was already known).Comment: In Proceedings GandALF 2011, arXiv:1106.081
Decidability of the interval temporal logic ABBar over the natural numbers
In this paper, we focus our attention on the interval temporal logic of the
Allen's relations "meets", "begins", and "begun by" (ABBar 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 point-based temporal
logic, such as the until operator, and to encode relevant metric constraints.
Then, we prove that the satisfiability problem for ABBar over natural numbers
is decidable by providing a small model theorem based on an original
contraction method. Finally, we prove the EXPSPACE-completeness of the proble
Crossing the Undecidability Border with Extensions of Propositional Neighborhood Logic over Natural Numbers
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 (NEXPTIME-complete) 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 first-order extension of the future fragment of PNL, obtained by replacing proposition letters by a restricted subclass of first-order formulae where only one variable is allowed, is undecidable (in contrast with the decidability of similar first-order extensions of point-based temporal logics)
Maximal decidable fragments of Halpern and Shoham's modal logic of intervals
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 (AAbarBBbar logic).
AAbarBBbar properly extends interesting interval temporal
logics recently investigated in the literature, such as the
logic BBbar of Allen's ``begun by/begins'' relations and
propositional neighborhood logic AAbar, in its many
variants (including metric ones). We prove that the satisfiability
problem for AAbarBBbar, interpreted over finite linear orders,
is decidable, but not primitive recursive (as a matter of fact,
AAbarBBbar turns out to be maximal with respect to decidability). Then, we show that it becomes undecidable when AAbarBBbar 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, Q, and R
Maximal decidable fragments of Halpern and Shoham's modal logic of intervals
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 (AAbarBBbar logic).
AAbarBBbar properly extends interesting interval temporal
logics recently investigated in the literature, such as the
logic BBbar of Allen's ``begun by/begins'' relations and
propositional neighborhood logic AAbar, in its many
variants (including metric ones). We prove that the satisfiability
problem for AAbarBBbar, interpreted over finite linear orders,
is decidable, but not primitive recursive (as a matter of fact,
AAbarBBbar turns out to be maximal with respect to decidability). Then, we show that it becomes undecidable when AAbarBBbar 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, Q, and R
Decidability and complexity of the fragments of the modal logic of Allen's relations over the rationals
Interval temporal logics provide a natural framework for temporal
reasoning about interval structures over linearly
ordered domains, where intervals are taken as first-class
citizens. Their expressive power and computational behaviour
mainly depend on two parameters: the set of modalities they feature and
the linear orders over which they are interpreted. In this paper, we consider
all fragments of Halpern and Shoham's interval temporal logic hs
with a decidable satisfiability problem over the rationals,
and we provide a complete classification of them in
terms of their expressiveness and computational complexity by solving the last few
open problems
Undecidability of the Logic of Overlap Relation over Discrete Linear Orderings
5The validity/satisfiability problem for most propositional interval temporal logics is (highly) undecidable, under very weak assumptions on the class of interval structures in which they are interpreted. That, in particular, holds for most fragments of Halpern and Shoham’s interval modal logic HS. Still, decidability
is the rule for the fragments of HS with only one modal operator, based on an Allen’s relation. In this paper, we show that the logic O of the Overlap relation, when interpreted over discrete linear orderings, is an exception. The proof is based on a reduction from the undecidable octant tiling problem. This is one of the sharpest undecidability result for fragments of HS.openopenBRESOLIN Davide; DELLA MONICA Dario; GORANKO Valentin; MONTANARI Angelo; SCIAVICCO GuidoBresolin, Davide; DELLA MONICA, Dario; Goranko, Valentin; Montanari, Angelo; Sciavicco, Guid
Bounded variability of metric temporal logic
Deciding validity of Metric Temporal Logic (MTL) formulas is generally very complex and even undecidable over dense time domains; bounded variability is one of the several restrictions that have been proposed to bring decidability back. A temporal model has bounded variability if no more than v events occur over any time interval of length V, for constant parameters v and V. Previous work has shown that MTL validity over models with bounded variability is less complex—and often decidable—than MTL validity over unconstrained models. This paper studies the related problem of deciding whether an MTL formula has intrinsic bounded variability, that is whether it is satisfied only by models with bounded variability. The results of the paper are mainly negative: over dense time domains, the problem is mostly undecidable (even if with an undecidability degree that is typically lower than deciding validity); over discrete time domains, it is decidable with the same complexity as deciding validity. As a partial complement to these negative results, the paper also identifies MTL fragments where deciding bounded variability is simpler than validity, which may provide for a reduction in complexity in some practical cases
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