91 research outputs found
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
Interval-based Synthesis
We introduce the synthesis problem for Halpern and Shoham's modal logic of
intervals extended with an equivalence relation over time points, abbreviated
HSeq. In analogy to the case of monadic second-order logic of one successor,
the considered synthesis problem receives as input an HSeq formula phi and a
finite set Sigma of propositional variables and temporal requests, and it
establishes whether or not, for all possible evaluations of elements in Sigma
in every interval structure, there exists an evaluation of the remaining
propositional variables and temporal requests such that the resulting structure
is a model for phi. We focus our attention on decidability of the synthesis
problem for some meaningful fragments of HSeq, whose modalities are drawn from
the set A (meets), Abar (met by), B (begins), Bbar (begun by), interpreted over
finite linear orders and natural numbers. We prove that the fragment ABBbareq
is decidable (non-primitive recursive hard), while the fragment AAbarBBbar
turns out to be undecidable. In addition, we show that even the synthesis
problem for ABBbar becomes undecidable if we replace finite linear orders by
natural numbers.Comment: In Proceedings GandALF 2014, arXiv:1408.556
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 fragments have an
undecidable satisfiability problem. This discouraged the research in this area
until recently, when a number non-trivial 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, After, Later and their inverses Abar, Bbar, and Lbar. We know
from previous works that the combination ABBbarAbar is decidable only when
finite domains are considered (and undecidable elsewhere), and that ABBbar is
decidable over the natural numbers. We extend these results by showing that
decidability of ABBar can be further extended to capture the language
ABBbarLbar, which lays in between ABBar and ABBbarAbar, 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 complexity class
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
Metric propositional neighborhood logic with an equivalence relation
The propositional interval logic of temporal neighborhood (PNL for short) features two modalities that make it possible to access intervals adjacent to the right (modality \u27e8 A\u27e9) and to the left (modality \u27e8 A\uaf \u27e9) of the current interval. PNL stands at a central position in the realm of interval temporal logics, as it is expressive enough to encode meaningful temporal conditions and decidable (undecidability rules over interval temporal logics, while PNL is NEXPTIME-complete). Moreover, it is expressively complete with respect to the two-variable fragment of first-order logic extended with a linear order FO 2[<]. Various extensions of PNL have been studied in the literature, including metric, hybrid, and first-order ones. Here, we study the effects of the addition of an equivalence relation 3c to Metric PNL (MPNL 3c). We first show that the finite satisfiability problem for PNL extended with 3c is still NEXPTIME-complete. Then, we prove that the same problem for MPNL 3c can be reduced to the decidable 0\u20130 reachability problem for vector addition systems and vice versa (EXPSPACE-hardness immediately follows)
A Quantitative Extension of Interval Temporal Logic over Infinite Words
Model checking (MC) for Halpern and Shoham’s interval temporal logic HS has been recently investigated in a systematic way, and it is known to be decidable under three distinct semantics (state-based, trace-based and tree-based semantics), all of them assuming homogeneity in the propositional valuation. Here, we focus on the trace-based semantics, where the main semantic entities are the infinite execution paths (traces) of the given Kripke structure. We introduce a quantitative extension of HS over traces, called Difference HS (DHS), allowing one to express timing constraints on the difference among interval lengths (durations). We show that MC and satisfiability of full DHS are in general undecidable, so, we investigate the decidability border for these problems by considering natural syntactical fragments of DHS. In particular, we identify a maximal decidable fragment DHSsimple of DHS proving in addition that the considered problems for this fragment are at least 2Expspace-hard. Moreover, by exploiting new results on linear-time hybrid logics, we show that for an equally expressive fragment of DHSsimple, the problems are Expspace-complete. Finally, we provide a characterization of HS over traces by means of the one-variable fragment of a novel hybrid logic
- …