9 research outputs found
A decidable weakening of Compass Logic based on cone-shaped cardinal directions
We introduce a modal logic, called Cone Logic, whose formulas describe
properties of points in the plane and spatial relationships between them.
Points are labelled by proposition letters and spatial relations are induced by
the four cone-shaped cardinal directions. Cone Logic can be seen as a weakening
of Venema's Compass Logic. We prove that, unlike Compass Logic and other
projection-based spatial logics, its satisfiability problem is decidable
(precisely, PSPACE-complete). We also show that it is expressive enough to
capture meaningful interval temporal logics - in particular, the interval
temporal logic of Allen's relations "Begins", "During", and "Later", and their
transposes
Satisfiability and Model Checking for the Logic of Sub-Intervals under the Homogeneity Assumption
The expressive power of interval temporal logics (ITLs) makes them really
fascinating, and one of the most natural choices as specification and planning
language. However, for a long time, due to their high computational complexity,
they were considered not suitable for practical purposes. The recent discovery
of several computationally well-behaved ITLs has finally changed the scenario.
In this paper, we investigate the finite satisfiability and model checking
problems for the ITL D featuring the sub-interval relation, under the
homogeneity assumption (that constrains a proposition letter to hold over an
interval if and only if it holds over all its points). First we prove that the
satisfiability problem for D, over finite linear orders, is PSPACE-complete;
then we show that its model checking problem, over finite Kripke structures, is
PSPACE-complete as well. The paper enrich the set of tractable interval
temporal logics with a meaningful representative.Comment: arXiv admin note: text overlap with arXiv:1901.0388
PSPACE-completeness of the temporal logic of sub-intervals and suffixes
In this paper, we prove PSPACE-completeness of the finite satisfiability and model checking problems for the fragment of Halpern and Shoham interval logic with modality 〈E〉, for the “suffix” relation on pairs of intervals, and modality 〈D〉, for the “sub-interval” relation, under the homogeneity assumption. The result significantly improves the EXPSPACE upper bound recently established for the same fragment, and proves the rather surprising fact that the complexity of the considered problems does not change when we add either the modality for suffixes (〈E〉) or, symmetrically, the modality for prefixes (〈B〉) to the logic of sub-intervals (featuring only 〈D〉)
A decidable weakening of Compass Logic based on cone-shaped cardinal directions
We introduce a modal logic, called Cone Logic, whose formulas describeproperties of points in the plane and spatial relationships between them.Points are labelled by proposition letters and spatial relations are induced bythe four cone-shaped cardinal directions. Cone Logic can be seen as a weakeningof Venema's Compass Logic. We prove that, unlike Compass Logic and otherprojection-based spatial logics, its satisfiability problem is decidable(precisely, PSPACE-complete). We also show that it is expressive enough tocapture meaningful interval temporal logics - in particular, the intervaltemporal logic of Allen's relations "Begins", "During", and "Later", and theirtransposes
Tableaux for Logics of Subinterval Structures over Dense Orderings
In this article, we develop tableau-based decision procedures for the logics of subinterval structures over dense linear orderings. In particular, we consider the two difficult cases: the relation of strict subintervals (with both endpoints strictly inside the current interval) and the relation of proper subintervals (that can share one endpoint with the current interval). For each of these logics, we establish a small pseudo-model property and construct a sound, complete and terminating tableau that searches systematically for existence of such a pseudo-model satisfying the input formulas. Both constructions are non-trivial, but the latter is substantially more complicated because of the presence of beginning and ending subintervals which require special treatment. We prove PSPACE completeness for both procedures and implement them in the generic tableau-based theorem prover Lotrec
Tableaux for logics of subinterval structures over dense orderings
In this article,we develop tableau-based decision procedures for the logics of subinterval structures over dense linear orderings. In particular, we consider the two dif\ufb01cult cases: the relation of strict subintervals (with both endpoints strictly inside the current interval) and the relation of proper subintervals (that can share one endpointwith the current interval). For each of these logics, we establish a small pseudo-model property and construct a sound, complete and terminating tableau that searches systematically for existence of such a pseudo-model satisfying the input formulas. Both constructions are non-trivial, but the latter is substantially more complicated because of the presence of beginning and ending subintervals which require special treatment. We prove PSPACE completeness for both procedures and implement them in the generic tableau-based theorem prover Lotrec