243 research outputs found
Modal Logics of Topological Relations
Logical formalisms for reasoning about relations between spatial regions play
a fundamental role in geographical information systems, spatial and constraint
databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's
modal logic of time intervals based on the Allen relations, we introduce a
family of modal logics equipped with eight modal operators that are interpreted
by the Egenhofer-Franzosa (or RCC8) relations between regions in topological
spaces such as the real plane. We investigate the expressive power and
computational complexity of logics obtained in this way. It turns out that our
modal logics have the same expressive power as the two-variable fragment of
first-order logic, but are exponentially less succinct. The complexity ranges
from (undecidable and) recursively enumerable to highly undecidable, where the
recursively enumerable logics are obtained by considering substructures of
structures induced by topological spaces. As our undecidability results also
capture logics based on the real line, they improve upon undecidability results
for interval temporal logics by Halpern and Shoham. We also analyze modal
logics based on the five RCC5 relations, with similar results regarding the
expressive power, but weaker results regarding the complexity
Horn fragments of the Halpern-Shoham Interval Temporal Logic
We investigate the satisfiability problem for Horn fragments of the Halpern-Shoham interval temporal logic depending on the type (box or diamond) of the interval modal operators, the type of the underlying linear order (discrete or dense), and the type of semantics for the interval relations (reflexive or irreflexive). For example, we show that satisfiability of Horn formulas with diamonds is undecidable for any type of linear orders and semantics. On the contrary, satisfiability of Horn formulas with boxes is tractable over both discrete and dense orders under the reflexive semantics and over dense orders under the irreflexive semantics but becomes undecidable over discrete orders under the irreflexive semantics. Satisfiability of binary Horn formulas with both boxes and diamonds is always undecidable under the irreflexive semantics
Complexity of ITL model checking: some well-behaved fragments of the interval logic HS
Model checking has been successfully used in many computer science fields,
including artificial intelligence, theoretical computer science, and databases.
Most of the proposed solutions make use of classical, point-based temporal
logics, while little work has been done in the interval temporal logic setting.
Recently, a non-elementary model checking algorithm for Halpern and Shoham's
modal logic of time intervals HS over finite Kripke structures (under the
homogeneity assumption) and an EXPSPACE model checking procedure for two
meaningful fragments of it have been proposed. In this paper, we show that more
efficient model checking procedures can be developed for some expressive enough
fragments of HS
The temporal logic of two-dimensional Minkowski spacetime with slower-than-light accessibility is decidable
We work primarily with the Kripke frame consisting of two-dimensional
Minkowski spacetime with the irreflexive accessibility relation 'can reach with
a slower-than-light signal'. We show that in the basic temporal language, the
set of validities over this frame is decidable. We then refine this to
PSPACE-complete. In both cases the same result for the corresponding reflexive
frame follows immediately. With a little more work we obtain
PSPACE-completeness for the validities of the Halpern-Shoham logic of intervals
on the real line with two different combinations of modalities.Comment: 20 page
Tractable interval temporal propositional and description logics
We design a tractable Horn fragment of the Halpern-Shaham temporal logic and extend it to interval-based temporal description logics, instance checking in which is P-complete for both combined and data complexity
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