35 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
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
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
Convolution algebras: Relational convolution, generalised modalities and incidence algebras
Convolution is a ubiquitous operation in mathematics and computing. The Kripke semantics for substructural and interval logics motivates its study for quantale-valued functions relative to ternary relations. The resulting notion of relational convolution leads to generalised binary and unary modal operators for qualitative and quantitative models, and to more conventional variants, when ternary relations arise from identities over partial semigroups. Convolution-based semantics for fragments of categorial, linear and incidence (segment or interval) logics are provided as qualitative applications. Quantitative examples include algebras of durations and mean values in the duration calculus
Acta Informatica manuscript No. (will be inserted by the editor) A Complete Classification of the Expressiveness of Interval Logics of Allen鈥檚 Relations The General and the Dense Cases
Abstract Interval temporal logics take time intervals, instead of time instants, as their primitive temporal entities. One of the most studied interval temporal logics is Halpern and Shoham鈥檚 modal logic of time intervals HS, which associates a modal operator with each binary relation between intervals over a linear order (the so-called Allen鈥檚 interval relations). In this paper, we compare and classify the expressiveness of all fragments of HS on the class of all linear orders and on the subclass of all dense linear orders. For each of these classes, we identify a complete set of definabilities between HS modalities, valid in that class, thus obtaining a complete classification of the family of all 4096 fragments of HS with respect to their expressiveness. We show that on the class of all linear orders there are exactly 1347 expressively different fragments of HS, while on the class of dense linear orders there are exactly 966 such expressively different fragments
The Temporal Logic of two dimensional Minkowski spacetime is decidable
We consider Minkowski spacetime, the set of all point-events of spacetime
under the relation of causal accessibility. That is, can access if an electromagnetic or (slower than light) mechanical signal could be
sent from to . We use Prior's tense language of
and representing causal accessibility and its converse relation. We
consider two versions, one where the accessibility relation is reflexive and
one where it is irreflexive.
In either case it has been an open problem, for decades, whether the logic is
decidable or axiomatisable. We make a small step forward by proving, for the
case where the accessibility relation is irreflexive, that the set of valid
formulas over two-dimensional Minkowski spacetime is decidable, decidability
for the reflexive case follows from this. The complexity of either problem is
PSPACE-complete.
A consequence is that the temporal logic of intervals with real endpoints
under either the containment relation or the strict containment relation is
PSPACE-complete, the same is true if the interval accessibility relation is
"each endpoint is not earlier", or its irreflexive restriction.
We provide a temporal formula that distinguishes between three-dimensional
and two-dimensional Minkowski spacetime and another temporal formula that
distinguishes the two-dimensional case where the underlying field is the real
numbers from the case where instead we use the rational numbers.Comment: 30 page
Probabilistic Interval Temporal Logic and Duration Calculus with Infinite Intervals: Complete Proof Systems
The paper presents probabilistic extensions of interval temporal logic (ITL)
and duration calculus (DC) with infinite intervals and complete Hilbert-style
proof systems for them. The completeness results are a strong completeness
theorem for the system of probabilistic ITL with respect to an abstract
semantics and a relative completeness theorem for the system of probabilistic
DC with respect to real-time semantics. The proposed systems subsume
probabilistic real-time DC as known from the literature. A correspondence
between the proposed systems and a system of probabilistic interval temporal
logic with finite intervals and expanding modalities is established too.Comment: 43 page