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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
Decidability of quantified propositional intuitionistic logic and S4 on trees
Quantified propositional intuitionistic logic is obtained from propositional
intuitionistic logic by adding quantifiers \forall p, \exists p over
propositions. In the context of Kripke semantics, a proposition is a subset of
the worlds in a model structure which is upward closed. Kremer (1997) has shown
that the quantified propositional intuitionistic logic H\pi+ based on the class
of all partial orders is recursively isomorphic to full second-order logic. He
raised the question of whether the logic resulting from restriction to trees is
axiomatizable. It is shown that it is, in fact, decidable. The methods used can
also be used to establish the decidability of modal S4 with propositional
quantification on similar types of Kripke structures.Comment: v2, 9 pages, corrections and additions; v1 8 page
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