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THE TEMPORAL LOGIC OF TWO DIMENSIONAL MINKOWSKI SPACETIME IS DECIDABLE

Published online by Cambridge University Press:  23 October 2018

ROBIN HIRSCH  and
MARK REYNOLDS
ROBIN HIRSCH
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY COLLEGE LONDON LONDON, UKE-mail:r.hirsch@ucl.ac.uk
MARK REYNOLDS
Affiliation:
SCHOOL OF COMPUTER SCIENCE AND SOFTWARE ENGINEERING THE UNIVERSITY OF WESTERN AUSTRALIA PERTH, AUSTRALIAE-mail:mark.reynolds@uwa.edu.au
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Abstract

We consider Minkowski spacetime, the set of all point-events of spacetime under the relation of causal accessibility. That is, x can access y if an electromagnetic or (slower than light) mechanical signal could be sent from x to y. We use Prior’s tense language of F and P 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, in each case, that the set of valid formulas over two-dimensional Minkowski spacetime is decidable and that the complexity of each 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.

Keywords

03B4483A05temporal logicdecidabilityMinkowski spacetimeinterval logic
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 3 , September 2018 , pp. 829 - 867
Copyright
Copyright © The Association for Symbolic Logic 2018 

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