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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:[email protected]
MARK REYNOLDS
Affiliation:
SCHOOL OF COMPUTER SCIENCE AND SOFTWARE ENGINEERING THE UNIVERSITY OF WESTERN AUSTRALIA PERTH, AUSTRALIAE-mail:[email protected]
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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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References

REFERENCES

Andréka, H., Madarász, J. X., and Németi, I., Logic of space-time and relativity theory, Handbook of Spatial Logics (Aiello, M., Pratt-Hartmann, I., and van Benthem, J., editors), Springer, New York, 2007, pp. 607–712.CrossRefGoogle Scholar
Andréka, H., Madarász, J. X., Németi, I., and Szekely, G., A logic road from special relativity to general relativity. Synthese , vol. 186 (2012), no. 3, pp. 633–649.CrossRefGoogle Scholar
Bresolin, D., Dela Monica, D., Goranko, V., Montanari, A., and Sciavicco, G., Decidabile and undecidable fragments of Halpern and Shoham’s interval temporal logic: Towards a complete classification. Annals of Pure and Applied Logic, vol. 161 (2008), pp. 289–304.CrossRefGoogle Scholar
Burgess, J. P., Basic tense logic, Handbook of Philosophical Logic, II: Extensions of Classical Logic, (Gabbay, D. M. and Guenthner, F., editors), Reidel, Dordrecht, 1984, pp. 89–113.CrossRefGoogle Scholar
Cantor, G., Über unendliche, lineare Punktmannigfaltigkeiten V [On infinite, linear point-manifolds (sets). Mathematische Annalen, vol. 21 (1883), pp. 545–591.CrossRefGoogle Scholar
French, T. N., McCabe-Dansted, J. C., and Reynolds, M., Synthesis for temporal logic over the reals, Advances in Modal Logic (Bolander, T., Braüner, T., Ghilardi, S., and Moss, L., editors), vol. 9, College Publications, London, 2012, pp. 217–238.Google Scholar
French, T. N., Synthesis for continuous time. Theoretical Computer Science, vol. 594 (2015), pp. 201–222.CrossRefGoogle Scholar
Goldblatt, R., Diodorean modality in Minkowski space-time. Studia Logica, vol. 39 (1980), pp. 219–236.CrossRefGoogle Scholar
Goldblatt, R., Mathematics of Modality, CSLI, Stanford, CA, 1993.Google Scholar
Hirsch, R., Hodkinson, I., Marx, M., Mikulás, S., and Reynolds, M., Mosaics and step-by-step. Remarks on “A modal logic of relations”, Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa (Orlowska, E., editor), Studies in Fuzziness and Soft Computing, vol. 24, Springer-Verlag, Berlin, 1999, pp. 158–167.Google Scholar
Halpern, J. and Shoham, Y., A propositional modal logic of intervals. Journal of the ACM, vol. 38 (1991), pp. 279–292.CrossRefGoogle Scholar
Hodkinson, I., Wolter, F., and Zakharyaschev, M., Decidable and undecidable fragments of first-order branching temporal logics, Proceedings of 17th Annual IEEE Symposium on Logic in Computer Science, IEEE, Los Alamitos, CA, 2002, pp. 393–402.CrossRefGoogle Scholar
Ladner, R., The computational complexity of provability in systems of modal propositional logic. SIAM Journal of Computing, vol. 6 (1977), pp. 467–480.CrossRefGoogle Scholar
Marcinkowski, J. and Michaliszyn, J., The ultimate undecidability result for the Halpern-Shoham logic. LICS, vol. 26 (2011), pp. 377–386.Google Scholar
Marx, M., Mikulas, S., and Reynolds, M., The mosaic method for temporal logics, Automated Reasoning with Analytic Tableaux and Related Methods (Dyckhoff, R., editor), LNAI 1847, Springer, New York, 2000, pp. 324–340.CrossRefGoogle Scholar
Müller, T., Prior’s tense-logical universalism. Logique et analyse, vol. 50 (2007), no. 199, pp. 223–252.Google Scholar
Montanari, A. and Sala, P., An optimal tableau system for the logic of temporal neighborhood over the reals, 2012 19th International Symposium on Temporal Representation and Reasoning, IEEE Computer Society, Los Alamitos, CA, 2012, pp. 39–46.CrossRefGoogle Scholar
Németi, I., Decidable versions of first order logic and cylindric-relativized set algebras, Logic Colloquium ’92 (Csirmaz, L., Gabbay, D., and de Rijke, M., editors), CSLI Publications, Stanford, CA, 1995, pp. 171–241.Google Scholar
Phillips, J., A note on the modal and temporal logics for n-dimensional spacetime. Notre Dame Journal of Formal Logic, vol. 39 (1998), no. 4, pp. 545–553.CrossRefGoogle Scholar
Phillips, J., Modal logics of succession for 2-dimensional integral spacetime. Journal of Philosophical Logic, vol. 30 (2001), pp. 1–25.CrossRefGoogle Scholar
Prior, A.. Past, Present and Future, Oxford University Press, Oxford, 1967.CrossRefGoogle Scholar
Prior, A., The notion of the present. Studium Generale, vol. 23 (1970), pp. 245–248.Google Scholar
Reynolds, M., A decidable temporal logic of parallelism. Notre Dame Journal of Formal Logic, vol. 38 (1997), no. 3, pp. 419–436.Google Scholar
Robb, A. A., A Theory of Time and Space, Cambridge University Press, Cambridge, 1914.Google Scholar
Reynolds, M. and Zakharyaschev, M., On the products of linear modal logics. Journal of Logic and Computation, vol. 11 (2001), pp. 909–931.CrossRefGoogle Scholar
Shapirovsky, I., On PSPACE-decidability in transitive modal logic, Advances in Modal Logic (Schmidt, R., Pratt-Hartmann, I., Reynolds, M., and Wansing, H., editors), vol. 5, King’s College Publications, London, 2005, pp. 269–287.Google Scholar
Shapirovsky, I., Simulation of two dimensions in unimodal logics, Advances in Modal Logic (Beklemishev, L., Goranko, V., and Shehtman, V., editors), vol. 8, College Publications, London, 2010, pp. 371–391.Google Scholar
Shehtman, V., Modal logics of the real plane. Studia Logica, vol. 42 (1983), pp. 63–80.CrossRefGoogle Scholar
Shapirovsky, I. and Shehtman, V. B., Chronological future modality in minkowski spacetime, Advances in Modal Logic (Balbiani, P., Suzuki, N.-Y., Wolter, F., and Zakharyaschev, M., editors), vol. 4, King’s College Publications, London, 2002, pp. 437–460.Google Scholar
Uckelman, S. L. and Uckelman, J., Modal and temporal logics for abstract space–time structures. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, vol. 38 (2007), no. 3, pp. 673–681.CrossRefGoogle Scholar
van Benthem, J., The Logic of Time, Reidel, Dordrecht, 1983.CrossRefGoogle Scholar
Venema, Y., Expressiveness and completeness of an interval tense logic. Notre Dame Journal of Formal Logic , vol. 31 (1990), pp. 529–547.CrossRefGoogle Scholar