Journal of Symbolic Logic

A logic for metric and topology

Frank Wolter and Michael Zakharyaschev

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper we devise some technical tools for dealing with problems connected with the philosophical view usually called mathematical instrumentalism. These tools are interesting in their own right, independently of their philosophical consequences. For example, we show that even though the fragment of Peano's Arithmetic known as IΣ₁ is a conservative extension of the equational theory of Primitive Recursive Arithmetic (PRA), IΣ₁ has a super-exponential speed-up over PRA. On the other hand, theories studied in the Program of Reverse Mathematics that formalize powerful mathematical principles have only polynomial speed-up over IΣ₁.

We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radius r,’ including or excluding the boundary. We supply the logic with both the intended metric space semantics and a natural relational semantics, and show that the latter (i) provides finite partial representations of (in general) infinite metric models and (ii) reduces the standard ‘ε-definitions’ of closure and interior to simple constraints on relations. These features of the relational semantics suggest a finite axiomatisation of the logic and provide means to prove its EXPTIME-completeness (even if the rational numerical parameters are coded in binary). An extension with metric variables satisfying linear rational (in)equalities is proved to be decidable as well. Our logic can be regarded as a ‘well-behaved’ common denominator of logical systems constructed in temporal, spatial, and similarity-based quantitative and qualitative representation and reasoning. Interpreted on the real line (with its Euclidean metric), it is a natural fragment of decidable temporal logics for specification and verification of real-time systems. On the real plane, it is closely related to quantitative and qualitative formalisms for spatial representation and reasoning, but this time the logic becomes undecidable.

Article information

Source
J. Symbolic Logic, Volume 70, Issue 3 (2005), 795-828.

Dates
First available in Project Euclid: 22 July 2005

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1122038915

Digital Object Identifier
doi:10.2178/jsl/1122038915

Mathematical Reviews number (MathSciNet)
MR2155267

Zentralblatt MATH identifier
1089.03021

Citation

Wolter, Frank; Zakharyaschev, Michael. A logic for metric and topology. J. Symbolic Logic 70 (2005), no. 3, 795--828. doi:10.2178/jsl/1122038915. https://projecteuclid.org/euclid.jsl/1122038915


Export citation