Automated Reasoning About Metric and Topology

In this paper we compare two approaches to automated reasoning about metric and topology in the framework of the logicMT introduced in [10].MT -formulas are built from set variables p1, p2, . . . (for arbitrary subsets of a metric space) using the Booleans ∧, ∨, →, and ¬, the distance operators ∃ and ∃≤a, for a ∈ Q, and the topological interior and closure operators I and C. Intended models for this logic are of the form I = (∆, d, p1 , p I 2 , . . . ) where (∆, d) is a metric space and pi ⊆ ∆. The extension φ ⊆ ∆ of an MT -formula φ in I is de ned inductively in the usual way, with I and C being interpreted as the interior and closure operators induced by the metric, and (∃φ) = {x ∈ ∆ | ∃y ∈ φ d(x, y) < a}. In other words, (Iφ) is the interior of φ, (∃φ) is the open a-neighbourhood of φ, and (∃≤aφ)I is the closed one. A formula φ is satis able if there is a model I such that φ 6= ∅; φ is valid if ¬φ is not satis able. InMT , one can represent various basic facts about metric and topology. For example, the validity of ∃p→ I∃p means that the open a-neighbourhood of any set is open. The non-validity of C∃p→ ∃≤ap means that there is a metric space with a subset X such that the closure of the open a-neighbourhood of X properly contains the closed a-neighbourhood of X. The logic MT as well as its metric fragment MS without the topological operators have been suggested as basic tools for reasoning about distances and similarity [9]. One obvious approach to automated reasoning with MT is to use the standard ε-de nition of the topological interior