Topological Logics over Euclidean Spaces

In this paper we prove some results on the computational complexity of standard quantifierfree spatial logics with the connectedness predicate interpreted over the Euclidean spaces R and R. Topological logics with connectedness. A topological logic is a formal language whose variables range over subsets of topological spaces, and whose non-logical primitives denote fixed topological properties and operations involving these subsets. For example, let the function symbols ∩, ∪ and ·− denote the operations of intersection, union and topological closure, respectively; let the constant 0 denote the empty set; let the unary predicate c denote the property of connectedness; and let the binary predicate ⊆ denote the subset relation. Then the formula c(r1) ∧ c(r2) ∧ ¬(r1 ∩ r2 ⊆ 0) → c(r1 ∪ r2) (1) states that the union of two intersecting connected sets r1 and r2 is connected; likewise, the formula c(r1) ∧ (r1 ⊆ r2) ∧ (r2 ⊆ r1) → c(r2) (2) states that, if r1 is a connected set, and r2 is sandwiched between r1 and its closure, then r2 is also connected. It is well known that these statements hold for any subsets r1, r2 of any topological space. As we might put it: formulas (1) and (2) are validities of the topological logic in question. Once the syntax of that logic has been made precise, it is natural to ask: what is the computational complexity of identifying such validities? Formally, let F be a set of function symbols with fixed interpretations as operations on subsets of a topological space; and let P be a set of predicates, again with fixed interpretations as relations between subsets of a topological space. We denote by L(F, P ) the set of quantifier-free first-order formulas over the signature (F, P ). Using the obvious abbreviations, we may regard formulas (1) and (2) as belonging to the language S4uc := L({∪,∩, ·− , · }, {c,=}), where the operator · denotes complementation with respect to the containing space, and = denotes the equality relation. An interpretation I for this language consists of a topological space T and a map r 7→ r taking every variable to a subset of T . This map is extended to terms in the obvious way, and truth-values are assigned to atomic formulas according to the rules: I |= τ1 = τ2 iff τ 1 = τ 2 and I |= c(τ) iff τ is connected. A formula φ is satisfiable if there exists an interpretation I such that I |= φ, and valid if I |= φ, for all interpretations I. The properties of validity and satisfiability are thus dual in the usual sense. If L is a topological language and K a class of interpretations, we write Sat(L,K) to denote the problem of determining whether a given L-formula is satisfied by some interpretation in K. It is shown in [2] that Sat(S4uc,All) is ExpTime-complete, where All denotes the class of all interpretations. Removing the connectedness predicates altogether, we obtain the language S4u := L({∪,∩, ·− , ·}, {=})—in essence the modal logic S4 with an additional universal modality, under the topological semantics of McKinsey and Tarski [4]. It is well-known that Sat(S4u,All) is PSpace-complete. By restricting the language in various ways, we obtain less expressive logics, having—in general—less complex satisfiability problems. A subset of a topological space is regular closed