Gap-definability and the gap closure operator were defined by S. Fenner, L. Fortnow and S. Kurth (J. Comput. System Sci. 48, 116 148 (1994)). Few complexity classes were known at that time to be gapdefinable. In this paper, we give simple characterizations of both gapdefinability and the gap-closure operator, and we show that many complexity classes are gap-definable, including P, P, PSPACE, EX...

In this paper we systematically explore questions of succinctness in modal logics employed in spatial reasoning. We show that the closure operator, despite being less expressive, is exponentially more succinct than the limit-point operator, and that the μ-calculus is exponentially more succinct than the equally-expressive tangled limit operator. These results hold for any class of spaces contai...

Closure operators are abundant in mathematics; here are a few examples. Given an algebraic structure, such as group, ring, field, lattice, vector space, etc., taking the substructure generated by a set, i.e., the least substructure which includes that set, is a closure operator. Given a binary relation, taking the relation with certain properties, such as reflexive, transitive, equivalence, etc...

We define strict closure space algebras as Boolean algebras augmented with a unary operation Int that satisfies conditions analogous to those of the interior operation from strict closure spaces. We also consider a first order modal logic SCS in which the necessity operator satisfies analogous conditions. Sikorski proved that every interior algebra can be represented as an algebra of subsets of...

We deene the notion of weak relative pseudo-complement, and we show that, for an arbitrary lattice, the property of weak relative pseudo-complementation is strictly weaker than relative pseudo-complementation, but stronger than pseudo-complementation. Our main interest for this notion is in relation with the theory of closure operators. We prove that if a complete lattice L is completely inf-di...

We study relations between predicate transformers and mul-tifunctions in a topological setting based on closure operators. We give topological deenitions of safety and liveness predicates and using these predicates we deene predicate transformers. State transformers are mul-tifunctions with values in the collection of xed points of a closure operator. We derive several isomorphisms between pred...

We consider a specific class of combinatorial search or optimization problems where the search space gives rise to a closure operator and essentially the hulls are the only relevant subsets that must be checked in a brute force approach. We suggest that such a closure property can help to reduce time complexities. Moreover we propose two types of (structural) parameterization of instance classe...

After a brief review of hidden algebra, we give behavioral speciications for set theory and closure operators, and then use these to give a behavioral speciication of an abstract protocol to support distributed cooperative work structured by dependencies in such a way as to form what we call a weak closure operator. We give some correctness proofs for this protocol, and then describe a concrete...

We study a cooperation structure in which the feasible coalitions are those belonging to a partition closure system. First, we develop the relation between the dividends of Harsanyi in the restricted game and the worth function in the original game. Next, we obtain an method for computing the Shapley and Banzhaf values of the closure-restricted games using the dividends formula. Finally, we ana...