The powerset operator, P, is compared with other operators of similar type and logical complexity. Namely we examine positive operators whose defining formula has a canonical form containing at most a string of universal quantifiers. We call them ∀-operators. The question we address in this paper is: How is the class of ∀-operators generated? It is shown that every positive ∀-operator Γ such th...

The Abiteboul and Beeri algebra for complex objects can express a query whose meaning is transitive closure, but the algorithm naturally associated to this query needs exponential space. We show that any other query in the algebra which expresses transitive closure needs exponential space. This proves that in general the powerset is an intractable operator for implementing xpoint queries.

motivated by the recent study on categorical properties of latticevalued topology, the paper considers a generalization of the notion of topological system introduced by s. vickers, providing an algebraic and a coalgebraic category of the new structures. as a result, the nature of the category   topsys   of s. vickers gets clari ed, and a metatheorem is stated, claiming that (latticevalu...

We propose a tableau-like decision procedure for deciding the satisfiability of set-theoretical formulae with restricted universal quantifiers and the powerset operator. Our result apply to a rather large class of set theories. The procedure we define can be used as a subroutine to decide the same class of formulae both in Set Theory and in non well-founded set theories, since we assume neither...

This report is based on the article [1] by Koetter and Kschischang. The authors introduced a novel approach for constructing codes by selecting a subset of a powerset of a vector space. With the code construction, they also defined an operator channel for code transmission. The codewords are vector subspaces. The data is transmitted by using one of the bases of such a subspace. The properties o...

We introduce the notion of operational resolution, i.e., an isotone map from a powerset to a poset that meets two additional conditions, which generalizes the description of states as the atoms in a property lattice (Piron, 1976 and Aerts, 1982) or as the underlying set of a closure operator (Aerts, 1994 and Moore, 1995). We study the structure preservance of the related state transitions and s...

domains (§3.3) are often complete lattices, i.e., algebras over partially ordered sets. We recall in this section the fundamental notions of lattice theory, which are the basis for the soundness of over-approximations of reachable state sets computed by abstract interpretation methods. For further details and proofs we refer to textbooks on lattice theory [DP90] and static analysis [NNH05]. Fro...

The Abiteboul and Beeri algebra for complex objects can express a query whose meaning is transitive closure, but the algorithm naturally associated to this query needs exponential space. We show that any other query in the algebra which expresses transitive closure needs exponential space, under a \call by value" evaluation strategy. This proves that in general the powerset is an intractable op...