The convexity of numerical domains such as polyhedra, octagons, intervals and linear equalities enables tractable analysis of software for buffer overflows, null pointer dereferences and floating point errors. However, convexity also causes the analysis to fail in many common cases. Powerset extensions can remedy this shortcoming by considering disjunctions of predicates. Unfortunately, analysi...

This paper deals with a broad question—to what extent is topology algebraic—using two specific questions: (1) what are the algebraic conditions on the underlying membership lattices which insure that categories for topology and fuzzy topology are indeed topological categories; and (2) what are the algebraic conditions which insure that algebraic theories in the sense of Manes are a foundation f...

This talk aims at presenting a new way of approaching topological structures, induced by recent developments in the eld of lattice-valued topology, and deemed to incorporate in itself both crisp and many-valued settings. Based on category theory and universal algebra, the new framework is called categorically-algebraic (catalg) [9-12], to underline its motivating theories, on one hand, and to d...

We examine double successive approximations on a set, which we denote by L2L1, U2U1, U2L1, L2U1 where L1, U1 and L2, U2 are based on generally non-equivalent equivalence relations E1 and E2 respectively, on a finite non-empty set V. We consider the case of these operators being given fully defined on its powerset P(V ). Then, we investigate if we can reconstruct the equivalence relations which ...

For any set-endofunctor T : Set→ Set there exists a largest subcartesian transformation μ to the filter functor F : Set → Set. Thus we can associate with every T -coalgebra A a certain filter-coalgebra AF. Precisely, when T weakly preserves preimages, μ is natural, and when T weakly preserves intersections, μ factors through the covariant powerset functor P, thus providing for every T -coalgebr...

In this paper we discuss Alpha Galois lattices (Alpha lattices for short) and the corresponding association rules. An alpha lattice is coarser than the related concept lattice and so contains fewer nodes, so fewer closed patterns, and a smaller basis of association rules. Coarseness depends on a a priori classification, i.e. a cover C of the powerset of the instance set I, and on a granularity ...

Counterexample-based Refinement for Powerset Domains R. Manevich, J. Field , T. A. Henzinger, G. Ramalingam, and M. Sagiv 1 Tel Aviv University, {rumster,msagiv}@tau.ac.il 2 IBM T.J. Watson Research Center, [email protected] 3 EPFL, [email protected] 4 Microsoft Research India, [email protected] Abstract. Counterexample-guided abstraction refinement (CEGAR) is a powerful technique to scale auto...

Kripke semantics plays a fundamental role for modal logics where it often provides modular semantics allowing a clear treatment of various logics related by the addition of axioms or connectives. Kripke semantics is also the “natural semantics” for classical modal logic in the sense that it is given by the representation theory for modal algebras. The algebraic structure corresponding to a trad...

A set A of positive integers is called a coprime Diophantine powerset if the shifted product ab + 1 of two different elements a and b of A is always a pure power, and the occuring pure powers are all coprime. We prove that each coprime Diophantine powerset A ⊂ {1, . . . , N} has |A| ≤ 8000 logN/ log logN for sufficiently large N . The proof combines results from extremal graph theory with numbe...

Paredaens and Van Gucht proved that the at relational algebra has the same expressive power as the nested relational algebra, as far as queries over at relations and with at results are concerned. We provide a new, very direct proof of this fact using a simulation technique. Our technique is also applied to partially answer a question posed by Suciu and Paredaens regarding the complexity of eva...