Generalizing CASL Specification Components and Preserving Rewrite Proofs

We propose the theoretical basis of a tool for the generation of reusable CASL specification components by generalization of existing ones. The underlying idea is, given a component and a set of semantic properties that it satisfies and that we want to preserve, to find a parameterized, more general, component satisfying the following conditions: the original component is one of its possible instantiations, and any of its instantiations satisfy the stated properties. We present here both the definition of the generalization operation for CASL and the problem of preserving properties in the generalized component. To guarantee the preservation of properties, we propose to preserve their proofs, concentrating on the use of rewrite proofs. This technique provides a simple way to find sufficient conditions for the preservation of the corresponding properties. This work is being integrated in the specification component development tool FERUS, under development for the CASL language, using ELAN as the rewrite proof engine

The maximization of submodular functions : old and new proofs for the correctness of the dichotomy algorithm

The first purpose of this paper is to make an old (Russian) theoretical results about the structure of local and global maxima of submodular functions, Cherenin’s excluding rules and his Dichotomy Algorithm more accessible for Western community. The second purpose of this paper is to present our main result which can be stated as follows. For any pair of embedded subsets, the difference of their function values is a lower bound for the difference between the unknown(!) optimal values of the corresponding partition defined by these subsets. A simple justification of Cherenin’s rules, the Dichotomy Algorithmand its generalization with the new branching rules from our main result are presented. The usefulness of our new branching rules is illustrated by means of a numerical example.