As interest in components and composition-related methods is growing rapidly, it is not always clear what the goals (and the corresponding difficulties) actually are. If composition is to become central in the future of software engineering, we need to better identify the fundamental issues that are related to it, before we attempt to solve them as they occur in object-oriented systems or in co...

This paper describes extensions of the B language concerning control flow breaks in implementations and specification of operations with exceptional behaviors. It does not claim to define those extensions in a pure formal and complete way. It is rather a presentation of what could be done and how it could be done. A syntax is proposed and proof obligations are defined using a weakest preconditi...

The indicators system and indicators normalization method for the condition assessment of Transformer is developed and membership function of the indicators is established. The establishment of Expert Weight is decided jointly by subjective weight and objective weight which is based on entropy weight thoughts, while the indicators weight is gained by the weight which is derived from the standar...

This paper can be seen as a continuation of the works contained in the recent preprints [Za], of the second author, and [Mi], of Juan Migliore. Our results are: 1). There exist codimension three artinian level algebras of type two which do not enjoy the Weak Lefschetz Property (WLP). In fact, for e ≫ 0, we will construct a codimension three, type two h-vector of socle degree e such that all the...

We present a purely syntactical but nevertheless handy definition of Dijkstra’s predicate transformer wp in weak second-order logic. This formalization allows us to prove a normal form theorem of wp, from which interesting properties can be inferred. In particular, it turns out that wp(P, .) is a homomorphism on the class of formulae in the formalism considered.

Although various proof rules for procedure calls in weakest precondition semantics have been proposed over the years [3, chapter 12] [4] [1], none of these is particularly suitable for calculational program construction. The problem is that they tend to yield a precondition that is tortuously expressed. For instance, let a procedure for finding the integer square root of a parameter y be specif...

Expectation-based probabilistic predicate transformers 15] provide a logic for probabilistic sequential programs, giving access to expressions such as`the probability that predicate A is achieved nally'. Using expectations more generally however, we can express-calculus formulae for the expected path-length of a computation tree. Moreover within an expectation-based-calculus such eeciency measu...

We study the underapproximation of the predicate transformers used to give semantics to the modalities in dynamic and temporal logic. Because predicate transformers operate on state sets, we define appropriate powerdomains for sound approximation. We study four such domains — two are based on “set inclusion” approximation, and two are based on “quantification” approximation — and we apply the d...

In a recent letter by Ying et al. [Inf. Process. Lett. 104 (2007) 152-158], it showed some sufficient conditions for commutativity of quantum weakest preconditions. This paper provides some simple characterizations for the commutativity of quantum weakest preconditions, i.e., Theorem 3, Theorem 4 and Proposition 5 in what follows. We also show that to characterize the commutativity of quantum w...

In this paper we present an algebraic construction of monotonic predicate transformers, using a categorical construction which is similar to the algebraic construction of the integers from the natural numbers. When applied to the category of sets and total functions once, it yields a category isomorphic to the category of sets and relations; a second application yields a category isomorphic to ...