Nondeterministic Relational Semantics of a while Program

A relational semantics is a mapping of programs to relations. We consider that the input-output semantics of a program is given by a relation on its set of states; in a nondeterministic context, this relation is calculated by considering the worst behavior of the program (demonic relational semantics). In this paper, we concentrate on while loops. Calculating the relational abstraction (semantics) of a loop is difficult, but showing the correctness of any candidate abstraction is much easier. For functional programs, Mills has described a checking method known as the while statement verification rule. A programming theorem for iterative constructs is proposed, proved, demonstrated and applied for an example. This theorem can be considered as a generalization of the while statement verification to nondeterministic loops.&nbsp

Harnessing a Refinement Theory to Compute Loop Functions

AbstractWe consider a while loop on some space S and we are interested in deriving the function that this loop defines between its initial states and its final states (when it terminates). Such a capability is useful in a wide range of applications, including reverse engineering, software maintenance, program comprehension, and program verification. In the absence of a general theoretical solution to the problem of deriving the function of a loop, we explore engineering solutions. In this paper we use a relational refinement calculus to approach this complex problem in a systematic manner. Our approach has many drawbacks, some surmountable and some not (being inherent to the approach); nevertheless, it offers a way to automatically derive the function of loops or an approximation thereof, under some conditions

Demonic fixed points

We deal with a relational model for the demonic semantics of programs. The demonic semantics of a while loop is given as a fixed point of a function involving the demonic operators. This motivates us to investigate the fixed points of these functions. We give the expression of the greatest fixed point with respect to the demonic ordering (demonic inclusion) of the semantic function. We prove that this greatest fixed coincides with the least fixed point with respect to the usual ordering (angelic inclusion) of the same function. This is followed by an example of application

Polynomial approximations of the relational semantics of imperative programs

AbstractWe present a static analysis that approximates the relational semantics of imperative programs by systems of low-degree polynomial equalities. Our method is based on Abstract Interpretation in a lattice of polynomial pseudo ideals — finite-dimensional vector spaces of degree-bounded polynomials that are closed under degree-bounded products. For a fixed degree bound, the sizes of bases of pseudo ideals and the lengths of chains in the lattice of pseudo ideals are bounded by polynomials in the number of program variables. Despite the approximate nature of our analysis, for several programs taken from the literature on non-linear polynomial invariant generation our method produces results that are as precise as those produced by methods based on polynomial ideals and Gröbner bases

An axiom system for sequence-based specification

AbstractThis paper establishes an axiomatic foundation and a representation theorem for the rigorous, constructive process, called sequence-based specification, of deriving precise specifications from ordinary (informal) statements of functional requirements. The representation theorem targets a special class of Mealy state machines, and algorithms are presented for converting from the set of sequences that define the specification to the equivalent Mealy machine, and vice versa. Since its inception, sequence-based specification has been effectively used in a variety of real applications, with gains reported in quality and productivity. This paper establishes the mathematical foundation independently of the process itself