Benny Godlin

Regression verification

Proving the equivalence of successive, closely related versions of a program has the potential of being easier in practice than functional verification, although both problems are undecidable. There are two main reasons for this claim: it circumvents the problem of specifying what the program should do, and in many cases it is computationally easier. We study theoretical and practical aspects of this problem, which we call regression verification.

Inference rules for proving the equivalence of recursive procedures

Inspired by Hoare’s rule for recursive procedures, we present three proof rules for the equivalence between recursive programs. The first rule can be used for proving partial equivalence of programs; the second can be used for proving their mutual termination; the third rule can be used for proving the equivalence of reactive programs. There are various applications to such rules, such as proving equivalence of programs after refactoring and proving backward compatibility.

Regression verification: proving the equivalence of similar programs

Proving the equivalence of successive, closely related versions of a program has the potential of being easier in practice than functional verification, although both problems are undecidable. There are three main reasons for this claim: (i) it circumvents the problem of specifying what the program should do; (ii) the problem can be naturally decomposed and hence is computationally easier; and (iii) there is an automatic invariant that enables to prove equivalence of loops and recursive functions in most practical cases. Theoretical and practical aspects of this problem are considered. Copyright

Computing graph polynomials on graphs of bounded clique-width

We discuss the complexity of computing various graph polynomials of graphs of fixed clique-width. We show that the chromatic polynomial, the matching polynomial and the two-variable interlace polynomial of a graph G of clique-width at most k with n vertices can be computed in time O(nf( k)), where f(k) ≤3 for the inerlace polynomial, f(k) ≤2k+1 for the matching polynomial and f(k) ≤3 2k+2 for the chromatic polynomial.

A Most General Edge Elimination Polynomial

We look for graph polynomials which satisfy recurrence relations on three kinds of edge elimination: edge deletion, edge contraction and edge extraction, i.e., deletion of edges together with their end points. Like in the case of deletion and contraction only (J.G. Oxley and D.J.A. Welsh 1979), it turns out that there is a most general polynomial satisfying such recurrence relations, which we call *** (G ,x ,y ,z ). We show that the new polynomial simultaneously generalizes the Tutte polynomial, the matching polynomial, and the recent generalization of the chromatic polynomial proposed by K.Dohmen, A.Ponitz and P.Tittman (2003), including also the independent set polynomial of I. Gutman and F. Harary, (1983) and the vertex-cover polynomial of F.M. Dong, M.D. Hendy, K.T. Teo and C.H.C. Little (2002). We give three definitions of the new polynomial: first, the most general recursive definition, second, an explicit one, using a set expansion formula, and finally, a partition function, using counting of weighted graph homomorphisms. We prove the equivalence of the three definitions. Finally, we discuss the complexity of computing *** (G ,x ,y ,z ).

Evaluations of Graph Polynomials

A graph polynomial

Graph Polynomials

p(G, \bar{X})

An extension of the bivariate chromatic polynomial

can code numeric information about the underlying graph G in various ways: as its degree, as one of its specific coefficients or as evaluations at specific points

Regression Verification - A Practical Way to Verify Programs

\bar{X}= \bar{x}_0

A most general edge elimination graph polynomial

. In this paper we study the question how to prove that a given graph parameter, say *** (G ), the size of the maximal clique of G , cannot be a fixed coefficient or the evaluation at any point of the Tutte polynomial, the interlace polynomial, or any graph polynomial of some infinite family of graph polynomials. Our result is very general. We give a sufficient condition in terms of the connection matrix of graph parameter f (G ) which implies that it cannot be the evaluation of any graph polynomial which is invariantly definable in CMSOL , the Monadic Second Order Logic augmented with modular counting quantifiers. This criterion covers most of the graph polynomials known from the literature.