How hard is it to verify flat affine counter systems with the finite monoid property ?

We study several decision problems for counter systems with guards defined by convex polyhedra and updates defined by affine transformations. In general, the reachability problem is undecidable for such systems. Decidability can be achieved by imposing two restrictions: (i) the control structure of the counter system is flat, meaning that nested loops are forbidden, and (ii) the set of matrix powers is finite, for any affine update matrix in the system. We provide tight complexity bounds for several decision problems of such systems, by proving that reachability and model checking for Past Linear Temporal Logic are complete for the second level of the polynomial hierarchy $\Sigma^P_2$, while model checking for First Order Logic is PSPACE-complete

Matrix Model Description of Baryonic Deformations

We investigate supersymmetric QCD with N_c+1 flavors using an extension of the recently proposed relation between gauge theories and matrix models. The impressive agreement between the two sides provides a beautiful confirmation of the extension of the gauge theory-matrix model relation to this case.Comment: 33pages, late

Program Verification with Separation Logic

International audienceSeparation Logic is a framework for the development of modular program analyses for sequential, inter-procedural and concurrent programs. The first part of the paper introduces Separation Logic first from a historical, then from a program verification perspective. Because program verification eventually boils down to deciding logical queries such as the validity of verification conditions, the second part is dedicated to a survey of decision procedures for Separation Logic, that stem from either SMT, proof theory or automata theory. Incidentally we address issues related to decidability and computational complexity of such problems, in order to expose certain sources of intractability

Underapproximation of Procedure Summaries for Integer Programs

We show how to underapproximate the procedure summaries of recursive programs over the integers using off-the-shelf analyzers for non-recursive programs. The novelty of our approach is that the non-recursive program we compute may capture unboundedly many behaviors of the original recursive program for which stack usage cannot be bounded. Moreover, we identify a class of recursive programs on which our method terminates and returns the precise summary relations without underapproximation. Doing so, we generalize a similar result for non-recursive programs to the recursive case. Finally, we present experimental results of an implementation of our method applied on a number of examples.Comment: 35 pages, 3 figures (this report supersedes the STTT version which in turn supersedes the TACAS'13 version

The Complexity of Prenex Separation Logic with One Selector

We first show that infinite satisfiability can be reduced to finite satisfiability for all prenex formulas of Separation Logic with $k\geq1$ selector fields (\seplogk{k}). Second, we show that this entails the decidability of the finite and infinite satisfiability problem for the class of prenex formulas of \seplogk{1}, by reduction to the first-order theory of one unary function symbol and unary predicate symbols. We also prove that the complexity is not elementary, by reduction from the first-order theory of one unary function symbol. Finally, we prove that the Bernays-Sch\"onfinkel-Ramsey fragment of prenex \seplogk{1} formulae with quantifier prefix in the language $\exists^*\forall^*$ is \pspace-complete. The definition of a complete (hierarchical) classification of the complexity of prenex \seplogk{1}, according to the quantifier alternation depth is left as an open problem

Deciding Conditional Termination

We address the problem of conditional termination, which is that of defining the set of initial configurations from which a given program always terminates. First we define the dual set, of initial configurations from which a non-terminating execution exists, as the greatest fixpoint of the function that maps a set of states into its pre-image with respect to the transition relation. This definition allows to compute the weakest non-termination precondition if at least one of the following holds: (i) the transition relation is deterministic, (ii) the descending Kleene sequence overapproximating the greatest fixpoint converges in finitely many steps, or (iii) the transition relation is well founded. We show that this is the case for two classes of relations, namely octagonal and finite monoid affine relations. Moreover, since the closed forms of these relations can be defined in Presburger arithmetic, we obtain the decidability of the termination problem for such loops.Comment: 61 pages, 6 figures, 2 table