Program Termination and Worst Time Complexity with Multi-Dimensional Affine Ranking Functions

A standard method for proving the termination of a flowchart program is to exhibit a ranking function, i.e., a function from the program states to a well-founded set, which strictly decreases at each program step. Our main contribution is to give an efficient algorithm for the automatic generation of multi-dimensional affine nonnegative ranking functions, a restricted class of ranking functions that can be handled with linear programming techniques. Our algorithm is based on the combination of the generation of invariants (a technique from abstract interpretation) and on an adaptation of multi-dimensional affine scheduling (a technique from automatic parallelization). We also prove the completeness of our technique with respect to its input and the class of rankings we consider. Finally, as a byproduct, by computing the cardinal of the range of the ranking function, we obtain an upper bound for the computational complexity of the source program, which does not depend on restrictions on the shape of loops or on program structure. This estimate is a polynomial, which means that we can handle programs with more than linear complexity. The method is tested on a large collection of test cases from the literature. We also point out future improvements to handle larger programs

Ranking and Repulsing Supermartingales for Reachability in Probabilistic Programs

Computing reachability probabilities is a fundamental problem in the analysis of probabilistic programs. This paper aims at a comprehensive and comparative account on various martingale-based methods for over- and under-approximating reachability probabilities. Based on the existing works that stretch across different communities (formal verification, control theory, etc.), we offer a unifying account. In particular, we emphasize the role of order-theoretic fixed points---a classic topic in computer science---in the analysis of probabilistic programs. This leads us to two new martingale-based techniques, too. We give rigorous proofs for their soundness and completeness. We also make an experimental comparison using our implementation of template-based synthesis algorithms for those martingales

Fundamental groups, Alexander invariants, and cohomology jumping loci

We survey the cohomology jumping loci and the Alexander-type invariants associated to a space, or to its fundamental group. Though most of the material is expository, we provide new examples and applications, which in turn raise several questions and conjectures. The jump loci of a space X come in two basic flavors: the characteristic varieties, or, the support loci for homology with coefficients in rank 1 local systems, and the resonance varieties, or, the support loci for the homology of the cochain complexes arising from multiplication by degree 1 classes in the cohomology ring of X. The geometry of these varieties is intimately related to the formality, (quasi-) projectivity, and homological finiteness properties of \pi_1(X). We illustrate this approach with various applications to the study of hyperplane arrangements, Milnor fibrations, 3-manifolds, and right-angled Artin groups.Comment: 45 pages; accepted for publication in Contemporary Mathematic

IST Austria Thesis

Hybrid automata combine finite automata and dynamical systems, and model the interaction of digital with physical systems. Formal analysis that can guarantee the safety of all behaviors or rigorously witness failures, while unsolvable in general, has been tackled algorithmically using, e.g., abstraction, bounded model-checking, assisted theorem proving. Nevertheless, very few methods have addressed the time-unbounded reachability analysis of hybrid automata and, for current sound and automatic tools, scalability remains critical. We develop methods for the polyhedral abstraction of hybrid automata, which construct coarse overapproximations and tightens them incrementally, in a CEGAR fashion. We use template polyhedra, i.e., polyhedra whose facets are normal to a given set of directions. While, previously, directions were given by the user, we introduce (1) the first method for computing template directions from spurious counterexamples, so as to generalize and eliminate them. The method applies naturally to convex hybrid automata, i.e., hybrid automata with (possibly non-linear) convex constraints on derivatives only, while for linear ODE requires further abstraction. Specifically, we introduce (2) the conic abstractions, which, partitioning the state space into appropriate (possibly non-uniform) cones, divide curvy trajectories into relatively straight sections, suitable for polyhedral abstractions. Finally, we introduce (3) space-time interpolation, which, combining interval arithmetic and template refinement, computes appropriate (possibly non-uniform) time partitioning and template directions along spurious trajectories, so as to eliminate them. We obtain sound and automatic methods for the reachability analysis over dense and unbounded time of convex hybrid automata and hybrid automata with linear ODE. We build prototype tools and compare—favorably—our methods against the respective state-of-the-art tools, on several benchmarks

Computer Aided Verification

This open access two-volume set LNCS 13371 and 13372 constitutes the refereed proceedings of the 34rd International Conference on Computer Aided Verification, CAV 2022, which was held in Haifa, Israel, in August 2022. The 40 full papers presented together with 9 tool papers and 2 case studies were carefully reviewed and selected from 209 submissions. The papers were organized in the following topical sections: Part I: Invited papers; formal methods for probabilistic programs; formal methods for neural networks; software Verification and model checking; hyperproperties and security; formal methods for hardware, cyber-physical, and hybrid systems. Part II: Probabilistic techniques; automata and logic; deductive verification and decision procedures; machine learning; synthesis and concurrency. This is an open access book