TOrPEDO: witnessing model correctness with topological proofs

Model design is not a linear, one-shot process. It proceeds through refinements and revisions. To effectively support developers in generating model refinements and revisions, it is desirable to have some automated support to verify evolvable models. To address this problem, we recently proposed to adopt topological proofs, which are slices of the original model that witness property satisfaction. We implemented , a framework that provides automated support for using topological proofs during model design. Our results showed that topological proofs are significantly smaller than the original models, and that, in most of the cases, they allow the property to be re-verified by relying only on a simple syntactic check. However, our results also show that the procedure that computes topological proofs, which requires extracting unsatisfiable cores of LTL formulae, is computationally expensive. For this reason, currently handles models with a small dimension. With the intent of providing practical and efficient support for flexible model design and wider adoption of our framework, in this paper, we propose an enhanced—re-engineered—version of . The new version of relies on a novel procedure to extract topological proofs, which has so far represented the bottleneck of performances. We implemented our procedure within by considering Partial Kripke Structures (PKSs) and Linear-time Temporal Logic (LTL): two widely used formalisms to express models with uncertain parts and their properties. To extract topological proofs, the new version of converts the LTL formulae into an SMT instance and reuses an existing SMT solver (e.g., Microsoft ) to compute an unsatisfiable core. Then, the unsatisfiable core returned by the SMT solver is automatically processed to generate the topological proof. We evaluated by assessing (i) how does the size of the proofs generated by compares to the size of the models being analyzed; and (ii) how frequently the use of the topological proof returned by avoids re-executing the model checker. Our results show that provides proofs that are smaller ($$\approx $$ ≈ 60%) than their respective initial models effectively supporting designers in creating model revisions. In a significant number of cases ($$\approx $$ ≈ 79%), the topological proofs returned by enable assessing the property satisfaction without re-running the model checker. We evaluated our new version of by assessing (i) how it compares to the previous one; and (ii) how useful it is in supporting the evaluation of alternative design choices of (small) model instances in applied domains. The results show that the new version of is significantly more efficient than the previous one and can compute topological proofs for models with less than 40 states within two hours. The topological proofs and counterexamples provided by are useful to support the development of alternative design choices of (small) model instances in applied domains.

The braid group and its presentation

In this paper, we recall the geometric definition of the braid group by Emil Artin and we give a complete, elementary geometric/topological proof of the standard presentation of the braid group on [Formula: see text] strings.

Geometrical realisations of the simple permutoassociahedron by Minkowski sums

This paper introduces a family of n-polytopes, PAn,c which is a geometrical realisation of simple permutoassociahedra. It has significant importance serving as a topological proof of Mac Lane's coherence. Polytopes in this family are defined as Minkowski sums of certain polytopes such that every summand produces exactly one truncation of the permutohedron, i.e. yields to the appropriate facet of the resulting sum. Additionally, it leads to the correlation between Minkowski sums and truncations, which gives a general procedure for similar geometrical realisation of a wider class of polytopes.

CONTINUITY OF ROOTS, REVISITED

The aim of this note is to give a simple topological proof of the well-known result concerning continuity of roots of polynomials. We also consider a more general case with polynomials of a higher degree approaching a given polynomial. We then examine the continuous dependence of solutions of linear differential equations with constant coefficients.