Team abstraction

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Section: New Results

Numerical Abstractions

Relational abstract domains used to be inefficient when implemented with rationals and unsound when implemented with machine floating point numbers. We have introduced new relational abstract domains which have efficient and sound implementations with floats.

An Abstract Domain to Infer Interval Linear Relationships

Participants : Liqian Chen, Antoine Miné, Ji Wang [National Laboratory for Parallel and Distributed Processing, Changsha, P. R. China] , Patrick Cousot.

We introduce in [20] a numerical abstract domain of systems of interval linear equalities, i.e., relations of the form Im1 ${\#8721 _k{[a_k;b_k]}x_k\#8712 {[c;d]}}$ . They encompass regular equalities (Karr's domain) as well as some (but not all) inequalities (polyhedra domain) and even some non-convex and unconnected properties (out of the scope of polyhedra). Systems are abstracted into a row echelon form through Gauss elimination algorithms to achieve a polynomial complexity, much lower than classic and interval [46] version of polyhedra. Interval arithmetic on coefficients makes it easy to implement the domain in a sound way using only floating-point arithmetics. The domain was implemented and tested within the Apron framework (see 5.3 ) with encouraging preliminary results.

Widenings for H -Polyhedra

Participants : Liqian Chen, Axel Simon [Technische Universität München, Garching, Germany] .

While the definition of the revised widening for polyhedra is defined in terms of inequalities, most implementations use the double description method as a means to an efficient implementation. In [30] , it is shown how standard widening can be implemented in a simple and efficient way using a normalized H -representation (constraint-only) which has become popular in recent approximations to polyhedral analysis. A novel heuristic for this representation is then tuned to capture linear transformations of the state space while ensuring quick convergence for non-linear transformations for which no precise linear invariants exist.

Linear Absolute Value Relation Analysis

Participants : Liqian Chen, Antoine Miné, Ji Wang [National Laboratory for Parallel and Distributed Processing, Changsha, P. R. China] , Patrick Cousot.

[21] proposes an abstract domain dealing with linear inequalities involving variables together with their absolute values. It is an extension of the classical linear relation analysis, which permits to deal with some non convex numerical sets. A first nice result states the equivalence between these “linear absolute value inequalities” (AVI) with “interval linear inequalities” (ILI), and “extended linear complementary inequalities” (XLCP, pairs of positive solutions whose pairwise components are not both not zero). The key contribution is the extension of the double-description of polyhedra to XLCP solutions, which is then used to define the standard operations on AVI. The method has been implemented, and experiments show interesting results, with reasonable performances with respect to linear relation analysis.

Graph-Based Weakly Relational Numerical Abstract Domains

Participant : Bouaziz Mehdi.

We introduce a new family of weakly relational numerical abstract domains that achieve improved performances by restricting a priori the set of variables related together. It is parametrized by a choice of a graph of variables to relate, and a numerical domain to represent constraints. We focus on variable graphs consisting of complete sub-graphs linked in a tree-structure, for which efficient complete (i.e. best precision) algorithms can be constructed. The internship report [39] describes our general construction, and studies in more details the case where the parameter numerical domain is the octagon one.


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