Xavier Allamigeon

Inferring Min and Max Invariants Using Max-Plus Polyhedra

We introduce a new numerical abstract domain able to infer min and max invariants over the program variables, based on max-plus polyhedra. Our abstraction is more precise than octagons, and allows to express non-convex properties without any disjunctive representations. We have defined sound abstract operators, evaluated their complexity, and implemented them in a static analyzer. It is able to automatically compute precise properties on numerical and memory manipulating programs such as algorithms on strings and arrays.

Reconstruction of attacks against cryptographic protocols

We study an automatic technique for the verification of cryptographic protocols based on a Horn clause model of the protocol. This technique yields proofs valid for an unbounded number of sessions of the protocol. However, up to now, it gave no definite information when the proof failed. In this paper, we present an algorithm for reconstructing an attack against the protocol when the desired security property does not hold. We have proved soundness, termination, as well as a partial completeness result for our algorithm. We have also implemented it in the automatic protocol verifier ProVerif. As an extreme example, we could reconstruct an attack involving 200 parallel sessions against f/sup 200/g/sup 200/ protocol (Millen, 1999).

The tropical double description method

We develop a tropical analogue of the classical double description method allowing one to compute an internal representation (in terms of vertices) of a polyhedron defined externally (by inequalities). The heart of the tropical algorithm is a characteri- zation of the extreme points of a polyhedron in terms of a system of constraints which define it. We show that checking the extremality of a point reduces to checking whether there is only one minimal strongly connected component in an hypergraph. The latter problem can be solved in almost linear time, which allows us to eliminate quickly redun- dant generators. We report extensive tests (including benchmarks from an application to static analysis) showing that the method outperforms experimentally the previous ones by orders of magnitude. The present tools also lead to worst case bounds which improve the ones provided by previous methods.

Computing the Vertices of Tropical Polyhedra Using Directed Hypergraphs

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.

Tropicalizing the Simplex Algorithm

We develop a tropical analogue of the simplex algorithm for linear programming. In particular, we obtain a combinatorial algorithm to perform one tropical pivoting step, including the computation of reduced costs, in

The number of extreme points of tropical polyhedra

O(n(m+n))

Tropical polar cones, hypergraph transversals, and mean payoff games☆

time, where

A scalable algebraic method to infer quadratic invariants of switched systems

m

On the Complexity of Strongly Connected Components in Directed Hypergraphs

is the number of constraints and

Solving Generic Nonarchimedean Semidefinite Programs Using Stochastic Game Algorithms

n