Precise Fixpoint Computation Through Strategy Iteration

Abstract

Abstract. We present a practical algorithm for computing least solutions of systems of equations over the integers with addition, multiplication with positive constants, maximum and minimum. The algorithm is based on strategy iteration. Its run-time (w.r.t. the uniform cost measure) is independent of the sizes of occurring numbers. We apply our technique to solve systems of interval equations. In particular, we show how arbitrary intersections as well as full interval multiplication in interval equations can be dealt with precisely.

Template-Based Unbounded Time Verification of Affine Hybrid Automata

“…, x n are variables that take values from R = R ∪ {−∞, ∞} and f is an operator of a special structure that is in particular monotone and concave (cf. Gawlitza and Seidl [10,11,12,14,15]). Our max-strategy improvement algorithm for solving these systems of inequalities performs at most exponentially many strategy improvement steps, each of which can be performed in polynomial-time through linear programming.…”

“…Strategy iteration can be seen as an alternative to the traditional widening/narrowing approach of Cousot and Cousot [7]. For more information regarding these approaches see Adjé et al [1,2], Costan et al [6], Gaubert et al [9], Gawlitza and Seidl [10,11,12], Gawlitza and Monniaux [13], Gawlitza and Seidl [14,15].…”

“…In this paper, we will adapt the max-strategy improvement algorithm of Gawlitza and Seidl [10,11,12,14,15] for computing V . We will find that abstract reachability is in co−NP.…”

“…We aim at adapting the max-strategy improvement algorithms of Gawlitza and Seidl [10,11,12,14,15] in order to obtain an algorithm for computing abstract semantics. For that we have to formulate the problem as a problem of finding the least fixpoint of a self-map that is a maximum of finitely many monotone and concave self-maps (cf.…”

A Sums-of-Squares extension of policy iterations

“…In all the considered cases, no common quadratic Lyapunov existed. In other words, not only the existing linear abstractions such as intervals or polyhedra would fail in computing a non trivial post-fixpoint, but also the existing analyses dedicated to digital filters such as [Fer04,GS07,AGG12,RJGF12]. The analysis has been implemented in Matlab and relies on the Mosek SDP solver [AA00], through the Yalmip [L04] SOS front-end.…”