Polynomial Precise Interval Analysis Revisited

Abstract

Abstract. We consider a class of arithmetic equations over the complete lattice of integers (extended with −∞ and ∞) and provide a polynomial time algorithm for computing least solutions. For systems of equations with addition and least upper bounds, this algorithm is a smooth generalization of the Bellman-Ford algorithm for computing the single source shortest path in presence of positive and negative edge weights. The method then is extended to deal with more general forms of operations as well as minima with constants. For the latter, a controlled widening is applied at loops where unbounded increase occurs. We apply this algorithm to construct a cubic time algorithm for the class of interval equations using least upper bounds, addition, intersection with constant intervals as well as multiplication.

Solving systems of rational equations through strategy iteration

“…The technique is fully compatible with the use of widening, and whenever it applies, it improves both the precision and the performance of the analysis. Leroux and Sutre [2007], and Gawlitza et al [2009] studied cases where interval analysis can be done in polynomial time with respect to a uniform cost measure, where arithmetic operations and memory accesses are counted for O(1).…”

Parameterized Construction of Program Representations for Sparse Dataflow Analyses

“…On the other hand, we know that echo v 2 is always safe, for variable v 2 is initialized with a constant value. The [17].…”

Widening with Thresholds for Programs with Complex Control Graphs

“…When the "then" branch is not taken, nothing happens in the loop body. It makes the transfer function on node 2 extensive: (2) . Hence, the descending sequence will never improve the invariant Y (2) = i ≥ 0 found by the ascending sequence.…”

“…Some techniques solves directly (1) in the case where concrete properties are invariants on numerical variables. In [1,2] classes of equations on intervals are identified, for which the least solution can be computed exactly. Policy iteration methods solve (1) by solving a succession of simpler equations Y = G π (Y ) indexed by a policy π [3,4].…”