# Precise Fixpoint Computation Through Strategy Iteration

@inproceedings{Gawlitza2007PreciseFC, title={Precise Fixpoint Computation Through Strategy Iteration}, author={Thomas Gawlitza and Helmut Seidl}, booktitle={ESOP}, year={2007} }

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…

## 82 Citations

Polynomial Precise Interval Analysis Revisited

- Mathematics, Computer ScienceEfficient Algorithms
- 2009

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 and applied 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.

Logico-Numerical Max-Strategy Iteration

- Computer ScienceVMCAI
- 2013

This paper proposes a method for applying max-strategy iteration to logico-numerical programs, i.e. programs with numerical and Boolean variables, without explicitly enumerating the Boolean state space, and gives experimental evidence about the efficiency and precision of the approach.

Derivation tree analysis for accelerated fixed-point computation

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2008

Improving Strategies via SMT Solving

- Computer ScienceESOP
- 2011

The algorithm has low polynomial space complexity and performs for contrived examples in the worst case exponentially many strategy improvement steps; this is unsurprising, since the associated abstract reachability problem is Pi-p-2-complete.

Computing Relaxed Abstract Semantics w.r.t. Quadratic Zones Precisely

- Computer ScienceSAS
- 2010

A relaxed abstract semantics is used and a practical strategy improvement algorithm is presented for precisely computing least solutions of fixpoint equation systems, whose right-hand sides use order-concave operators and the maximum operator.

Computing the smallest xed point of nonexpansive mappings arising in game theory and static analysis of programs

- Mathematics
- 2009

The problem of computing the smallest xed point of a monotone map arises classically in the study of zero-sum repeated games. It also arises in static analysis of programs by abstract interpretation.…

When the Decreasing Sequence Fails

- Computer ScienceSAS
- 2012

This paper proposes a method to improve a fixpoint after its computation that consists in projecting the solution onto well-chosen components and to start again increasing and decreasing sequences from the result of the projection.

Template-Based Unbounded Time Verification of Affine Hybrid Automata

- Computer ScienceAPLAS
- 2011

A max-strategy improvement algorithm is used for computing an abstract semantics for affine hybrid automata that is based on template polyhedra and safely over-approximates the concrete semantics and shows that the corresponding abstract reachability problem is in co-NP.

Practical policy iterations

- Computer ScienceFormal Methods Syst. Des.
- 2015

The rationale behind policy iteration is recalled and required steps towards an automatic use of it are addressed: synthesis of numerical templates, floating point semantics of the analyzed program and issues with the accuracy of numerical solvers.

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