Coupling policy iteration with semi-definite relaxation to compute accurate numerical invariants in static analysis

@inproceedings{Adj2010CouplingPI,
  title={Coupling policy iteration with semi-definite relaxation to compute accurate numerical invariants in static analysis},
  author={Assal{\'e} Adj{\'e} and St{\'e}phane Gaubert and {\'E}ric Goubault},
  booktitle={Log. Methods Comput. Sci.},
  year={2010}
}
We introduce a new domain for finding precise numerical invariants of programs by abstract interpretation. This domain, which consists of level sets of non-linear functions, generalizes the domain of linear “templates” introduced by Manna, Sankaranarayanan, and Sipma. In the case of quadratic templates, we use Shor's semi-definite relaxation to derive computable yet precise abstractions of semantic functionals, and we show that the abstract fixpoint equation can be solved accurately by coupling… 
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References

SHOWING 1-10 OF 58 REFERENCES
A Policy Iteration Algorithm for Computing Fixed Points in Static Analysis of Programs
TLDR
A policy iteration algorithm for monotone self-maps of complete lattices for lattices arising in the interval abstraction of values of variables is introduced and analyzed.
Proving Program Invariance and Termination by Parametric Abstraction, Lagrangian Relaxation and Semidefinite Programming
TLDR
This new approach exploits the recent progress in the numerical resolution of linear or bilinear matrix inequalities by semidefinite programming using efficient polynomial primal/dual interior point methods generalizing those well-known in linear programming to convex optimization.
Computing Relaxed Abstract Semantics w.r.t. Quadratic Zones Precisely
TLDR
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.
Static Analysis by Policy Iteration on Relational Domains
We give a new practical algorithm to compute, in finite time, a fixpoint (and often the least fixpoint) of a system of equations in the abstract numerical domains of zones and templates used for
Precise Relational Invariants Through Strategy Iteration
TLDR
A practical algorithm is presented for computing exact least solutions of systems of equations over the rationals with addition, multiplication with positive constants, minimum and maximum and applied to compute the abstract least fixpoint semantics of affine programs over the relational template constraint matrix domain.
Rigorous Error Bounds for the Optimal Value in Semidefinite Programming
TLDR
It turns out that in many cases the computational costs for postprocessing the output of a linear or semidefinite programming solver are small compared to the effort required by the solver.
Generation of Basic Semi-algebraic Invariants Using Convex Polyhedra
A technique for generating invariant polynomial inequalities of bounded degree is presented using the abstract interpretation framework. It is based on overapproximating basic semi-algebraic sets,
Reachability Analysis of Hybrid Systems Using Support Functions
TLDR
An approach for computing over-approximations of the set of reachable states based on the notion of support function that allows us to consider invariants, guards and constraints on continuous inputs and initial states defined by arbitrary compact convex sets.
Numerical Abstract Domains for Digital Filters ⋆
TLDR
This work proposes a systematic method for designing the abstract domain by using intervals and ellipsoidal constraints for designing digital filters, and gives a framework to deal with filter iteration, filter reinitialization, branching, loop, and so on.
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