22 research outputs found
Optimal Analysis of Discrete-time Affine Systems
Our very first concern is the resolution of the verification problem for the
class of discrete-time affine dynamical systems. This verification problem is
turned into an optimization problem where the constraint set is the reachable
values set of the dynamical system. To solve this optimization problem, we
truncate the infinite sequences belonging to the reachable values set at some
step which is uniform with respect to the initial conditions. In theory, the
best possible uniform step is the optimal solution of a non-convex
semi-definite program. In practice, we propose a methodology to compute a
uniform step that over-approximate the best solution.Comment: 16 page
A Sums-of-Squares Extension of Policy Iterations
In order to address the imprecision often introduced by widening operators in
static analysis, policy iteration based on min-computations amounts to
considering the characterization of reachable value set of a program as an
iterative computation of policies, starting from a post-fixpoint. Computing
each policy and the associated invariant relies on a sequence of numerical
optimizations. While the early research efforts relied on linear programming
(LP) to address linear properties of linear programs, the current state of the
art is still limited to the analysis of linear programs with at most quadratic
invariants, relying on semidefinite programming (SDP) solvers to compute
policies, and LP solvers to refine invariants.
We propose here to extend the class of programs considered through the use of
Sums-of-Squares (SOS) based optimization. Our approach enables the precise
analysis of switched systems with polynomial updates and guards. The analysis
presented has been implemented in Matlab and applied on existing programs
coming from the system control literature, improving both the range of
analyzable systems and the precision of previously handled ones.Comment: 29 pages, 4 figure
Quadratic Zonotopes:An extension of Zonotopes to Quadratic Arithmetics
Affine forms are a common way to represent convex sets of using
a base of error terms . Quadratic forms are an
extension of affine forms enabling the use of quadratic error terms .
In static analysis, the zonotope domain, a relational abstract domain based
on affine forms has been used in a wide set of settings, e.g. set-based
simulation for hybrid systems, or floating point analysis, providing relational
abstraction of functions with a cost linear in the number of errors terms.
In this paper, we propose a quadratic version of zonotopes. We also present a
new algorithm based on semi-definite programming to project a quadratic
zonotope, and therefore quadratic forms, to intervals. All presented material
has been implemented and applied on representative examples.Comment: 17 pages, 5 figures, 1 tabl
Set-based value operators for non-stationary Markovian environments
This paper analyzes finite state Markov Decision Processes (MDPs) with
uncertain parameters in compact sets and re-examines results from robust MDP
via set-based fixed point theory. To this end, we generalize the Bellman and
policy evaluation operators to contracting operators on the value function
space and denote them as \emph{value operators}. We lift these value operators
to act on \emph{sets} of value functions and denote them as \emph{set-based
value operators}. We prove that the set-based value operators are
\emph{contractions} in the space of compact value function sets. Leveraging
insights from set theory, we generalize the rectangularity condition in classic
robust MDP literature to a containment condition for all value operators, which
is weaker and can be applied to a larger set of parameter-uncertain MDPs and
contracting operators in dynamic programming. We prove that both the
rectangularity condition and the containment condition sufficiently ensure that
the set-based value operator's fixed point set contains its own extrema
elements. For convex and compact sets of uncertain MDP parameters, we show
equivalence between the classic robust value function and the supremum of the
fixed point set of the set-based Bellman operator. Under dynamically changing
MDP parameters in compact sets, we prove a set convergence result for value
iteration, which otherwise may not converge to a single value function.
Finally, we derive novel guarantees for probabilistic path-planning problems in
planet exploration and stratospheric station-keeping.Comment: 17 pages, 11 figures, 1 tabl
Computing the smallest fixed point of order-preserving nonexpansive mappings arising in positive stochastic games and static analysis of programs
The problem of computing the smallest fixed point of an order-preserving map
arises in the study of zero-sum positive stochastic games. It also arises in
static analysis of programs by abstract interpretation. In this context, the
discount rate may be negative. We characterize the minimality of a fixed point
in terms of the nonlinear spectral radius of a certain semidifferential. We
apply this characterization to design a policy iteration algorithm, which
applies to the case of finite state and action spaces. The algorithm returns a
locally minimal fixed point, which turns out to be globally minimal when the
discount rate is nonnegative.Comment: 26 pages, 3 figures. We add new results, improvements and two
examples of positive stochastic games. Note that an initial version of the
paper has appeared in the proceedings of the Eighteenth International
Symposium on Mathematical Theory of Networks and Systems (MTNS2008),
Blacksburg, Virginia, July 200
Polynomial Template Generation using Sum-of-Squares Programming
19 pages, 3 figures, 1 tableTemplate abstract domains allow to express more interesting properties than classical abstract domains. However, template generation is a challenging problem when one uses template abstract domains for program analysis. In this paper, we relate template computation with the program properties that we want to prove. We focus on one-loop programs with a conditional branch and assume that all the functions involved in these programs are polynomials. We formally define the notion of well-representative template basis with respect to a given property. The definition relies on the fact that template abstract domains produce inductive invariants. We show that these invariants can be obtained by solving certain systems of functional inequalities. Then, such systems can be strengthened using a hierarchy of sum-of-squares feasibility problems. Each step of the SOS hierarchy can possibly provide a solution which in turn yields feasible invariant bound together with a certificate that the desired property holds. The interest of this approach is illustrated on nontrivial program examples in polynomial arithmetic