Title: Finding polynomial loop invariants for probabilistic programs,
Abstract: Quantitative loop invariants are an essential element in the verification of
probabilistic programs. Recently, multivariate Lagrange interpolation has been
applied to synthesizing polynomial invariants. In this paper, we propose an
alternative approach. First, we fix a polynomial template as a candidate of a
loop invariant. Using Stengle's Positivstellensatz and a transformation to a
sum-of-squares problem, we find sufficient conditions on the coefficients.
Then, we solve a semidefinite programming feasibility problem to synthesize the
loop invariants. If the semidefinite program is unfeasible, we backtrack after
increasing the degree of the template. Our approach is semi-complete in the
sense that it will always lead us to a feasible solution if one exists and
numerical errors are small. Experimental results show the efficiency of our
approach.