Finding Polynomial Loop Invariants for Probabilistic Programs

Abstract

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.

Automatic Generation of Moment-Based Invariants for Prob-Solvable Loops

“…In more detail, for each monomial M = x αj j from S, we substitute x αi i in M by its probabilistic behaviour. That is, the update of x i in the Prob-solvable loop P is rewritten, according to (8), into the sum of its two probabilistic updates, weighted by their respective probabilities (lines 5-7 of Alg. 1).…”

“…We used 7 programs from works [4,6,8,14,18] on invariant generation. These examples are given in lines 1-7 of Table 1; we note though that BINOMIAL("p") represents our generalisation of a binomial distribution example taken from [6,8,14] to a probabilistic program with parametrised probability p. We further crafted 6 examples of our own, illustrating the distinctive features of our work. These examples are listed in lines 8-13 of Table 1: lines 8-11 correspond to the examples of Fig.…”

Aiming low is harder: induction for lower bounds in probabilistic program verification

“…Amongst others, [Jones 1990], , [McIver and Morgan 2005], and [Hehner 2011] have furthered this line of research, e.g., by considering nondeterminism and proof rules for bounding preexpectations in the presence of loops. Work towards automation of weakest preexpectation reasoning was carried out, amongst others, by [Chen et al 2015], [Cock 2014], [Katoen et al 2010], and [Feng et al 2017]. Abstract interpretation of probabilistic programs was studied in this setting by [Monniaux 2005].…”

Formal Specification of Memory Coherence Protocol

“…The main difference in Central Manager and Improved Central Manager is the elimination of confirmation operation to manager [2]. The locking mechanism not only deals with local requests but also with remote requests.…”

“…As the shared virtual memory on the loosely coupled multiprocessors has no physically shared memory and the communication cost between processors is nontrivial. Thus the conflicts are not likely to be solved with negligible delay [2]. The problem that Kai Li faced in building the shared virtual memory was memory coherence problem and in [3] Kai Li et al are focusing on memory coherence problem for shared virtual memory and they provide a number of algorithms as a to solve memory coherence problem.…”