Counterexample-Guided Polynomial Loop Invariant Generation by Lagrange Interpolation

Abstract

We apply multivariate Lagrange interpolation to synthesizing polynomial quantitative loop invariants for probabilistic programs. We reduce the computation of a quantitative loop invariant to solving constraints over program variables and unknown coefficients. Lagrange interpolation allows us to find constraints with less unknown coefficients. Counterexample-guided refinement furthermore generates linear constraints that pinpoint the desired quantitative invariants. We evaluate our technique by several case studies with polynomial quantitative loop invariants in the experiments.

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

“…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].…”

Finding Polynomial Loop Invariants for Probabilistic Programs

“…The program in this example originally served as a running example in [6]. There, after transforming the constraints into the form above, Lagrange interpolation is applied to synthesize the coefficients in the template.…”

“…The method is applied to several case studies taken from [6]. The technique usually solves the problem within one second, which is about one tenth of the time taken by the tool described in [6]. Our tool supports real variables rather than discrete ones, and supports programs that require polynomial invariants.…”

“…Moreover, we conduct a sequence of trials on parameterized probabilistic programs to show that the main influence factor on the running time of our method is the degree of the invariant template. We compare our results on these examples with the Lagrange Interpolation Prototype (LIP) in [6], Prinsys [15] and the tool for super-martingales (TM) [2].…”