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

Abstract

One of the main challenges in the analysis of probabilistic programs is to compute invariant properties that summarise loop behaviours. Automation of invariant generation is still at its infancy and most of the times targets only expected values of the program variables, which is insufficient to recover the full probabilistic program behaviour. We present a method to automatically generate moment-based invariants of a subclass of probabilistic programs, called Prob-solvable loops, with polynomial assignments over random variables and parametrised distributions. We combine methods from symbolic summation and statistics to derive invariants as valid properties over higher-order moments, such as expected values or variances, of program variables. We successfully evaluated our work on several examples where full automation for computing higher-order moments and invariants over program variables was not yet possible.

Mora - Automatic Generation of Moment-Based Invariants

“…A proof-of-concept implementation, together with initial experiments, were already given in our work on generating moment-based invariants [2]. MORA comes however with a new design and re-implementation of [2], significantly improving the experimental setting and evaluations of [2]. Table 1 compares MORA against the experiments of [2], on a subset of Prob-solvable loops from [2], evidencing that MORA is faster than our initial proof-of-concept implementation.…”

“…Input programs to MORA are PP loops that are Prob-solvable [2]. In Figure 1, we give an example of a Prob-solvable loop and use this example as a running example to guide the potential users of MORA in the rest of this paper.…”

“…Note, that even if 1 was omitted from the aforementioned input goal, MORA would still need to compute some of the first moments of the variables, as they are required for computing the second-order moments. In the sequel, we show-case the MORA behaviour for: mora("running", [1,2]).…”

“…Moreover, the invariants inferred by MORA are not restricted to expected values but are quantitative invariants over the higher-order moments of program variables. We refer to such invariants as moment-based invariants [2]. To the best of our knowledge, no other approach can so far automatically compute higher-order moments of PPs, not even for the restricted yet expressive enough class of Prob-solvable loop.…”

“…We focus on the usage and implementation aspects of MORA. For details on theoretical foundations and algorithmic aspects of MORA for computing moment-based invariants, we refer to [2]. We note however that, when compared to the experimental setup of [2], MORA comes with a completely new design, fully implemented in python and supporting an easy installation and use by even non-experts in PPs.…”