Automated Termination Analysis of Polynomial Probabilistic Programs

Abstract

The termination behavior of probabilistic programs depends on the outcomes of random assignments. Almost sure termination (AST) is concerned with the question whether a program terminates with probability one on all possible inputs. Positive almost sure termination (PAST) focuses on termination in a finite expected number of steps. This paper presents a fully automated approach to the termination analysis of probabilistic while-programs whose guards and expressions are polynomial expressions. As proving (positive) AST is undecidable in general, existing proof rules typically provide sufficient conditions. These conditions mostly involve constraints on supermartingales. We consider four proof rules from the literature and extend these with generalizations of existing proof rules for (P)AST. We automate the resulting set of proof rules by effectively computing asymptotic bounds on polynomials over the program variables. These bounds are used to decide the sufficient conditions – including the constraints on supermartingales – of a proof rule. Our software tool Amber can thus check AST, PAST, as well as their negations for a large class of polynomial probabilistic programs, while carrying out the termination reasoning fully with polynomial witnesses. Experimental results show the merits of our generalized proof rules and demonstrate that Amber can handle probabilistic programs that are out of reach for other state-of-the-art tools.

Learning Probabilistic Termination Proofs

“…We compare our framework against three existing tools. The first is Amber [39]: where possible, we translate instances from the WTC suite into the language of Amber, but this is not possible for some programs where the loop predicate is a logical conjunction or disjunction of predicates (indicated by dashes in Table 1). Second, we compare against a tool for synthesising affine lexicographic RSMs (referred to as Farkas' lemma) for affine programs (i.e.…”

“…Our method naturally generalises to deeper networks, but whether these are necessary in practice remains an open question; notably, neural networks with one hidden layer were sufficient to solve our examples. We have exclusively tackled the PAST question, and techniques for almost-sure (but not necessarily PAST) termination and non-termination exist [16,37,39]. Our results pose the basis for future research in machine learning (and CEGIS) for the formal verification of probabilistic programs.…”

“…Since the program cannot be encoded as a prob-solvable loop (due to the disjunctive guard predicate which cannot be replaced by a polynomial inequality), it cannot be handled by another recent tool, Amber [39]. However, this program admits the following piecewise-linear NRSM:…”

“…These methods have lent themselves to various generalisations, including polynomial programs, programs with non-determinism, lexicographic and modular termination arguments, and persistence properties [2,[14][15][16]20,25]. Recently, for special classes of probabilistic programs or term rewriting systems, novel automated proof techniques that leverage computer algebra systems and satisfiability modulo theories (SMT) have been introduced [5,6,38,39,41]. All the above methods are sound and, under specific assumptions, complete; they represent the state of the art for the class of programs they have been designed for.…”

“…This includes joint distributions, state-dependent distributions, distributions whose parameters are in turn random (hierarchical models), and distributions with undefined moments (e.g., the Cauchy distribution). We compare our method with a tool based on Farkas' lemma and with the tools Amber and Absynth [2,39,41]; whilst our software prototype is slower than these alternatives, it covers the widest range of benchmark single-loop programs.…”