A modular cost analysis for probabilistic programs

Abstract

We present a novel methodology for the automated resource analysis of non-deterministic, probabilistic imperative programs, which gives rise to a modular approach. Program fragments are analysed in full independence. Moreover, the established results allow us to incorporate sampling from dynamic distributions, making our analysis applicable to a wider class of examples, for example the Coupon Collector's problem. We have implemented our contributions in the tool eco-imp, exploiting a constraint-solver over iterative refineable cost functions facilitated by off-the-shelf SMT solvers. We provide ample experimental evidence of the prototype's algorithmic power. Our experiments show that our tool runs typically at least one order of magnitude faster than comparable tools. On more involved examples, it may even be the case that execution times of seconds become milliseconds. At the same time we retain the precision of existing tools. The extensions in applicability and the greater efficiency of our prototype, yield scalability of sorts. This effects into a wider class of examples, whose expected cost analysis can be thus be performed fully automatically.CCS Concepts: • Theory of computation → Program analysis; Automated reasoning.

On continuation-passing transformations and expected cost analysis

“…2 Via a telescoping-sum argument, it can be shown that ecost ( ) coincides with the mean cost emitted along all probabilistic reduction paths, compare e.g. [Avanzini et al 2020], thereby matching its intended meaning. In a similar spirit, nf : Λ → D(V ) can be defined as the least function such that:…”

“…Operational Semantics. In order to capture the two effects, probabilistic sampling and cost, we express the operational semantics of the language Λ by a (probabilistic) reduction relation − − → ⊆ D(Λ )×R +∞ ×D(Λ ), defined through a (weighted) probabilistic abstract reduction system [Avanzini et al 2020;Bournez and Garnier 2005] over terms Λ . Triples ( , , ) ∈ − − →, written as − − → , signify that the term distribution evolves in one-step to a reduct-distribution , producing an expected cost equal to .…”

“…It can be shown that this cost-transformer is continuous and monotone, hence well-defined, see also [Avanzini et al 2020] where this operator is defined in a more general setting.…”

On Probabilistic Term Rewriting

“…Besides, the authors also investigate lower bounds on expected runtimes and tail-probabilities, topics not covered in our work. 1 While it is possible to translate such programs into probabilistic term rewrite systems [4], our method will not capture typical termination arguments on probabilistic integer programs, since properties of integers will be unavailable after the encoding. 2 We thank Luis María Ferrer Fioriti for this analysis.…”

Learning Probabilistic Termination Proofs

“…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.…”