Categorical liveness checking by corecursive algebras

Abstract

Abstract-Final coalgebras as "categorical greatest fixed points" play a central role in the theory of coalgebras. Somewhat analogously, most proof methods studied therein have focused on greatest fixed-point properties like safety and bisimilarity. Here we make a step towards categorical proof methods for least fixed-point properties over dynamical systems modeled as coalgebras. Concretely, we seek a categorical axiomatization of well-known proof methods for liveness, namely ranking functions (in nondeterministic settings) and ranking supermartingales (in probabilistic ones). We find an answer in a suitable combination of coalgebraic simulation (studied previously by the authors) and corecursive algebra as a classifier for (non-)well-foundedness.

Ranking and Repulsing Supermartingales for Reachability in Randomized Programs

“…Ranking supermartingales are amenable to template-based synthesis [10,11,13], making them appealing from the automatic analysis point of view. Recently, methods for quantitatively under-approximating reachability probabilities are also proposed in References [15,41].…”

Ranking and Repulsing Supermartingales for Reachability in Probabilistic Programs

“…The notion of γ-SclSubM is first introduced in [33], as an instance of a categorical abstraction of ranking functions. The current paper's contribution lies in the following: i) the theoretical developments about γ-SclSubM in concrete (non-categorical) terms; ii) introduction of nondeterminism (the setting of [33] is purely probabilistic); and iii) template-based synthesis of γ-SclSubM. Definition 6.1 (γ-SclSubM for pCFG, [33]).…”

“…[24,8] γ-scaled submartingale (γ-SclSubM, §6) P(reach C) ≥ ? this paper for PP, following categorical observations in [33] for MC repulsing supermartingale for over-approximation ε-decreasing supermartingale (ε-RepSupM, §3) P(reach C) ≤ ? P(reach C) < ?…”

“…Ranking supermartingales are amenable to template-based synthesis [8,11,9], making them appealing from the automatic analysis point of view. Recently, methods for quantitatively underapproximating reachability probabilities are also proposed in [13,33]. Table 1.…”

“…-We introduce two martingale-based techniques that seem to be new, namely γ-SclSubM and NNRepSupM in Table 1. Their purely probabilistic versions have been in the literature: γ-SclSubM is from a category-theoretic account in [33], and…”