Tree rules in probabilistic transition system specifications with negative and quantitative premises

TL;DR

This paper extends probabilistic transition system specifications to include nested convex combinations, removing the need for well-foundedness, and proves that bisimilarity remains a congruence in this broader setting.

Contribution

It introduces an extended PTSS format with nested convex combinations and shows bisimilarity is a congruence without the well-foundedness restriction.

Findings

Extended PTSS format guarantees bisimilarity as a congruence.

Constructs equivalent well-founded transition systems from non-well-founded PTSS.

Develops a proof-theoretic notion matching stratification-based semantics.

Abstract

Probabilistic transition system specifications (PTSSs) in the ntmufnu/ntmuxnu format provide structural operational semantics for Segala-type systems that exhibit both probabilistic and nondeterministic behavior and guarantee that isimilarity is a congruence.Similar to the nondeterministic case of rule format tyft/tyxt, we show that the well-foundedness requirement is unnecessary in the probabilistic setting. To achieve this, we first define an extended version of the ntmufnu/ntmuxnu format in which quantitative premises and conclusions include nested convex combinations of distributions. This format also guarantees that bisimilarity is a congruence. Then, for a given (possibly non-well-founded) PTSS in the new format, we construct an equivalent well-founded transition system consisting of only rules of the simpler (well-founded) probabilistic ntree format. Furthermore, we develop a proof-theoretic notion for these PTSSs that coincides with the existing stratification-based meaning in case the PTSS is stratifiable. This continues the line of research lifting structural operational semantic results from the nondeterministic setting to systems with both probabilistic and nondeterministic behavior.