Probabilistic Abstract Interpretation: From Trace Semantics to DTMC’s and Linear Regression

Abstract

Abstract. In order to perform probabilistic program analysis we need to consider probabilistic languages or languages with a probabilistic semantics, as well as a corresponding framework for the analysis which is able to accomodate probabilistic properties and properties of probabilistic computations. To this purpose we investigate in this paper the relationship between three different types of probabilisitic semantics for a core imperative language, namely Kozen's Fixpoint Semantics (KFS), our Linear Operator Semantics (LOS) as well as probabilistic versions of Maximal Trace Semantics (MTS). We also discuss the relationship between Probabilistic Abstract Interpretation (PAI) and statistical or linear regression analysis. While classical Abstract Interpretation, based on Galois connection, allows only for worst-case analyses, the use of the Moore-Penrose pseudo inverse in PAI opens the possibility of exploiting statistical and noisy observations in order to analyse and identify various system properties.

Relatively complete verification of probabilistic programs: an expressive language for expectation-based reasoning

“…We consider Kozen's semantics the reference forward semantics. More operational semantics are provided in the form of probabilistic transition systems [Gretz et al 2014;Kaminski 2019], where programs describe potentially infinite Markov chains whose state spaces comprise of program states, or trace semantics [Cousot and Monerau 2012;Di Pierro and Wiklicky 2016;, where the traces are sequences of program states and each trace is assigned a certain probability.…”