Probabilistic Model Checking of Recursive Programs with Conditioning

Core Concepts

This paper introduces a new class of probabilistic pushdown automata, called Probabilistic Operator Precedence Automata (pOPA), that can model recursive probabilistic programs with conditioning. It develops a model checking algorithm for verifying temporal logic specifications expressed in a fragment of Precedence Oriented Temporal Logic (POTLf X) on pOPA in single exponential time.

Abstract

The paper addresses the problem of model checking temporal logic specifications for probabilistic programs with recursive procedures, nested queries, and conditioning expressed with observe statements. Key highlights: Introduces pOPA, a new class of probabilistic pushdown automata suitable to model constructs and behaviors of probabilistic programs. Develops a model checking algorithm that can verify requirements expressed in a fragment of POTLf X on a pOPA in single EXPTIME. POTLf X is a temporal logic based on Operator Precedence Languages, which features modalities that interact with the context-free structure of program traces. Provides the first probabilistic model checking implementation of context-free language properties for probabilistic pushdown systems. Introduces the concept of separated ωOPBA, which allows for efficient model checking by exploiting a backward-deterministic support graph. Defines the support chain, a finite Markov chain that encodes the behavior of non-terminating pOPA runs while preserving the probability distribution. Implements the model checking algorithm using both an SMT-based approach and a numerical semi-algorithm, and evaluates it on several case studies.

Stats

global p := Categorical(0, 0.1 … 1); aliceLoc := Bernoulli(0.55); bobLoc := Bernoulli(0.55);

Quotes

"Probabilistic programming languages (PPLs) [29] provide primitives to sample from probability distributions and to perform inference tasks (queries) based on evidence of observed events (conditioning)." "Verifying formal properties on such programs is a very challenging task and, despite significant theoretical contributions [23,10,19,47], there are not yet model checking approaches and tools directly addressing these constructs all together."

Key Insights Distilled From

Model Checking Recursive Probabilistic Programs with Conditioning

by Francesco Po... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2404.03515.pdf

Deeper Inquiries

How can the expressiveness of the POTLf X logic be further extended to capture a wider range of properties for recursive probabilistic programs

The expressiveness of the POTLf X logic can be further extended by introducing additional modalities or operators that can capture more complex properties of recursive probabilistic programs. One approach could be to incorporate temporal operators that allow for reasoning about probabilistic transitions over multiple steps or iterations. This could involve defining operators that track the evolution of probabilities over time or across different branches of recursion. Additionally, introducing modalities that can handle probabilistic dependencies or correlations between different program states could enhance the logic's ability to model intricate probabilistic behaviors in recursive programs.

What are the limitations of the pOPA model in terms of the types of probabilistic programs it can accurately represent, and how could it be generalized

The pOPA model has limitations in terms of the types of probabilistic programs it can accurately represent, particularly when it comes to handling complex dependencies or interactions between different program components. To generalize the model, one could consider extending the state space to incorporate more sophisticated probabilistic behaviors, such as non-Markovian processes or continuous distributions. Additionally, enhancing the transition functions to support more intricate probabilistic operations, like conditional probabilities or dynamic parameter updates, could broaden the model's applicability to a wider range of probabilistic programs.

What are the potential applications of the probabilistic model checking techniques developed in this work beyond the domain of recursive probabilistic programs, such as in other areas of formal verification or probabilistic systems analysis

The probabilistic model checking techniques developed for recursive probabilistic programs have potential applications beyond this specific domain. These techniques could be utilized in the verification of various probabilistic systems, such as stochastic models in biology, finance, or cybersecurity. They could also be applied to analyze the reliability and performance of distributed systems, where probabilistic behaviors play a crucial role. Furthermore, the methods could be adapted for use in machine learning systems to verify the correctness and robustness of probabilistic algorithms or models. Overall, the developed techniques offer a versatile framework for probabilistic systems analysis and formal verification in diverse application domains.