[Paper Review] The compositional construction of Markov processes

L. de Francesco Albasini, N. Sabadini+1 more|arXiv (Cornell University)|Jan 16, 2009

Logic, Reasoning, and Knowledge3 citations

TL;DR

This paper introduces a compositional algebra for modeling probabilistic systems using Markov automata, where transitions are governed by probabilities associated with signals on left and right interfaces. It demonstrates how to compute the probability of reaching deadlock in the dining philosophers problem, proving via Perron-Frobenius theory that this probability approaches 1 as steps increase, offering a mathematically elegant and expressive alternative to non-compositional models.

ABSTRACT

We describe an algebra for composing automata in which the actions have probabilities. We illustrate by showing how to calculate the probability of reaching deadlock in k steps in a model of the classical Dining Philosopher problem, and show, using the Perron-Frobenius Theorem, that this probability tends to 1 as k tends to infinity.

Motivation & Objective

• To develop a compositional framework for modeling probabilistic systems using Markov automata with probabilistic transitions on dual interfaces.
• To address limitations of existing non-compositional or interleaved models in concurrency theory by incorporating conditional probabilities and signal-based transitions.
• To provide a mathematically rigorous and expressive model capable of capturing complex probabilistic behaviors, such as those in the dining philosophers problem.
• To demonstrate the utility of the framework through a concrete analysis of deadlock probability in a probabilistic dining philosophers system.

Proposed method

• Define Markov automata as weighted automata with non-negative transition matrices indexed by signals from left and right interfaces, where the total transition matrix has row sums equal to 1.
• Introduce normalization to convert weighted automata into Markov automata by renormalizing transition probabilities based on total outgoing weight from each state.
• Use the power construction to model sequences of k transitions, forming k-step transition matrices as products of individual transition matrices.
• Apply the Perron-Frobenius theorem to analyze long-term behavior, particularly convergence of state distributions in irreducible Markov chains.
• Model the dining philosophers problem using state-dependent transitions for forks and philosophers, with probabilistic actions for picking up and putting down forks.
• Use conditional probability reasoning to account for dependencies introduced by composition, especially in synchronized actions like multiple philosophers attempting to pick up forks simultaneously.

Experimental results

Research questions

• RQ1 Can a compositional algebra be developed for probabilistic automata that supports both parallel and sequential composition while preserving probabilistic semantics?
• RQ2 How can conditional probabilities be properly modeled in composite systems where transitions depend on signals from multiple components?
• RQ3 What is the long-term behavior of probabilistic systems like the dining philosophers under general probabilistic action rules, particularly regarding deadlock occurrence?
• RQ4 Can the Perron-Frobenius theorem be applied to prove convergence to certain absorbing states (e.g., deadlock) in such systems?
• RQ5 How does the proposed model compare in expressiveness and mathematical elegance to existing models like Rabin’s probabilistic automata or broadcast-based concurrency models?

Key findings

• The probability of reaching deadlock in the dining philosophers problem tends to 1 as the number of steps k tends to infinity, under the given probabilistic transition rules.
• The model proves that in any non-initial, non-deadlock configuration of the dining philosophers system, there exists a transition that increases the number of forks in state 1, implying eventual absorption into deadlock.
• The result holds regardless of the specific positive probabilities assigned to actions, as long as all transition probabilities are strictly positive.
• The framework supports the composition of probabilistic systems through signal-based synchronization, enabling modeling of simultaneous actions such as multiple philosophers picking up their left forks in one step.
• The normalization procedure ensures that weighted automata can be converted into Markov automata while preserving the relative probabilities of transitions.
• The model is more expressive than prior models such as Rabin’s probabilistic automata or the broadcast-based model in [5], as it supports true simultaneous actions and conditional probabilities arising from composition.