[Paper Review] Bisimulations for Nondeterministic Labeled Markov Processes

Pedro R. D’Argenio, Pedro Sánchez Terraf+1 more|arXiv (Cornell University)|Nov 15, 2010

Formal Methods in Verification20 references4 citations

TL;DR

This paper introduces nondeterministic labeled Markov processes (NLMPs) to model systems with both continuous probabilistic transitions and internal nondeterminism. It defines three bisimulation notions—traditional, state-based, and event-based—and proves that the largest state bisimulation coincides with event bisimulation. The key contribution is a Hennessy-Milner-style logic that characterizes event bisimulation and, under image-finite and analytic Borel space conditions, all bisimulation notions collapse to equivalence.

ABSTRACT

We extend the theory of labeled Markov processes with internal nondeterminism, a fundamental concept for the further development of a process theory with abstraction on nondeterministic continuous probabilistic systems. We define nondeterministic labeled Markov processes (NLMP) and provide three definition of bisimulations: a bisimulation following a traditional characterization, a state based bisimulation tailored to our "measurable" non-determinism, and an event based bisimulation. We show the relation between them, including that the largest state bisimulation is also an event bisimulation. We also introduce a variation of the Hennessy-Milner logic that characterizes event bisimulation and that is sound w.r.t. the other bisimulations for arbitrary NLMP. This logic, however, is infinitary as it contains a denumerable $\lor$. We then introduce a finitary sublogic that characterize all bisimulations for image finite NLMP whose underlying measure space is also analytic. Hence, in this setting, all notions of bisimulation we deal with turn out to be equal. Finally, we show that all notions of bisimulations are different in the general case. The counterexamples that separate them turn to be non-probabilistic NLMP.

Motivation & Objective

• To extend labeled Markov processes (LMPs) to include internal nondeterminism, enabling abstraction in continuous probabilistic systems.
• To define and compare three notions of bisimulation—traditional, state-based, and event-based—for nondeterministic labeled Markov processes (NLMPs).
• To develop a modal logic that characterizes event bisimulation and is sound for other bisimulation types.
• To identify conditions under which all bisimulation notions coincide, particularly in image-finite NLMPs over analytic Borel spaces.
• To demonstrate that in the general case, the three bisimulation notions are distinct, using counterexamples over uncountable branching spaces.

Proposed method

• Define NLMPs as systems with a state space, labels, and a transition function assigning a measurable set of probability measures to each state and label.
• Introduce a σ-algebra on Giry’s space of probability measures to ensure measurability of transition functions.
• Define three bisimulation relations: traditional (based on relation between states), state-based (measurable invariance), and event-based (sub-σ-algebra refinement).
• Construct a two-sorted Hennessy-Milner logic with state formulas and measure formulas, allowing denumerable disjunctions to capture internal nondeterminism.
• Prove that the logic fully characterizes event bisimulation and is sound for traditional and state bisimulations.
• Establish that for image-finite NLMPs on analytic Borel spaces, all three bisimulation notions are equivalent.

Experimental results

Research questions

• RQ1 How can labeled Markov processes be extended to incorporate internal nondeterminism while preserving measurable structure?
• RQ2 What is the relationship between traditional, state-based, and event-based bisimulations in NLMPs?
• RQ3 Can a modal logic characterize event bisimulation in NLMPs, and how does it relate to other bisimulation notions?
• RQ4 Under what conditions do traditional, state, and event bisimulations coincide in NLMPs?
• RQ5 Are there natural counterexamples where the three bisimulation notions differ, and what do they reveal about the semantics of nondeterministic probabilistic systems?

Key findings

• The largest state bisimulation in NLMPs is also an event bisimulation, generalizing a key result from LMP theory.
• A Hennessy-Milner-style logic with denumerable disjunctions fully characterizes event bisimulation in NLMPs.
• For image-finite NLMPs over analytic Borel spaces, all three bisimulation notions (traditional, state, event) are equivalent.
• In the general case, counterexamples exist where the three bisimulation notions are distinct, with the most significant difference arising in non-probabilistic NLMPs over uncountable branching spaces.
• The traditional bisimulation is strictly finer than the state bisimulation in general, indicating that state bisimulation may be more appropriate for observable behavior.
• The logic cannot distinguish certain non-measurable or measurable sets in counterexamples, suggesting limitations in capturing all behavioral distinctions without additional label structure.