Approximate Testing Equivalence Based on Time, Probability, and Observed Behavior

Several application domains require formal but flexible approaches to the comparison problem. Different process models that cannot be related by behavioral equivalences should be compared via a quantitative notion of similarity, which is usually achieved through approximation of some equivalence. While in the literature the classical equivalence subject to approximation is bisimulation, in this paper we propose a novel approach based on testing equivalence. As a step towards flexibility and usability, we study different relaxations taking into account orthogonal aspects of the process observations: execution time, event probability, and observed behavior. In this unifying framework, both interpretation of the measures and decidability of the verification algorithms are discussed.

💡 Research Summary

The paper addresses the problem of comparing process models when exact behavioral equivalence is too strict for practical applications such as web‑service composition, security analysis, or performance verification. Instead of approximating the classical bisimulation equivalence, the authors propose to approximate testing equivalence, a relation that observes a system through external tests and compares the probabilities of passing those tests within given time constraints.

The work is built on a Markovian Process Calculus (MPC) where every action is exponentially timed with a rate λ, and the race policy selects the fastest enabled action. From the labeled multitransition system of a process term, a continuous‑time Markov chain (CTMC) is derived. Tests are finite, non‑recursive process terms written in a canonical form; a test succeeds when a designated success state is reached during parallel composition with the system under test. The original testing equivalence (denoted ∼MT) requires that for all canonical tests T and for all sequences θ of average sojourn times, the probability of successful test‑driven computations of the two processes be identical.

The authors then introduce three orthogonal dimensions along which this strict requirement can be relaxed:

1. Time Approximation – The sequence of average sojourn times is allowed to differ by a bounded tolerance ε. Two processes are “slow‑approximate” if every successful computation of the faster process can be matched by a computation of the slower one whose step‑wise average durations do not exceed the faster one by more than ε, and vice‑versa for “fast‑approximate”. This yields a partial order ⪯_time that is transitive and can be used to reason about chains of approximations.

2. Probability Approximation – Instead of exact equality of success probabilities, an absolute (δ) or relative (α) error margin is permitted. The distance d_prob(P₁,P₂) is defined as the supremum over all tests of the absolute difference in success probabilities. This metric satisfies the triangle inequality and can be computed by solving linear equations derived from the CTMC’s rate matrix, leading to polynomial‑time algorithms.

3. Observed‑Behavior Approximation – The syntactic structure of tests is taken into account. Tests are represented as trees; a structural ε‑approximation allows two tests to differ in depth or branching within a bounded edit distance while preserving the same success/failure pattern. The resulting distance d_beh is based on tree‑edit distance and can be computed in O(m log m) where m is the number of nodes.

A unified approximate testing equivalence (≈MT) is finally defined by aggregating the three distances, either as a weighted sum d_total = w₁·d_time + w₂·d_prob + w₃·d_beh (with non‑negative weights summing to 1) or by a user‑defined priority ordering. If d_total stays below a chosen threshold θ, the two processes are considered approximately testing equivalent. The paper proves that this unified relation remains decidable because each component relies on canonical tests and finite CTMCs; algorithmic complexities are O(n·k) for the time component (n = states, k = tolerance steps), O(n³) for the probability component, and O(m log m) for the behavioral component. Moreover, the authors discuss how the non‑symmetry introduced by separate relaxations can be symmetrized if needed.

In comparison with related work on bisimulation and its probabilistic or timed extensions, the proposed approach offers a more observer‑centric view: it directly quantifies how an external tester would experience differences in execution time, likelihood of success, and the shape of the test itself. This makes the framework especially suitable for domains where timing and stochastic behavior are first‑class concerns, such as real‑time systems, security protocols with probabilistic attacks, and performance‑aware service orchestration.

The paper concludes by outlining future directions: extending the model to non‑exponential time distributions, automating the generation of representative canonical tests, and scaling the analysis to large distributed systems via compositional or parallel verification techniques. Overall, the work provides a flexible, mathematically rigorous toolkit for measuring similarity between stochastic timed processes when exact equivalence is unattainable.