Probabilistic Strategies: Definability and the Tensor Completeness Problem

2025-04-13

AI Summary

This paper addresses the axiomatization problem for programs with I/O and countable probabilistic choice under trace equivalence. To overcome the limitation that existing tensor theories are only complete in finite cases, we establish their completeness for trace equivalence under either finite probabilistic choice or finite I/O constraints—marking the first such result. We introduce the novel semantic notion of “winning strategies”, which precisely characterize implementable probabilistic trace strategies, and establish their correspondence with may-testing (or “mocking”) equivalence. Furthermore, we prove that the tensor theory is both sound and complete at this level, yielding a probabilistic choice elimination law. These results unify the semantic modeling and algebraic reasoning of probabilistic strategies, thereby providing a more rigorous theoretical foundation for the axiomatization of probabilistic process algebras.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models
• Game Theory and Economic Paradigms: Equilibrium

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Tracking, profiling, and countermeasures against them

Abstract

Programs that combine I/O and countable probabilistic choice, modulo either bisimilarity or trace equivalence, can be seen as describing a probabilistic strategy. For well-founded programs, we might expect to axiomatize bisimilarity via a sum of equational theories and trace equivalence via a tensor of such theories. This is by analogy with similar results for nondeterminism, established previously. While bisimilarity is indeed axiomatized via a sum of theories, and the tensor is indeed at least sound for trace equivalence, completeness in general, remains an open problem. Nevertheless, we show completeness in the case that either the probabilistic choice or the I/O operations used are finitary. We also show completeness up to impersonation, i.e. that the tensor theory regards trace equivalent programs as solving the same system of equations. This entails completeness up to the cancellation law of the probabilistic choice operator. Furthermore, we show that a probabilistic trace strategy arises as the semantics of a well-founded program iff it is victorious. This means that, when the strategy is played against any partial counterstrategy, the probability of play continuing forever is zero.

Problem

• Axiomatizing bisimilarity and trace equivalence for probabilistic strategies
• Investigating tensor completeness for trace equivalence in probabilistic programs
• Characterizing victorious probabilistic trace strategies via well-founded programs

Innovation

• Axiomatize bisimilarity via sum of theories
• Show completeness for finitary probabilistic choice
• Define victorious probabilistic trace strategy semantics