TREBL -- A Relative Complete Temporal Event-B Logic. Part I: Theory

2025-09-01

AI Summary

Event-B’s standard logic lacks expressiveness for trace-level liveness properties. To address this, we propose TREBL (Trace-Extended Event-B Logic), a trajectory-extension of Event-B that formalizes and verifies liveness conditions—such as fairness and responsiveness—over execution paths, while retaining a state-based semantics (i.e., interpretations remain grounded in individual states, not full traces). Our key contribution is a syntactically restricted, yet expressive fragment of TREBL capable of capturing essential liveness properties, together with a relatively complete deductive proof system; its soundness and relative completeness rely on the definability of variant terms throughout refinement. Leveraging Event-B’s state-machine model and the principle that each state determines its possible future trajectories, TREBL reduces trace-level reasoning to iterative state-assertion reasoning. We validate TREBL’s expressiveness and verification efficacy through case studies drawn from safety-critical systems.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Representing, Reasoning, and Using Provenance, Trust

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

The verification of liveness conditions is an important aspect of state-based rigorous methods. This article addresses the extension of the logic of Event-B to a powerful logic, in which properties of traces of an Event-B machine can be expressed. However, all formulae of this logic are still interpreted over states of an Event-B machine rather than traces. The logic exploits that for an Event-B machine $M$ a state $S$ determines all traces of $M$ starting in $S$. We identify a fragment called TREBL of this logic, in which all liveness conditions of interest can be expressed, and define a set of sound derivation rules for the fragment. We further show relative completeness of these derivation rules in the sense that for every valid entailment of a formula $varphi$ one can find a derivation, provided the machine $M$ is sufficiently refined. The decisive property is that certain variant terms must be definable in the refined machine. We show that such refinements always exist. Throughout the article several examples from the field of security are used to illustrate the theory.

Problem

• Extending Event-B logic to express trace properties
• Verifying liveness conditions in state-based systems
• Developing sound derivation rules for temporal logic

Innovation

• Extends Event-B logic for trace properties
• Defines TREBL fragment for liveness conditions
• Provides sound derivation rules with relative completeness