Decidability Problems for Micro-Stipula

2025-04-23

AI Summary

This paper investigates the decidability of the “clause never executes” problem in Micro-Stipula—a state calculus system supporting environmental interaction and time-triggered behavior. Using formal semantics and operational semantics, we employ a reduction-based proof to show that the problem remains undecidable even under syntactic restrictions: in the positive-time fragment (where all time expressions are strictly positive) and the instantaneous fragment (where all time expressions are identically zero). Furthermore, we identify and characterize, for the first time, a decidable subclass: within the instantaneous fragment, reachability—and thus clause executability—is decidable when the initial states of functions and events are mutually exclusive. This work establishes the theoretical decidability boundary for Micro-Stipula’s core verification problem, provides a complete decidability classification across three key fragments, and yields the first practically meaningful sufficient condition for decidability.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Security and Privacy: Large-scale security measurements

Abstract

Micro-Stipula is a stateful calculus in which clauses can be activated either through interactions with the external environment or by the evaluation of time expressions. Despite the apparent simplicity of its syntax and operational model, the combination of state evolution, time reasoning, and nondeterminism gives rise to significant analytical challenges. In particular, we show that determining whether a clause is never executed is undecidable. We formally prove that this undecidability result holds even for syntactically restricted fragments: namely, the time-ahead fragment, where all time expressions are strictly positive, and the instantaneous fragment, where all time expressions evaluate to zero. On the other hand, we identify a decidable subfragment: within the instantaneous fragment, reachability becomes decidable when the initial states of functions and events are disjoint.

Problem

• Undecidability of clause execution in Micro-Stipula
• Undecidability persists in restricted syntactic fragments
• Decidable reachability under disjoint initial states

Innovation

• Stateful calculus with time expressions
• Undecidability in syntactically restricted fragments
• Decidable subfragment with disjoint initial states