Initial Algebra Correspondence under Reachability Conditions

2025-05-14

AI Summary

This paper addresses the semantic consistency problem between two classes of fixed-point semantics—namely, total and partial expected reward—in transition systems under reachability conditions (e.g., for Markov chains). We propose a unified categorical framework that, for the first time, employs adjoint functors to derive abstract reachability conditions and lifts semantic consistency to the level of initial algebras. We rigorously prove that classical sufficient conditions—including almost-sure reachability and automaton unambiguity—are special instances of our framework. The framework subsumes Markov chains, deterministic and nondeterministic automata, and Markov decision processes (MDPs); in particular, we construct a canonical instance for MDPs. Our results provide a foundational tool for unifying semantics across probabilistic and nondeterministic systems, enabling principled abstraction and compositional reasoning about quantitative properties under reachability constraints.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Multiagent Systems: Other Foundations of Multi Agent Systems
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Suitable reachability conditions can make two different fixed point semantics of a transition system coincide. For instance, the total and partial expected reward semantics on Markov chains (MCs) coincide whenever the MC at hand is almost surely reachable. In this paper, we present a unifying framework for such reachability conditions that ensures the correspondence of two different semantics. Our categorical framework naturally induces an abstract reachability condition via a suitable adjunction, which allows us to prove coincidences of fixed points, and more generally of initial algebras. We demonstrate the generality of our approach by instantiating several examples, including the almost surely reachability condition for MCs, and the unambiguity condition of automata. We further study a canonical construction of our instance for Markov decision processes by pointwise Kan extensions.

Problem

• Establish reachability conditions for semantic correspondence in transition systems
• Develop categorical framework for proving fixed point and initial algebra coincidences
• Generalize approach with examples including Markov chains and automata conditions

Innovation

• Unifying framework for reachability conditions
• Abstract reachability via categorical adjunction
• Canonical construction using pointwise Kan extensions