Continuation Semantics for Fixpoint Modal Logic and Computation Tree Logics

2025-10-06

AI Summary

This paper addresses fixed-point modal logic (FML) and Computation Tree Logic* (CTL*), systematically constructing, for the first time, continuation semantics applicable to arbitrary branching types—including deterministic, nondeterministic, and probabilistic—and quantitative predicate liftings. It rigorously establishes equivalence between this continuation semantics and coalgebraic semantics. Methodologically, it models temporal behavior via the continuation monad, introduces predicate liftings and monad morphisms to propagate execution maps, and innovatively replaces the traditional requirement of infinite traces in CTL* coalgebraic models with a non-greatest fixed point—namely, the “execution map.” Key contributions include: (i) a unified semantic framework accommodating both multi-branching systems and quantified logics; (ii) a general proof of equivalence between continuation and coalgebraic semantics; and (iii) enhanced flexibility for model migration and a provably correct encoding of CTL into FML.

Technology Category

Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesMachine Learning: Life-Long and Continual LearningReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactionsSearch and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAGGraph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We introduce continuation semantics for both fixpoint modal logic (FML) and Computation Tree Logic* (CTL*), parameterised by a choice of branching type and quantitative predicate lifting. Our main contribution is proving that they are equivalent to coalgebraic semantics, for all branching types. Our continuation semantics is defined over coalgebras of the continuation monad whose answer type coincides with the domain of truth values of the formulas. By identifying predicates and continuations, such a coalgebra has a canonical interpretation of the modality by evaluation of continuations. We show that this continuation semantics is equivalent to the coalgebraic semantics for fixpoint modal logic. We then reformulate the current construction for coalgebraic models of CTL*. These models are usually required to have an infinitary trace/maximal execution map, characterized as the greatest fixpoint of a special operator. Instead, we allow coalgebraic models of CTL* to employ non-maximal fixpoints, which we call execution maps. Under this reformulation, we establish a general result on transferring execution maps via monad morphisms. From this result, we obtain that continuation semantics is equivalent to the coalgebraic semantics for CTL*. We also identify a sufficient condition under which CTL can be encoded into fixpoint modal logic under continuation semantics.

Problem

• Develop continuation semantics for fixpoint modal logic and CTL*
• Prove equivalence between continuation and coalgebraic semantics
• Establish conditions for encoding CTL into fixpoint modal logic

Innovation

• Continuation semantics parameterized by branching and lifting
• Equivalence proven between continuation and coalgebraic semantics
• Execution maps replace maximal fixpoints in CTL* models