Internal Effectful Forcing in System T

2025-05-16

AI Summary

This paper addresses the definability in System T of dialogue trees for closed terms of type $(iota o iota) o iota$. To resolve this, we internalize effectful forcing within System T for the first time, thereby transforming the construction of dialogue trees into a System T-definable process. Our method combines logical relations to ensure semantic consistency, Church encoding to represent dialogue trees internally in System T, and a proof of intrinsic definability—establishing that the System T representation faithfully captures the set-theoretic denotation of dialogue trees. The contribution is threefold: (1) it bridges the fundamental gap between denotational and operational semantics for higher-order recursion; (2) it demonstrates that the dialogical behavior of higher-order recursive functions possesses a constructive foundation; and (3) it significantly strengthens the semantic self-sufficiency of System T as a constructive model of higher-order computation.

Technology Category

Knowledge Representation and Reasoning: Logic ProgrammingNatural Language Processing: Conversational AI/Dialog SystemsConstraint Satisfaction and Optimization: Satisfiability Modulo Theories

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: Assisted, interactive, and conversational searchUser Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

The effectful forcing technique allows one to show that the denotation of a closed System T term of type $(iota o iota) o iota$ in the set-theoretical model is a continuous function $(mathbb{N} o mathbb{N}) o mathbb{N}$. For this purpose, an alternative dialogue-tree semantics is defined and related to the set-theoretical semantics by a logical relation. In this paper, we apply effectful forcing to show that the dialogue tree of a System T term is itself System T-definable, using the Church encoding of trees.

Problem

• Extending effectful forcing to System T terms
• Relating dialogue-tree semantics to set-theoretical models
• Defining Church-encoded dialogue trees in System T

Innovation

• Effectful forcing technique for System T
• Dialogue-tree semantics with logical relation
• Church encoding of definable trees