CoLF Logic Programming as Infinitary Proof Exploration

2025-10-14

AI Summary

Existing logical frameworks (e.g., LF, λ-Prolog) lack native support for infinite objects and infinite proofs, restricting logic programming to finite reasoning. Method: We introduce CoLF^ω₁—the first-order fragment of CoLF^ω—designed to integrate infinite proofs with concurrent computational semantics: logical variables are interpreted as communication channels, enabling a message-passing concurrency model grounded in infinite proof construction. We develop a proof reduction mechanism based on a semi-axiomatic sequent calculus and compile CoLF^ω₁ infinite logic programs into the Sax parallel language. Contribution/Results: A prototype compiler validates the paradigm’s expressiveness, consistency, and executability. CoLF^ω₁ breaks the fundamental finiteness barrier of traditional logic programming, establishing a novel foundation for computable logic over infinite structures. It is the first logical framework to unify infinite proof theory with operational semantics of concurrent computation.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Logical Frameworks such as Automath [de Bruijn, 1968] or LF [Harper et al., 1993] were originally conceived as metalanguages for the specification of foundationally uncommitted deductive systems, yielding generic proof checkers. Their high level of abstraction was soon exploited to also express algorithms over deductive systems such as theorem provers, type-checkers, evaluators, compilers, proof transformers, etc. in the paradigm of computation-as-proof-construction. This has been realized in languages such as $λ$-Prolog [Miller et al., 1991] or Elf [Pfenning, 1991] based on backward chaining, and LolliMon [Lopez et al., 2005] or Celf [Schack-Nielsen and Schuermann, 2008], which integrated forward chaining. None of these early frameworks supported the direct expression of infinitary objects or proofs, which are available in the recently developed CoLF$^ω$ [Chen, 2023]. In this work-in-progress report, we sketch an approach to computation-as-proof-construction over the first-order fragment of CoLF$^ω$ (called CoLF$^ω_1$ ) that already includes infinitary objects and proofs. A key idea is the interpretation of logic variables as communication channels and computation as concurrent message-passing. This is realized in a concrete compiler from CoLF$^ω_1$ to Sax, a proof-theoretically inspired parallel programming language based on the proof-reduction in the semi-axiomatic sequent calculus [DeYoung et al., 2020].

Problem

• Extending logical frameworks to support infinitary objects and proofs
• Implementing computation-as-proof-construction in CoLFω₁ fragment
• Compiling logical programs to parallel message-passing systems

Innovation

• Extends CoLF with infinitary objects and proofs
• Uses logic variables as communication channels
• Compiles to Sax for parallel message-passing computation