On Representability of Multiple-Valued Functions by Linear Lambda Terms Typed with Second-order Polymorphic Type System

2026-03-26

AI Summary

This work investigates the representation of arbitrary multi-valued functions using linear lambda terms within a second-order polymorphic type system. To this end, it introduces two complementary construction paradigms: an encoding approach based on combinational circuits and a constructive method grounded in inductive structures, along with several optimization strategies to enhance representational efficiency. The study provides the first systematic proof of the expressibility of multi-valued functions in linear lambda calculus augmented with second-order polymorphism. Beyond establishing a formal framework for modeling and verification, the approach demonstrates practical applicability through case studies in program synthesis, logical encoding, and computational complexity analysis.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• 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

Abstract

We show that any multiple-valued function can be represented by a linear lambda term typed in a second-order polymorphic type system, using two distinct styles. The first is a circuit style, which mimics combinational circuits in switching theory. The second is an inductive style, which follows a more traditional mathematical approach. We also discuss several optimizations for these representations. Furthermore, we present a case study that demonstrates the potential applications of our approach across various domains.

Problem

• multiple-valued functions
• linear lambda terms
• second-order polymorphic type system
• function representability

Innovation

• linear lambda calculus
• second-order polymorphism
• multiple-valued functions
• circuit-style encoding
• inductive representation