Simple Types for Polymorphic Functions

2026-04-13

AI Summary

This work proposes a concise type system for combinatory logic that achieves expressive polymorphism without relying on explicit quantified types. The system assigns at most one type to each combinator, with polymorphism emerging dynamically during application and precisely capturing the structure of values. In contrast to Hindley-Milner typing, the approach supports a broader notion of polymorphism while avoiding complex type syntax. An efficient type inference algorithm is developed, preserving the formal simplicity of the system and providing a solid foundation for static program analysis.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

This paper introduces a simple type system for combinatory logic in which combinators have at most one type, whose polymorphism is revealed by application. The combinatory types exactly describe the structure of their values, which may be hidden by abstract types, such as list types and function types. Even without any quantified types, it supports polymorphism beyond that of the Hindley-Milner type system that underpins functional programming, and an effective type inference algorithm. Also, the simplicity of the formalism should make other static program analyses easier.

Problem

• simple types
• polymorphic functions
• combinatory logic
• type inference
• Hindley-Milner

Innovation

• combinatory logic
• simple type system
• polymorphism
• type inference
• Hindley-Milner