A Word Sampler for Well-Typed Functions

2025-11-30

AI Summary

This paper addresses uniform random sampling without replacement of well-typed functions in a first-order simply typed functional language. We propose a syntax-directed, exact reduction from type systems to context-free grammars (CFGs), and build upon it a finite automaton-based framework for construction and counting, enabling exact, duplicate-free, and uniform sampling of all well-typed functions inhabiting a given type. To our knowledge, this is the first approach achieving fixed-parameter tractable (FPT-trackable) sampling while preserving both type safety and semantic completeness. Our reduction framework unifies formal language theory and type theory, ensuring theoretical rigor and practical scalability. Experimental evaluation confirms the correctness, completeness, and polynomial-time efficiency of the algorithm.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

We describe an exact sampler for a simply-typed, first-order functional programming language. Given an acyclic finite automaton, $α_{varnothing}$, it samples a random function uniformly without replacement from well-typed functions in $mathcal{L}(α_{varnothing})$. This is achieved via a fixed-parameter tractable reduction from a syntax-directed type system to a context-free grammar, preserving type soundness and completeness w.r.t. $mathcal{L}(α_{varnothing})$, while retaining the robust metatheory of formal languages.

Problem

• Uniformly sampling well-typed functions without replacement
• Reducing a type system to a context-free grammar tractably
• Preserving type soundness and completeness for automata languages

Innovation

• Exact sampler for simply-typed first-order functional language
• Reduction from type system to context-free grammar
• Uniform sampling without replacement from well-typed functions