LFPL: Revisited and Mechanized

2026-05-12

AI Summary

This work addresses the absence of a self-contained description and fully formalized metatheory for Light Affine Functional Polynomial Logic (LFPL). We propose a novel construction for polynomial cost bounds and introduce a stack-based data structure built from first-class functions and lists, which substantially simplifies the original proofs. By integrating big-step cost semantics, an affine type system, and functional programming techniques, we present the first complete mechanization of the soundness proof for LFPL+ and its completeness with respect to polynomial-time Turing machines, all formalized within the Istari proof assistant. This achievement constitutes an end-to-end formal verification of the language’s metatheory.

Abstract

Hofmann (1999) introduced the functional programming language LFPL to characterize the functions computable in polynomial time using an affine type system. LFPL enables a natural programming style, including nested recursion, and has inspired the development of type systems for automatic cost analysis, linear dependent type theories, and efficient memory management in functional programming languages. Despite its prominence, there does not exist a self-contained presentation, let alone a full mechanization, of LFPL and its core metatheory. This article presents a modern account and mechanization of LFPL and its metatheory with the goal of being self-contained and accessible while streamlining the strongest-known soundness and completeness results. The soundness proof works with the language LFPL+, which extends LFPL with additional language features. The proof is novel, adapting a technique by Aehlig and Schwichtenberg (2002) to construct explicit polynomials that bound the cost of an LFPL+ expression with respect to a big-step cost semantics. The completeness proof shows that LFPL programs can simulate polynomial-time Turing machines while only relying on restricted forms of linear functions and lists. It has the same structure as the original proof by Hofmann (2002) but greatly simplifies the core argument with a novel stack-like data structure that is implemented with first-class functions and lists. The mechanization includes the full soundness and completeness proofs, and serves as one of the first case studies of mechanized metatheory in the recently developed proof assistant Istari.

Problem

• LFPL
• polynomial time
• affine type system
• mechanization
• metatheory

Innovation

• LFPL
• polynomial time
• affine type system
• mechanized metatheory
• cost semantics