[Paper Review] Simulating Linear Logic in 1-Only Linear Logic

Max Kanovich|arXiv (Cornell University)|Sep 9, 2017

Logic, programming, and type systems3 citations

TL;DR

This paper demonstrates that the simplest fragments of Linear Logic—specifically, one-literal, bottom-only, and unit-only fragments—are computationally as expressive as their full counterparts. By constructing a computational encoding using acyclic programs with stack, the authors prove that these minimal fragments capture the same complexity classes: NP-completeness for multiplicatives, PSPACE-completeness with additives, and undecidability when extended with the storage modality !.

ABSTRACT

Linear Logic was introduced by Girard as a resource-sensitive refinement of classical logic. It turned out that full propositional Linear Logic is undecidable (Lincoln, Mitchell, Scedrov, and Shankar) and, hence, it is more expressive than (modalized) classical or intuitionistic logic. In this paper we focus on the study of the simplest fragments of Linear Logic, such as the one-literal and constant-only fragments (the latter contains no literals at all). Here we demonstrate that all these extremely simple fragments of Linear Logic (one-literal, $\bot$-only, and even unit-only) are exactly of the same expressive power as the corresponding full versions. We present also a complete computational interpretation (in terms of acyclic programs with stack) for bottom-free Intuitionistic Linear Logic. Based on this interpretation, we prove the fairness of our encodings and establish the foregoing complexity results.

Motivation & Objective

• To investigate the expressive power of the most basic fragments of Linear Logic, including one-literal, bottom-only, and unit-only fragments.
• To determine whether these minimal fragments can simulate the full expressive power of propositional Linear Logic.
• To establish precise complexity classifications for these fragments using a computational encoding framework.
• To provide a complete computational interpretation of ⊥-free Intuitionistic Linear Logic via acyclic programs with stack.
• To prove the correctness and fairness of the encoding through technical lemmas and modular reasoning.

Proposed method

• Constructs a computational interpretation of ⊥-free Intuitionistic Linear Logic using acyclic programs with stack, enabling simulation of logical derivations.
• Introduces a syntactic encoding of Linear Logic formulas into a restricted language of stack-based programs, preserving logical structure and derivability.
• Employs modular proof techniques based on inductive reasoning over sequent derivations and congruence relations modulo 9N to analyze encoding behavior.
• Uses key lemmas (e.g., Lemma 4.3 and Lemma 4.4) to analyze the behavior of derived sequents under the encoding, particularly focusing on modular arithmetic constraints on exponents.
• Applies the rule L−ˆ (left implication) to derive sequents from sub-derivations, tracking the interaction between structural components and formula contexts.
• Establishes fairness of the encoding by showing that no spurious derivations are introduced, relying on the uniqueness of modular solutions in the derived congruences.

Experimental results

Research questions

• RQ1 Can the one-literal fragment of Linear Logic simulate the full multiplicative fragment in terms of expressive power?
• RQ2 What is the computational complexity of the ⊥-only and unit-only fragments of Linear Logic?
• RQ3 Does the addition of the storage modality ! to minimal fragments lead to undecidability, and if so, why?
• RQ4 Can a computational model based on acyclic programs with stack faithfully interpret ⊥-free Intuitionistic Linear Logic?
• RQ5 Is the encoding of Linear Logic into 1-only fragments sound and complete, and how is fairness ensured?

Key findings

• The one-literal fragment of Linear Logic, restricted to only the multiplicative connectives ⊗ and −ˆ, is NP-complete.
• Adding the additive connectives &per; and ⊕ to the one-literal fragment increases its expressive power to PSPACE-completeness.
• The inclusion of the storage modality ! in any of the minimal fragments (one-literal, ⊥-only, or unit-only) results in undecidability.
• The ⊥-free fragment of Intuitionistic Linear Logic admits a complete and fair computational interpretation via acyclic programs with stack.
• The encoding is proven sound and complete through a series of technical lemmas showing that all derived sequents must satisfy strict modular constraints, ruling out spurious derivations.
• The proof of correctness relies on contradiction arguments derived from modular arithmetic constraints on exponents in the encoding, particularly modulo 9N, which ensures that only valid derivations are preserved.