[Paper Review] Linear lambda Calculus with Explicit Substitutions as Proof-Search in Deep Inference

Luca Roversi|arXiv (Cornell University)|Nov 16, 2010

Logic, programming, and type systems33 references3 citations

TL;DR

This paper introduces $σ$BVr, a deep inference system extending $σ$BV with a self-dual atom-renaming operator to model explicit substitutions in linear lambda calculus. It proves cut elimination and shows that proof search in $σ$BVr simulates linear $β$-reduction with explicit substitutions, establishing completeness and soundness, and demonstrating that proof search is at least PTIME-complete.

ABSTRACT

SBV is a deep inference system that extends the set of logical operators of multiplicative linear logic with the non commutative operator Seq. We introduce the logical system SBVr which extends SBV by adding a self-dual atom-renaming operator to it. We prove that the cut elimination holds on SBVr. SBVr and its cut free subsystem BVr are complete and sound with respect to linear lambda calculus with explicit substitutions. Under any strategy, a sequence of evaluation steps of any linear lambda-term M becomes a process of proof-search in SBVr (BVr) once M is mapped into a formula of SBVr. Completeness and soundness follow from simulating linear beta-reduction with explicit substitutions as processes. The role of the new renaming operator of SBVr is to rename channel-names on-demand. This simulates the substitution that occurs in a beta-reduction. Despite SBVr is a minimal extension of SBV its proof-search can compute all boolean functions, as linear lambda-calculus with explicit substitutions can compute all boolean functions as well. So, proof search of SBVr and BVr is at least ptime-complete.

Motivation & Objective

• To bridge deep inference proof theory with functional computation by modeling linear lambda calculus with explicit substitutions as proof search in a deep inference system.
• To extend the deep inference system $σ$BV with a self-dual atom-renaming operator to capture the dynamics of explicit substitutions.
• To establish completeness and soundness of the system $σ$BVr with respect to linear lambda calculus with explicit substitutions.
• To demonstrate that proof search in $σ$BVr simulates the evaluation of linear $λ$-terms under any strategy.
• To show that the proof search in $σ$BVr is at least PTIME-complete, reflecting the computational complexity of linear lambda calculus with explicit substitutions.

Proposed method

• Extend $σ$BV with a self-dual atom-renaming operator to form $σ$BVr, enabling on-demand renaming of channel names during proof construction.
• Define a mapping from linear $λ$-terms into formulas of $σ$BVr such that evaluation steps correspond to proof-search transitions.
• Prove cut elimination for $σ$BVr using induction on derivation structure, handling rules such as $σ$r, lft, and tra.
• Establish completeness by showing that every $β$-reduction step in linear lambda calculus with explicit substitutions is simulated by a sequence of inference steps in $σ$BVr.
• Use the deep inference mechanism to allow rules to apply at any depth in the formula structure, enabling expressive proof construction.
• Leverage the correspondence between proof search and computation to show that $σ$BVr can compute all boolean functions, implying PTIME-completeness.

Experimental results

Research questions

• RQ1 Can deep inference systems model the dynamics of linear lambda calculus with explicit substitutions through proof search?
• RQ2 How can a self-dual atom-renaming operator be integrated into a deep inference system to simulate explicit substitutions?
• RQ3 Is the proof search in the extended system $σ$BVr complete and sound with respect to linear $λ$-calculus with explicit substitutions?
• RQ4 What is the computational complexity of proof search in $σ$BVr, and how does it relate to the complexity of functional computation?
• RQ5 Can the deep inference framework support the full expressiveness of linear lambda calculus with explicit substitutions, including boolean function computation?

Key findings

• The system $σ$BVr admits cut elimination, ensuring that every provable formula has a cut-free proof.
• Proof search in $σ$BVr faithfully simulates the evaluation of any linear $λ$-term with explicit substitutions under any strategy.
• The completeness and soundness of $σ$BVr with respect to linear lambda calculus with explicit substitutions are established via simulation of $β$-reduction steps.
• The self-dual atom-renaming operator in $σ$BVr enables on-demand renaming of channel names, directly simulating substitution in $β$-reduction.
• Proof search in $σ$BVr is at least PTIME-complete, matching the computational complexity of linear lambda calculus with explicit substitutions.
• The system $σ$BVr can compute all boolean functions, confirming its computational adequacy and expressive power.