A Linear-Time Abstract Machine for Cut Elimination in Intuitionistic Multiplicative Exponential Linear Logic

Core Concepts

The authors design an environment-based abstract machine that implements a good cut elimination strategy for the Exponential Substitution Calculus (ESC), which is an untyped calculus of proof terms for Intuitionistic Multiplicative Exponential Linear Logic (IMELL). They prove that the machine has a bi-linear overhead, providing the first linear-time cost model for an expressive fragment of linear logic.

Abstract

The paper introduces the Exponential Substitution Calculus (ESC), an untyped calculus of proof terms for the sequent calculus proofs of Intuitionistic Multiplicative Exponential Linear Logic (IMELL). The authors then design an environment-based abstract machine, called SESAME (Strong Exponential Substitution Abstract Machine without Erasure), that implements a good cut elimination strategy for ESC. The key aspects are: The good strategy is designed to have the sub-term property, which is crucial for time analyses. It forbids reducing redexes inside cut values or subtraction values that may later become cuts, in order to prevent the breaking of the sub-term property. The SESAME machine is deterministic and never erases. It implements the non-deterministic but diamond good strategy. A simple garbage collection pass is then performed on the output of SESAME to produce a cut-free term. The authors prove that SESAME has a bi-linear overhead, that is, the number of machine steps is linear in both the length of the cut elimination sequence and the size of the initial term. This provides the first linear-time cost model for an expressive fragment of linear logic. The machine design differs from previous environment-based machines in several aspects, such as working on a sequent calculus rather than natural deduction, being micro-step rather than small-step, and performing strong evaluation without backtracking. An OCaml implementation of the SESAME machine is provided in the appendix, verifying the complexity analysis.

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IMELL Cut Elimination with Linear Overhead

by Beniamino Ac... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.03669.pdf

Deeper Inquiries

How can the techniques used to design the SESAME machine be extended to other fragments of linear logic beyond IMELL

The techniques used to design the SESAME machine can be extended to other fragments of linear logic beyond IMELL by following a similar approach of abstract machine design tailored to the specific features and rules of the target logic. Here are some key steps to extend these techniques: Identifying the Logic: The first step is to understand the specific rules and characteristics of the target fragment of linear logic. This involves analyzing the syntax, inference rules, and operational semantics of the logic to determine how cut elimination can be effectively implemented. Adapting the Machine: Once the logic is identified, the abstract machine design needs to be adapted to accommodate the unique features of the new logic. This may involve modifying the transition rules, states, and initialization process to align with the requirements of the new logic. Ensuring Correctness: It is crucial to ensure that the abstract machine accurately reflects the operational semantics of the logic and that it implements cut elimination in a sound and complete manner. This may involve rigorous testing and verification to guarantee the correctness of the machine. Optimizing Performance: Depending on the complexity of the new logic, optimizations may be required to ensure efficient cut elimination with linear overhead. This could involve refining the transition rules, data structures, or search strategies to improve performance. By following these steps and adapting the design principles of the SESAME machine to the specific requirements of the target fragment of linear logic, it is possible to extend the techniques to other contexts effectively.

What are the potential applications of having a linear-time cost model for cut elimination in linear logic, for example in the context of implicit computational complexity

Having a linear-time cost model for cut elimination in linear logic, especially in the context of implicit computational complexity, offers several potential applications: Complexity Analysis: A linear-time cost model provides a precise understanding of the computational complexity of cut elimination in linear logic. This information is valuable for analyzing the efficiency of proof search algorithms and evaluating the complexity of logical systems. Implicit Computational Complexity: In the field of implicit computational complexity, linear logic is used to characterize complexity classes. A linear-time cost model for cut elimination enables the study of complexity properties of logical systems and their relationship to computational complexity classes. Algorithm Optimization: Understanding the linear overhead of cut elimination can lead to the development of optimized algorithms for proof search and manipulation in linear logic. This can improve the efficiency of automated theorem provers and proof assistants based on linear logic. Resource-Bounded Reasoning: Linear-time cost models can be used to analyze the resource consumption of proof search processes in linear logic under resource constraints. This is valuable for ensuring efficient reasoning in resource-bounded environments. Overall, a linear-time cost model for cut elimination in linear logic opens up avenues for advanced complexity analysis, algorithm optimization, and resource-bounded reasoning in logical systems.

Can the approach of using a deterministic machine to implement a non-deterministic but diamond strategy be applied to other calculi beyond ESC, and what are the general principles behind this technique

The approach of using a deterministic machine to implement a non-deterministic but diamond strategy can be applied to other calculi beyond ESC by following a similar design methodology tailored to the specific features of the target calculus. The general principles behind this technique include: Mapping Strategies to Machines: The key idea is to map the non-deterministic strategy to a deterministic abstract machine by defining transitions that correspond to the steps of the strategy. This mapping ensures that the machine faithfully implements the strategy's behavior. Maintaining Equivalence: The deterministic machine should provide a big-step simulation of the non-deterministic strategy, ensuring that the outcomes of the strategy and the machine align. This requires careful design to capture the essence of the strategy in the machine's transitions. Diamond Property: Ensuring that the strategy is diamond, meaning that non-determinism does not affect the final outcome, is crucial for the deterministic machine to accurately replicate the strategy's behavior. This property guarantees that the machine's transitions lead to consistent results. Verification and Validation: Rigorous testing and verification are essential to confirm that the deterministic machine correctly implements the non-deterministic strategy. This involves checking for equivalence in outcomes, handling of non-determinism, and overall correctness of the implementation. By applying these general principles and adapting the design to the specific characteristics of the target calculus, the approach of using a deterministic machine to implement a non-deterministic strategy can be successfully extended to other calculi beyond ESC.