An estimation for the lengths of reduction sequences of the $\lambda\mu\rho\theta$-calculus

TL;DR

This paper provides an upper bound estimate for the length of reduction sequences in an extended $\lambda\mu ho heta$-calculus, showing that added rules do not increase computational complexity.

Contribution

It introduces a method to estimate reduction sequence lengths in the extended calculus, extending previous results to include new rules without increasing complexity.

Findings

Upper bounds for reduction sequences established

New rules do not increase computational complexity

Extension maintains efficiency despite unbounded $\mu$-abstraction

Abstract

Since it was realized that the Curry-Howard isomorphism can be extended to the case of classical logic as well, several calculi have appeared as candidates for the encodings of proofs in classical logic. One of the most extensively studied among them is the $\lambda\mu$-calculus of Parigot. In this paper, based on the result of Xi presented for the $\lambda$-calculus Xi, we give an upper bound for the lengths of the reduction sequences in the $\lambda\mu$-calculus extended with the $\rho$- and $\theta$-rules. Surprisingly, our results show that the new terms and the new rules do not add to the computational complexity of the calculus despite the fact that $\mu$-abstraction is able to consume an unbounded number of arguments by virtue of the $\mu$-rule.