How big is BCI fragment of BCK logic

TL;DR

This paper analyzes the proportion of provable formulas and proofs in BCI versus BCK logics, showing that as formulas and proofs grow longer, the likelihood of a BCK proof also being BCI tends to zero.

Contribution

It provides an asymptotic comparison of provability in BCI and BCK logics, revealing the diminishing overlap as formula and proof length increase.

Findings

Probability of a BCK formula being provable in BCI tends to zero as variables increase.

Asymptotic behavior shows negligible overlap between BCK and BCI proofs for large proof lengths.

The proportion of BCK proofs that are also BCI proofs approaches zero with increasing proof length.

Abstract

We investigate quantitative properties of BCI and BCK logics. The first part of the paper compares the number of formulas provable in BCI versus BCK logics. We consider formulas built on implication and a fixed set of $k$ variables. We investigate the proportion between the number of such formulas of a given length $n$ provable in BCI logic against the number of formulas of length $n$ provable in richer BCK logic. We examine an asymptotic behavior of this fraction when length $n$ of formulas tends to infinity. This limit gives a probability measure that randomly chosen BCK formula is also provable in BCI. We prove that this probability tends to zero as the number of variables tends to infinity. The second part of the paper is devoted to the number of lambda terms representing proofs of BCI and BCK logics. We build a proportion between number of such proofs of the same length $n$ and we investigate asymptotic behavior of this proportion when length of proofs tends to infinity. We demonstrate that with probability 0 a randomly chosen BCK proof is also a proof of a BCI formula.