On Universality of the S Combinator

TL;DR

This paper investigates whether the S combinator alone is computationally universal, addressing a question in combinatory logic and expanding understanding of the expressive power of fundamental combinators.

Contribution

It provides a formal analysis demonstrating that the S combinator alone cannot achieve universality, clarifying the limitations of single combinator systems.

Findings

S combinator alone is not universal

K combinator cannot be expressed solely in terms of S

Addresses a question raised by Stephen Wolfram

Abstract

In combinatory logic it is known that the set of two combinators K and S are universal; in the sense that any other combinator can be expressed in terms of these two. K combinator can not be expressed only in terms of the S combinator. This will answer a question raised by Stephen Wolfram as ``Is the S combinator on its own computation universal?''