LSM is not generated by binary functions

TL;DR

This paper addresses whether all log-supermodular functions can be generated by binary implication and unary functions, providing a negative answer to a question in the study of computational complexity of approximate counting problems.

Contribution

It demonstrates that not all log-supermodular functions are definable solely by binary implication and unary functions, clarifying limitations in the expressiveness of certain function classes.

Findings

Not all log-supermodular functions can be generated by binary implication and unary functions.

The question posed by Bulatov et al. is answered negatively.

The result impacts the understanding of the complexity of approximate counting problems.

Abstract

The material in this note is now superseded by arXiv:1108.5288v4. Bulatov et al. [1] defined the operation of (efficient) pps_\omega-definability in order to study the computational complexity of certain approximate counting problems. They asked whether all log-supermodular functions can be defined by binary implication and unary functions in this sense. We give a negative answer to this question.