# Polynomial time estimates for #SAT

## TL;DR

This paper derives polynomial-time bounds on the number of satisfying assignments for CNF SAT instances, relating these bounds to distribution properties like mean and variance, especially for large formulas.

## Contribution

It introduces new polynomial-time computable bounds on #SAT counts based on distribution metrics, improving understanding of satisfiability counts for large instances.

## Key findings

- Bounds depend on distribution variance and mean.
- Some bounds are computable in polynomial time.
- Bounds are more relevant for formulas with many solutions.

## Abstract

Limits on the number of satisfying assignments for CNS instances with n variables and m clauses are derived from various inequalities. Some bounds can be calculated in polynomial time, sharper bounds demand information about the distribution of the number of unsatisfied clauses. Quite generally, the number of satisfying assignments involve variance and mean of this distribution. For large formulae, m>>1, bounds vary with 2**n/n, so they may be of use only for instances with a large number of satisfying assignments.

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Source: https://tomesphere.com/paper/1705.05452