# The complexity of the first-order theory of pure equality

## TL;DR

This paper establishes a lower bound on the recognition complexity of nontrivial first-order theories relative to equivalence relations, demonstrating that the theory of Boolean algebra with two elements has a complexity beyond polynomial time.

## Contribution

It introduces a method to code Turing machine computations into first-order formulas, linking the complexity of the theory of Boolean algebra to computational complexity classes.

## Key findings

- The first-order theory of Boolean algebra with two elements is not solvable in polynomial time.
- Polynomial time is a proper subset of polynomial space, as shown by the complexity bounds.
- The paper provides a new lower bound on the recognition complexity of certain logical theories.

## Abstract

We will find a lower bound on the recognition complexity of the theories that are nontrivial relative to some equivalence relation (this relation may be equality), namely, each of these theories is consistent with the formula, whose sense is that there exist two non-equivalent elements. However, at first, we will obtain a lower bound on the computational complexity for the first-order theory of Boolean algebra that has only two elements. For this purpose, we will code the long-continued deterministic Turing machine computations by the relatively short-length quantified Boolean formulae; the modified Stockmeyer and Meyer method will appreciably be used for this simulation. Then, we will transform the modeling formulae of the theory of this Boolean algebra to the simulation ones of the first-order theory of the only equivalence relation in polynomial time. Since the computational complexity of these theories is not polynomial, we obtain that the class $\mathbf{P}$ is a proper subclass of $\mathbf{PSPACE}$ (Polynomial Time is a proper subset of Polynomial Space). Keywords: Computational complexity, the theory of equality, the coding of computations, simulation by means formulae, polynomial time, polynomial space, lower complexity bound