# Intersection Non-Emptiness Problem

The intersection non-emptiness problem (also known as finite automaton intersection problem and non-emptiness of intersection problem) is a PSPACE-complete decision problem from the field of automata theory.

## Definitions

A non-emptiness decision problem is defined as follows. Given an automaton as input, the goal is to determine whether or not the automaton's language is non-empty. In other words, the goal is to determine if there exists a string that is accepted by the automaton.

Non-emptiness problems have been studied in the field of automata theory for many years. Several common non-emptiness problems have been shown to be complete for complexity classes ranging from Deterministic Logspace up to PSPACE.

The intersection non-emptiness decision problem is concerned with whether the intersection of given languages is non-empty. In particular, the intersection non-emptiness problem is defined as follows. Given a list of deterministic finite automata as input, the goal is to determine whether or not their associated regular languages have a non-empty intersection. In other, the goal is to determine if there exists a string that is accepted by all of the automata in the list.

## Algorithm

There is a common exponential time algorithm that solves the intersection non-emptiness problem based on the Cartesian product construction introduced by Michael O. Rabin and Dana Scott. The idea is that all of the automata together form a product automaton such that a string is accepted by all of the automata if and only if it is accepted by the product automaton. Therefore, a breadth first search (or depth first search) within the product automaton's state diagram will determine whether there exists a path from the product start state to one of the product final states. Whether or not such a path exists is equivalent to determining if any string is accepted by all of the automata in the list.

Note: The product automaton does not need to be fully constructed. The automata together provide sufficient information so that transitions can be determined as needed.

## Hardness

The intersection non-emptiness problem was shown to be PSPACE-complete in a work by Dexter Kozen in 1977. Since then, many additional hardness results have been shown. Yet, it is still an open problem to determine whether any faster algorithms exist.