# Complexity of Infimal Observable Superlanguages

## TL;DR

This paper investigates the state complexity of infimal prefix-closed, controllable, and observable superlanguages, proving exponential bounds and showing that polynomial algorithms are unlikely to exist for their computation.

## Contribution

It establishes tight exponential bounds on the state complexity of these superlanguages and demonstrates the computational hardness of their exact computation.

## Key findings

- Upper bound of $2^n + 1$ on state complexity for the superlanguage
- Proves no polynomial-time algorithm exists for exact computation
- Shows the exponential time complexity $O(2^n)$ for the construction

## Abstract

The infimal prefix-closed, controllable and observable superlanguage plays an essential role in the relationship between controllability, observability and co-observability -- the central notions of supervisory control theory. Existing algorithms for its computation are exponential and it is not known whether a polynomial algorithm exists. In this paper, we study the state complexity of this language. State complexity of a language is the number of states of the minimal DFA for the language. For a language of state complexity $n$, we show that the upper-bound state complexity on the infimal prefix-closed and observable superlanguage is $2^n + 1$ and that this bound is asymptotically tight. It proves that there is no algorithm computing a DFA of the infimal prefix-closed and observable superlanguage in polynomial time. Our construction further shows that such a DFA can be computed in time $O(2^n)$. The construction involves NFAs and a computation of the supremal prefix-closed sublanguage. We study the computation of the supremal prefix-closed sublanguage and show that there is no polynomial-time algorithm that computes an NFA of the supremal prefix-closed sublanguage of a language given as an NFA even if the language is unary.