# Approximate Learning of Limit-Average Automata

## TL;DR

This paper investigates the learnability of limit-average automata, revealing that passive learning requires exponential samples and is NP-complete, while active learning can achieve almost-exact or approximate solutions efficiently.

## Contribution

It provides the first complexity and learnability results for limit-average automata in both passive and active settings, highlighting fundamental limitations and possibilities.

## Key findings

- Passive learning requires exponential sample size.
- Finding a fitting automaton is NP-complete.
- Active learning can produce almost-exact automata in polynomial time.

## Abstract

Limit-average automata are weighted automata on infinite words that use average to aggregate the weights seen in infinite runs. We study approximate learning problems for limit-average automata in two settings: passive and active. In the passive learning case, we show that limit-average automata are not PAC-learnable as samples must be of exponential-size to provide (with good probability) enough details to learn an automaton. We also show that the problem of finding an automaton that fits a given sample is NP-complete. In the active learning case, we show that limit-average automata can be learned almost-exactly, i.e., we can learn in polynomial time an automaton that is consistent with the target automaton on almost all words. On the other hand, we show that the problem of learning an automaton that approximates the target automaton (with perhaps fewer states) is NP-complete. The abovementioned results are shown for the uniform distribution on words. We briefly discuss learning over different distributions.