On Continuous Weighted Finite Automata

TL;DR

This paper studies the continuity properties of functions defined by weighted finite automata, especially average preserving ones, providing characterizations, decision algorithms, and methods for constructing continuous functions.

Contribution

It characterizes continuous -functions for average preserving WFA, links continuity decision problems to matrix stability, and offers construction methods for continuous functions.

Findings

Every continuous -function definable by WFA can be represented by an average preserving WFA.

Deciding continuity of WFA functions is computationally equivalent to matrix stability.

Methods for constructing WFA that compute continuous real functions are provided.

Abstract

We investigate the continuity of the \omega-functions and real functions defined by weighted finite automata (WFA). We concentrate on the case of average preserving WFA. We show that every continuous \omega-function definable by some WFA can be defined by an average preserving WFA and then characterize minimal average preserving WFA whose \omega-function or \omega-function and real function are continuous. We obtain several algorithmic reductions for WFA-related decision problems. In particular, we show that deciding whether the \omega-function and real function of an average preserving WFA are both continuous is computationally equivalent to deciding stability of a set of matrices. We also present a method for constructing WFA that compute continuous real functions.