Lipschitz Robustness of Finite-state Transducers

TL;DR

This paper studies the Lipschitz robustness of finite-state transducers, showing undecidability results for general cases and providing polynomial-time decision procedures for specific classes of transducers.

Contribution

It introduces a formal framework for Lipschitz robustness in transducers, proves undecidability in general, and identifies classes with decidable robustness.

Findings

K-robustness is undecidable for deterministic transducers.

Polynomial-time decision procedure for certain functional transducers.

Undecidability of robustness in nondeterministic transducers.

Abstract

We investigate the problem of checking if a finite-state transducer is robust to uncertainty in its input. Our notion of robustness is based on the analytic notion of Lipschitz continuity --- a transducer is K-(Lipschitz) robust if the perturbation in its output is at most K times the perturbation in its input. We quantify input and output perturbation using similarity functions. We show that K-robustness is undecidable even for deterministic transducers. We identify a class of functional transducers, which admits a polynomial time automata-theoretic decision procedure for K-robustness. This class includes Mealy machines and functional letter-to-letter transducers. We also study K-robustness of nondeterministic transducers. Since a nondeterministic transducer generates a set of output words for each input word, we quantify output perturbation using set-similarity functions. We show that K-robustness of nondeterministic transducers is undecidable, even for letter-to-letter transducers. We identify a class of set-similarity functions which admit decidable K-robustness of letter-to-letter transducers.