Regular Transducer Expressions for Regular Transformations

Abstract

Functional MSO transductions, deterministic two-way transducers, as well as streaming string transducers are all equivalent models for regular functions. In this paper, we show that every regular function, either on finite words or on infinite words, captured by a deterministic two-way transducer, can be described with a regular transducer expression (RTE). For infinite words, the transducer uses Muller acceptance and ω-regular look-ahead. RTEs are constructed from constant functions using the combinators if-then-else (deterministic choice), Hadamard product, and unambiguous versions of the Cauchy product, the 2-chained Kleene-iteration and the 2chained omega-iteration. Our proof works for transformations of both finite and infinite words, extending the result on finite words of Alur et al. in LICS'14. In order to construct an RTE associated with a deterministic two-way Muller transducer with look-ahead, we introduce the notion of transition monoid for such two-way transducers where the look-ahead is captured by some backward deterministic Büchi automaton. Then, we use an unambiguous version of Imre Simon's famous forest factorization theorem in order to derive a "good" (ω-)regular expression for the domain of the two-way transducer. "Good" expressions are unambiguous and Kleene-plus as well as ω-iterations are only used on subexpressions corresponding to idempotent elements of the transition monoid. The combinator expressions are finally constructed by structural induction on the "good" (ω-)regular expression describing the domain of the transducer.

Computing and classifying reduplication with 2-way finite-state transducers

“…Subclasses of Regular Functions a. C-Seq function: In terms of function combinatorics for regular string transformations (Alur et al 2014;Dave et al 2018), the class of C-Seq functions involves the use of a 'sum combinator'…”

The influence of Imre Simon’s work in the theory of automata, languages and semigroups

“…There are aspects of Simon's work that remain to be explored. For instance, very recently an unambiguous version of the Factorization Theorem has been announced in [15,18].…”

From Two-Way Transducers to Regular Function Expressions

“…The very recent work [11] adopts an approach similar to ours in order to lift the results of [3] to infinite words. However, to deal with Kleene iteration, they resort to an unambiguous version of Simon's factorization forest theorem.…”