Origin-equivalence of two-way word transducers is in PSPACE

We consider equivalence and containment problems for word transductions. These problems are known to be undecidable when the transductions are relations between words realized by non-deterministic transducers, and become decidable when restricting to functions from words to words. Here we prove that decidability can be equally recovered the origin semantics, that was introduced by Bojanczyk in 2014. We prove that the equivalence and containment problems for two-way word transducers in the origin semantics are PSPACE-complete. We also consider a variant of the containment problem where two-way transducers are compared under the origin semantics, but in a more relaxed way, by allowing distortions of the origins. The possible distortions are described by means of a resynchronization relation. We propose MSO-definable resynchronizers and show that they preserve the decidability of the containment problem under resynchronizations. {

Logical and Algebraic Characterizations of Rational Transductions

Rational word languages can be defined by several equivalent means: finite state automata, rational expressions, finite congruences, or monadic second-order (MSO) logic. The robust subclass of aperiodic languages is defined by: counter-free automata, star-free expressions, aperiodic (finite) congruences, or first-order (FO) logic. In particular, their algebraic characterization by aperiodic congruences allows to decide whether a regular language is aperiodic. We lift this decidability result to rational transductions, i.e., word-to-word functions defined by finite state transducers. In this context, logical and algebraic characterizations have also been proposed. Our main result is that one can decide if a rational transduction (given as a transducer) is in a given decidable congruence class. We also establish a transfer result from logic-algebra equivalences over languages to equivalences over transductions. As a consequence, it is decidable if a rational transduction is first-order definable, and we show that this problem is PSPACE-complete

The many facets of string transducers

Regular word transductions extend the robust notion of regular languages from a qualitative to a quantitative reasoning. They were already considered in early papers of formal language theory, but turned out to be much more challenging. The last decade brought considerable research around various transducer models, aiming to achieve similar robustness as for automata and languages. In this paper we survey some older and more recent results on string transducers. We present classical connections between automata, logic and algebra extended to transducers, some genuine definability questions, and review approaches to the equivalence problem

On Equivalence and Uniformisation Problems for Finite Transducers

Transductions are binary relations of finite words. For rational transductions, i.e., transductions defined by finite transducers, the inclusion, equivalence and sequential uniformisation problems are known to be undecidable. In this paper, we investigate stronger variants of inclusion, equivalence and sequential uniformisation, based on a general notion of transducer resynchronisation, and show their decidability. We also investigate the classes of finite-valued rational transductions and deterministic rational transductions, which are known to have a decidable equivalence problem. We show that sequential uniformisation is also decidable for them

A Regular and Complete Notion of Delay for Streaming String Transducers

The notion of delay between finite transducers is a core element of numerous fundamental results of transducer theory. The goal of this work is to provide a similar notion for more complex abstract machines: we introduce a new notion of delay tailored to measure the similarity between streaming string transducers (SST). We show that our notion is regular: we design a finite automaton that can check whether the delay between any two SSTs executions is smaller than some given bound. As a consequence, our notion enjoys good decidability properties: in particular, while equivalence between non-deterministic SSTs is undecidable, we show that equivalence up to fixed delay is decidable. Moreover, we show that our notion has good completeness properties: we prove that two SSTs are equivalent if and only if they are equivalent up to some (computable) bounded delay. Together with the regularity of our delay notion, it provides an alternative proof that SSTs equivalence is decidable. Finally, the definition of our delay notion is machine-independent, as it only depends on the origin semantics of SSTs. As a corollary, the completeness result also holds for equivalent machine models such as deterministic two-way transducers, or MSO transducers

One-way definability of sweeping transducers

Two-way finite-state transducers on words are strictly more expressive than one-way transducers. It has been shown recently how to decide if a two-way functional transducer has an equivalent one-way transducer, and the complexity of the algorithm is non-elementary. We propose an alternative and simpler characterization for sweeping functional transducers, namely, for transducers that can only reverse their head direction at the extremities of the input. Our algorithm works in 2EXPSPACE and, in the positive case, produces an equivalent one-way transducer of doubly exponential size. We also show that the bound on the size of the transducer is tight, and that the one-way definability problem is undecidable for (sweeping) non-functional transducers

One-way resynchronizability of word transducers

The origin semantics for transducers was proposed in 2014, and it led to various characterizations and decidability results that are in contrast with the classical semantics. In this paper we add a further decidability result for characterizing transducers that are close to one-way transducers in the origin semantics. We show that it is decidable whether a non-deterministic two-way word transducer can be resynchro-nized by a bounded, regular resynchronizer into an origin-equivalent one-way transducer. The result is in contrast with the usual semantics, where it is undecidable to know if a non-deterministic two-way transducer is equivalent to some one-way transducer

On synthesis of resynchronizers for transducers

We study two formalisms that allow to compare transducers over words under origin semantics: rational and regular resynchronizers, and show that the former are captured by the latter. We then consider some instances of the following synthesis problem: given transducers T_1,T_2, construct a rational (resp. regular) resynchronizer R, if it exists, such that T_1 is contained in R(T_2) under the origin semantics. We show that synthesis of rational resynchronizers is decidable for functional, and even finite-valued, one-way transducers, and undecidable for relational one-way transducers. In the two-way setting, synthesis of regular resynchronizers is shown to be decidable for unambiguous two-way transducers. For larger classes of two-way transducers, the decidability status is open