Transductions: resources and characterizations

Transducers define word-to-word transformations by extending automata with outputs. We study some decision problems related to transducers.First, we characterize some resources required by any functional transducer implementing a given transformation. We begin with two algorithms determining whether a two-way functional transducer has an equivalent one-way transducer, and synthesizing it in this case. If the transducer is not one-way definable, another algorithm permits to decide whether it can perform its reversals only at the borders of the input word (sweeping transducers), and determine the minimal number of passes over the input. A side result is the minimization of the number of registers of a particular class of streaming string transducers, a model of one-way transducers with registers. We also study the memory required when evaluating visibly pushdown transducers, in particular whether the stack is required, and if so, whether the memory can be bounded by the degree of nesting of the input word.Second, we study the algebraic properties of functional transductions. A central result is an algorithm that takes a one-way transducer (or a bimachine) as input, and decides whether it belongs to a given decidable congruence class (for instance, aperiodic congruences). A transfer theorem between algebra and logic permits to relate congruence classes with logics. For instance, aperiodic congruences characterize exactly transductions definable in first-order logic. We extend this result to infinite words for the special case of aperiodic transductions. As a consequence, it is decidable whether a rational transduction is first-order definable, for the cases of finite and infinite words