Nathan Lhote

First-order definability of rational transductions: An algebraic approach

The algebraic theory of rational languages has provided powerful decidability results. Among them, one of the most fundamental is the definability of a rational language in the class of aperiodic languages, i.e., languages recognized by finite automata whose transition relation defines an aperiodic congruence. An important corollary of this result is the first-order definability of monadic second-order formulas over finite words.Our goal is to extend these results to rational transductions, i.e. word functions realized by finite transducers. We take an algebraic approach and consider definability problems of rational transductions in a given variety of congruences (or monoids).The strength of the algebraic theory of rational languages relies on the existence of a congruence canonically attached to every language, the syntactic congruence. In a similar spirit, Reutenauer and Schützenberger have defined a canonical device for rational transductions, that we extend to establish our main contribution: an effective characterization of V-transductions, i.e. rational transductions realizable by transducers whose transition relation defines a congruence in a (decidable) variety V. In particular, it provides an algorithm to decide the definability of a rational transduction by an aperiodic finite transducer.Using those results, we show that the FO-definability of a rational transduction is decidable, where FO-definable means definable in a first-order restriction of logical transducers à la Courcelle.Categories and Subject Descriptors F.4.2 [Mathematical Logic and Formal Languages]: Formal Languages

On reversible transducers

Deterministic two-way transducers define the robust class of regular functions which is, among other good properties, closed under composition. However, the best known algorithms for composing two-way transducers cause a double exponential blow-up in the size of the inputs. In this paper, we introduce a class of transducers for which the composition has polynomial complexity. It is the class of reversible transducers, for which the computation steps can be reversed deterministically. While in the one-way setting this class is not very expressive, we prove that any two-way transducer can be made reversible through a single exponential blow-up. As a consequence, we prove that the composition of two-way transducers can be done with a single exponential blow-up in the number of states. A uniformization of a relation is a function with the same domain and which is included in the original relation. Our main result actually states that we can uniformize any non-deterministic two-way transducer by a reversible transducer with a single exponential blow-up, improving the known result by de Souza which has a quadruple exponential complexity. As a side result, our construction also gives a quadratic transformation from copyless streaming string transducers to two-way transducers, improving the exponential previous bound.

On delay and regret determinization of max-plus automata

Decidability of the determinization problem for weighted automata over the semiring (ℤ∪{−∞}, max; +), WA for short, is a long-standing open question. We propose two ways of approaching it by constraining the search space of deterministic WA: k-delay and r-regret. A WA N is k-delay determinizable if there exists a deterministic automaton D that defines the same function as N and for all words α in the language of N, the accepting run of D on α is always at most k-away from a maximal accepting run of N on α. That is, along all prefixes of the same length, the absolute difference between the running sums of weights of the two runs is at most k. A WA N is r-regret determinizable if for all words α in its language, its non-determinism can be resolved on the fly to construct a run of N such that the absolute difference between its value and the value assigned to α by N is at most r. We show that a WA is determinizable if and only if it is k-delay determinizable for some k. Hence deciding the existence of some k is as difficult as the general determinization problem. When k and r are given as input, the k-delay and r-regret determinization problems are shown to be EXPTIME-complete. We also show that determining whether a WA is r-regret determinizable for some r is in EXPTIME.

Aperiodicity of Rational Functions Is PSPACE-Complete

It is known that a language of finite words is definable in monadic second-order logic - MSO - (resp. first-order logic - FO -) iff it is recognized by some finite automaton (resp. some aperiodic finite automaton). Deciding whether an automaton A is equivalent to an aperiodic one is known to be PSPACE-complete. This problem has an important application in logic: it allows one to decide whether a given MSO formula is equivalent to some FO formula. In this paper, we address the aperiodicity problem for functions from finite words to finite words (transductions), defined by finite transducers, or equivalently by bimachines, a transducer model studied by Schutzenberger and Reutenauer. Precisely, we show that the problem of deciding whether a given bimachine is equivalent to some aperiodic one is PSPACE-complete.

Logics for Word Transductions with Synthesis

We introduce a logic, called ℒT, to express properties of transductions, i.e. binary relations from input to output (finite) words. In ℒT, the input/output dependencies are modelled via an origin function which associates to any position of the output word, the input position from which it originates. ℒT is well-suited to express relations (which are not necessarily functional), and can express all regular functional transductions, i.e. transductions definable for instance by deterministic two-way transducers. Despite its high expressive power, ℒT has decidable satisfiability and equivalence problems, with tight non-elementary and elementary complexities, depending on specific representation of ℒT-formulas. Our main contribution is a synthesis result: from any transduction R defined in ℒT, it is possible to synthesise a regular functional transduction f such that for all input words u in the domain of R, f is defined and (u, f(u)) ∈ R. As a consequence, we obtain that any functional transduction is regular iff it is ℒT-definable. We also investigate the algorithmic and expressiveness properties of several extensions of ℒT, and explicit a correspondence between transductions and data words. As a side-result, we obtain a new decidable logic for data words.