We show that the minimization of visibly pushdown automata is NP-complete. This result is obtained by introducing immersions, that recognize multiple languages (over a usual, non-visible alphabet) using a common deterministic transition graph, such that each language is associated with an initial state and a set of final states. We show that minimizing immersions is NP-complete, and reduce this problem to the minimization of visibly pushdown automata.

We consider the problem of evaluating in streaming (i.e., in a single left-to-right pass) a nested word transduction with a limited amount of memory. A transduction T is said to be height bounded memory (HBM) if it can be evaluated with a memory that depends only on the size of T and on the height of the input word. We show that it is decidable in coNPTime for a nested word transduction defined by a visibly pushdown transducer (VPT), if it is HBM. In this case, the required amount of memory may depend exponentially on the height of the word. We exhibit a sufficient, decidable condition for a VPT to be evaluated with a memory that depends quadratically on the height of the word. This condition defines a class of transductions that strictly contains all determinizable VPTs.

This paper investigates canonical transducers for rational functions over infinite words, i.e., functions of infinite words defined by finite transducers. We first consider sequential functions, defined by finite transducers with a deterministic underlying automaton. We provide a Myhill-Nerode-like characterization, in the vein of Choffrut's result over finite words, from which we derive an algorithm that computes a transducer realizing the function which is minimal and unique (up to the automaton for the domain). The main contribution of the paper is the notion of a canonical transducer for rational functions over infinite words, extending the notion of canonical bimachine due to Reutenauer and Schützenberger from finite to infinite words. As an application, we show that the canonical transducer is aperiodic whenever the function is definable by some aperiodic transducer, or equivalently, by a first-order transduction. This allows to decide whether a rational function of infinite words is first-order definable.

Functional transductions realized by two-way transducers (or, equally, by streaming transducers or MSO transductions) are the natural and standard notion of "regular" mappings from words to words. It was shown in 2013 that it is decidable if such a transduction can be implemented by some one-way transducer, but the given algorithm has non-elementary complexity. We provide an algorithm of different flavor solving the above question, that has doubly exponential space complexity. In the special case of sweeping transducers the complexity is one exponential less. We also show how to construct an equivalent one-way transducer, whenever it exists, in doubly or triply exponential time, again depending on whether the input transducer is sweeping or two-way. In the sweeping case our construction is shown to be optimal.

Functional transductions realized by two-way transducers (equivalently, by streaming transducers and by MSO transductions) are the natural and standard notion of “regular” mappings from words to words. It was shown recently (LICS'13) that it is decidable if such a transduction can be implemented by some one-way transducer, but the given algorithm has non-elementary complexity. We provide an algorithm of different flavor solving the above question, that has double exponential space complexity. We further apply our technique to decide whether the transduction realized by a two-way transducer can be implemented by a sweeping transducer, with either known or unknown number of passes.

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 Schützenberger and Reutenauer. Precisely, we show that the problem of deciding whether a given bimachine is equivalent to some aperiodic one is PSPACE-complete.

We consider minimization problems for natural parameters of word transducers: the number of passes performed by two-way transducers and the number of registers used by streaming transducers. We show how to compute in ExpSpace the minimum number of passes needed to implement a transduction given as sweeping transducer, and we provide effective constructions of transducers of (worst-case optimal) doubly exponential size. We then consider streaming transducers where concatenations of registers are forbidden in the register updates. Based on a correspondence between the number of passes of sweeping transducers and the number of registers of equivalent concatenation-free streaming transducers, we derive a minimization procedure for the number of registers of concatenation-free streaming transducers.

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.

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.

Any two-way finite state automaton is equivalent to some one-way finite state automaton. This well-known result, shown by Rabin and Scott and independently by Shepherdson, states that two-way finite state automata (even non-deterministic) characterize the class of regular languages. It is also known that this result does not extend to finite string transductions: (deterministic) two-way finite state transducers strictly extend the expressive power of (functional) one-way transducers. In particular deterministic two-way transducers capture exactly the class of MSO-transductions of finite strings. In this paper, we address the following definability problem: given a function defined by a two-way finite state transducer, is it definable by a one-way finite state transducer? By extending Rabin and Scott's proof to transductions, we show that this problem is decidable. Our procedure builds a one-way transducer, which is equivalent to the two-way transducer, whenever one exists.

We consider the problem of evaluating in streaming (i.e. in a single left-to-right pass) a nested word transduction with a limited amount of memory. A transduction T is said to be height bounded memory (HBM) if it can be evaluated with a memory that depends only on the size of T and on the height of the input word. We show that it is decidable in coNPTime for a nested word transduction defined by a visibly pushdown transducer (VPT), if it is HBM. In this case, the required amount of memory may depend exponentially on the height of the word. We exhibit a sufficient, decidable condition for a VPT to be evaluated with a memory that depends quadratically on the height of the word. This condition defines a class of transductions that strictly contains all determinizable VPTs.

Two families of conciliation processes for intelligent agents based on an iterated merge-then-revise change function for belief profiles are introduced and studied. The processes from the first family are sceptical in the sense that at any revision step, each agent considers that her current beliefs are more important than the current beliefs of the group, while the processes from the other family are credulous. Some key features of such conciliation processes are pointed out for several merging operators; especially, the stationarity issue, the existence of consensus and the properties of the induced iterated merging operators are investigated.