Automata-logic connections are pillars of the theory of regular languages. Such connections are harder to obtain for transducers, but important results have been obtained recently for word-to-word transformations, showing that the three following models are equivalent: deterministic two-way transducers, monadic second-order (MSO) transducers, and deterministic one-way automata equipped with a finite number of registers. Nested words are words with a nesting structure, allowing to model unranked trees as their depth-first-search linearisations. In this paper, we consider transformations from nested words to words, allowing in particular to produce unranked trees if output words have a nesting structure. The model of visibly pushdown transducers allows to describe such transformations, and we propose a simple deterministic extension of this model with two-way moves that has the following properties: i) it is a simple computational model, that naturally has a good evaluation complexity; ii) it is expressive: it subsumes nested word-to-word MSO transducers, and the exact expressiveness of MSO transducers is recovered using a simple syntactic restriction; iii) it has good algorithmic/closure properties: the model is closed under composition with a unambiguous one-way letter-to-letter transducer which gives closure under regular look-around, and has a decidable equivalence problem
Abstract. We consider the fragments FO, and ∆ 2 of first-order logic FO[<] over finite and infinite words. For all four fragments, we give characterizations in terms of rankers. In particular, we generalize the notion of a ranker to infinite words in two possible ways. Both extensions are natural in the sense that over finite words, they coincide with classical rankers and over infinite words, they both have the full expressive power of FO 2 . Moreover, the first extension of rankers admits a characterization of Σ 2 ∩ FO 2 while the other leads to a characterization of Π 2 ∩ FO 2 . Both versions of rankers yield characterizations of the fragment ∆ 2 = Σ 2 ∩ Π 2 . As a byproduct, we also obtain characterizations based on unambiguous temporal logic and unambiguous interval temporal logic.
We investigate the decidability of the definability problem for fragments of first order logic over finite words enriched with regular numerical predicates. In this paper, we focus on the quantifier alternation hierarchies of first order logic. We obtain that deciding this problem for each level of the alternation hierarchy of both first order logic and its twovariable fragment when equipped with all regular numerical predicates is not harder than deciding it for the corresponding level equipped with only the linear order. Relying on some recent results, this proves the decidability for each level of the alternation hierarchy of the two-variable first order fragment while in the case of the first order logic the question remains open for levels greater than two. The main ingredients of the proofs are syntactic transformations of firstorder formulas as well as the infinitely testable property, a new algebraic notion on varieties that we define.
Abstract. Regular string-to-string functions enjoy a nice triple characterization through deterministic two-way transducers (2DFT), streaming string transducers (SST) and MSO definable functions. This result has recently been lifted to FO definable functions, with equivalent representations by means of aperiodic 2DFT and aperiodic 1-bounded SST, extending a well-known result on regular languages. In this paper, we give three direct transformations: i) from 1-bounded SST to 2DFT, ii) from 2DFT to copyless SST, and iii) from k-bounded to 1-bounded SST. We give the complexity of each construction and also prove that they preserve the aperiodicity of transducers. As corollaries, we obtain that FO definable string-to-string functions are equivalent to SST whose transition monoid is finite and aperiodic, and to aperiodic copyless SST.
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