Two-Way Visibly Pushdown Automata and Transducers

Abstract

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

Two-Way Parikh Automata with a Visibly Pushdown Stack

“…The single-use property is more general than the more classical finite-visit restriction, used for instance in [9,19]: it requires to visit any input position a (machine-dependent) constant number of times, while single-useness only bounds the number of visits by producing transitions. Although, consequently to our results, 2VPPA single-use and finite-visit have the same expressive power, this extra modelling feature is desirable, for instance when using 2VPPA to test properties of 2VPT: single-use 2VPT are strictly more expressive than finite-visit ones, and this relaxation is crucial to capture MSO transductions [8]. Moreover, we somehow get it for free: we show that the NEx-pTime lower bound also holds for finite-visit 2VPPA.…”

“…VPA have been extended to two-way VPA (2VPA) [8] with the following stack constraints: in a backward reading mode, the role of the return and call symbols regarding the stack are inverted: when reading a call, exactly one symbol is popped from the stack and when reading a return, one symbol is pushed. It was shown in [8] that adding this visibly condition to two-way pushdown automata allows one to recover decidability for the emptiness problem. However, for Parikh acceptance, this restriction is not sufficient.…”

“…The problem is that by using the combination of twowayness and a pushdown stack, it is possible to encode polynomially, and even exponentially large counter values, with respect to the length of the input word. We consider therefore the single-use restriction, which appears in several transducer models [6,8,10], by which it is possible to keep a linear behaviour for the counters. Informally, a single-use two-way machine bounds the size of the production per input positions.…”

“…As an application, we give an elementary upper-bound (NExpTime) for the equivalence problem of functional single-use two-way visibly pushdown transducers [8], while an ExpTime lower bound was known. This transducer model defines transductions from well-nested words to words and, as shown in [8], they are well-suited to define XML transformations, have the same expressive power as Courcelle's MSO-transducers [6] (casted to well-nested words), and admit a memory-efficient evaluation algorithm. We also provide two other new results on single-use 2VPT (not necessarily functional).…”

“…To the best of our knowledge, this is the first model of unranked tree-to-word transducers for which k-valuedness and codomain well-nestedness is shown to be decidable. Another model, introduced in [1], is known to be expressively equivalent to 2VPT su [8], and in the functional case, has decidable equivalence problem in NExpTime. However, translating 2VPT su to this model requires an exponential blow-up, yielding a worst complexity for equivalence testing.…”