Finite-valued Streaming String Transducers

Emmanuel Filiot, Ismaël Jecker, Gabriele Puppis, Christof Löding, Anca Muscholl, Sarah Winter
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Abstract

A transducer is finite-valued if for some bound k, it maps any given input to at most k outputs. For classical, one-way transducers, it is known since the 80s that finite valuedness entails decidability of the equivalence problem. This decidability result is in contrast to the general case, which makes finite-valued transducers very attractive. For classical transducers, it is also known that finite valuedness is decidable and that any k-valued finite transducer can be decomposed as a union of k single-valued finite transducers. In this paper, we extend the above results to copyless streaming string transducers (SSTs), answering questions raised by Alur and Deshmukh in 2011. SSTs strictly extend the expressiveness of one-way transducers via additional variables that store partial outputs. We prove that any k-valued SST can be effectively decomposed as a union of k (single-valued) deterministic SSTs. As a corollary, we obtain equivalence of SSTs and two-way transducers in the finite-valued case (those two models are incomparable in general). Another corollary is an elementary upper bound for checking equivalence of finite-valued SSTs. The latter problem was already known to be decidable, but the proof complexity was unknown (it relied on Ehrenfeucht's conjecture). Finally, our main result is that finite valuedness of SSTs is decidable. The complexity is PSpace, and even PTime when the number of variables is fixed.
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有限值流字符串转换器
如果对于某个界值 k,一个转换器最多能将任何给定输入映射到 k 个输出,那么这个转换器就是有限值的。对于经典的单向变换器,自 20 世纪 80 年代以来,人们就知道有限有值性意味着等价问题的可解性。对于经典变换器,人们也知道有限有值性是可解的,而且任何 k 值有限变换器都可以分解为 k 个单值有限变换器的联合。在本文中,我们将上述结果扩展到了无副本流串变换器(SST),回答了阿卢尔和德什穆克在 2011 年提出的问题。我们证明,任何 k 值 SST 都可以有效地分解为 k 个(单值)确定性 SST 的联合。作为推论,我们得到了 SST 和双向变换器在有限值情况下的等价性(这两种模型在一般情况下是不可比的)。另一个推论是检查无穷值 SST 等价性的基本上界。最后,我们的主要结果是 SST 的有限有值性是可解的。最后,我们的主要结果是,SST 的有限值性是可解的。其复杂度是 PS 空间,当变量数固定时甚至是 PT 时间。
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