Decision Problems of Tree Transducers with Origin

Abstract

Abstract. A tree transducer with origin translates an input tree into a pair of output tree and origin info. The origin info maps each node in the output tree to the unique input node that created it. In this way, the implementation of the transducer becomes part of its semantics. We show that the landscape of decidable properties changes drastically when origin info is added. For instance, equivalence of nondeterministic top-down and MSO transducers with origin is decidable. Both problems are undecidable without origin. The equivalence of deterministic topdown tree-to-string transducers is decidable with origin, while without origin it is a long standing open problem. With origin, we can decide if a deterministic macro tree transducer can be realized by a deterministic top-down tree transducer; without origin this is an open problem.Tree transducers were invented in the early 1970's as a formal model for compilers and linguistics [24,23]. They are being applied in many fields of computer science, such as syntax-directed translation [13], databases [22,15], linguistics [19,4], programming languages [27,21], and security analysis [16]. The most essential feature of tree transducers is their good balance between expressive power and decidability.Bojańczyk [3] introduces (string) transducers with origin. For "regular" stringto-string transducers with origin he presents a machine independent characterization which admits Angluin-style learning and the decidability of natural subclasses. These results indicate that classes of translations with origin are mathematically better behaved than their origin-less counter parts.We initiate a rigorous study of tree transducers with origin by investigating the decidability of equivalence, injectivity and query determinacy on the following models: top-down tree-to-tree transducers [24,23], top-down tree-to-string transducers [11], and mso definable tree-to-string transducers (see, e.g., [10]).The authors are grateful to Joost Engelfriet for its remarks for improvements and corrections on a preliminary of this work.

Look-ahead removal for total deterministic top-down tree transducers

“…For total deterministic transducers, this problem can be reduced to our problem by first proving the decidability of the following related question: Is it decidable for a given macro tree transducer whether it is equivalent to a top-down tree transducer with regular look-ahead? This is an open problem of which it was recently shown in [14] that the answer is 'yes' when the transducers have origin-semantics, which means that they should associate the same origins in the input tree to the nodes of the output tree.…”

Query Rewriting for Nondeterministic Tree Transducers

“…In [8], query preservation was shown to be decidable if a given transducer is a single-valued extended linear top-down tree transducer with regular look-ahead and a given query is a run-based n-ary node-selecting query. The result in [19] is for deterministic tree transducers that require "origin" information.…”

One-way Resynchronizability of Word Transducers

“…An open problem here is to characterize origin graphs generated by aperiodic streaming string transducers [10,16]. Going beyond words, [17] investigates decision problems of tree transducers with origin, and regains the decidability of the equivalence problem for non-deterministic top-down and MSO transducers by considering the origin semantics. An open problem for tree transducers with origin is that of synthesizing resynchronizers as in the word case.…”