Equivalence of Deterministic Top-Down Tree-to-String Transducers Is Decidable

Abstract

We show that equivalence of deterministic top-down tree-to-string transducers is decidable, thus solving a long standing open problem in formal language theory. We also present efficient algorithms for subclasses: polynomial time for total transducers with unary output alphabet (over a given top-down regular domain language), and corandomized polynomial time for linear transducers; these results are obtained using techniques from multi-linear algebra. For our main result, we prove that equivalence can be certified by means of inductive invariants using polynomial ideals. This allows us to construct two semi-algorithms, one searching for a proof of equivalence, one for a witness of non-equivalence. Furthermore, we extend our result to deterministic top-down tree-to-string transducers which produce output not in a free monoid but in a free group.sets. More precisely, for a Parikh language L (this means L, if the order of symbols is ignored, is equivalent to a regular language) it is decidable whether there exists an output string with equal number of a's and b's (for given letters a = b). The idea of the proof is to construct L which contains a n b m if and only if, on input t, transducer M 1 outputs a at position n and transducer M 2 outputs b at position m.Our main result generalizes the result of [34] by proving that equivalence of unrestricted deterministic topdown tree-to-string transducers is decidable. By that, it solves an intriguing problem which has been open for at least thirty-five years [29]. The difficulty of the problem may perhaps become apparent as it encompasses not only the equivalence problem for MSO definable transductions, but also the famous HDT0L sequence equivalence problem [36], [37], [38], the latter is the sub-case when the input is restricted to monadic trees [39]. Opposed to the attempts, e.g., in [30], we refrain from any arguments based on the combinatorial structure of finite state devices or output strings. We also do not follow the line of arguments in [34] based on semi-linear sets. Instead, we proceed in two stages. In the first stage, we consider transducers with unary output alphabets only (Sections III-V). In this case, a produced output string can be represented by its length, thus turning the transducers effectively into treeto-integer transducers. For a given input tree, the output behavior of the states of such a unary yDT transducer is collected into a vector. Interestingly, the output vector for an input tree turns out to be a multi-affine transformation of the corresponding output vectors of the input subtrees. As the property we are interested in can be expressed as an affine equality to be satisfied by output vectors, we succeed in replacing the sets of reachable output vectors of the transducer by their affine closures. This observation allows us to apply exact fixpoint techniques as known from abstract interpretation of programs [40], to effectively compute these affine closures and thus to decide equivalence. In the next step, we generalize these techniques to a l...

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