Polynomial-time inverse computation for accumulative functions with multiple data traversals

Matsuda, Kazutaka; Inaba, Kazuhiro; Nakano, Kazushi

doi:10.1145/2103746.2103752

Cited by 9 publications

(1 citation statement)

References 22 publications

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“…Transformations of structured data are at the heart of functional programming [1], [2], [3], [4], [5] and also application areas such as compiling [6], document processing [7], [8], [9], [10], [11], [12], [13], automatic translation of natural languages [14], [15], [16], [17] or even cryptographic protocols [18]. The most fundamental model of such transformations is given by (finite-state tree) transducers [19], [6].…”

Section: Introductionmentioning

confidence: 99%

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

Seidl

Maneth

Kemper

2018

J. ACM

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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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Section: Introductionmentioning

confidence: 99%

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

Seidl

Maneth

Kemper

2018

J. ACM

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Proving Correctness of Compilers Using Structured Graphs

Bahr

2014

Lecture Notes in Computer Science

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Decision Problems of Tree Transducers with Origin

Filiot

Maneth

Reynier

et al. 2015

Automata, Languages, and Programming

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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.

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Polynomial-time inverse computation for accumulative functions with multiple data traversals

Cited by 9 publications

References 22 publications

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

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

Proving Correctness of Compilers Using Structured Graphs

Decision Problems of Tree Transducers with Origin

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