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

Matsuda, Kazutaka; Inaba, Kazuhiro; Nakano, Kazushi

doi:10.1007/s10990-013-9097-8

Cited by 12 publications

(6 citation statements)

References 35 publications

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“…For example, Matsuda et al [23] and Nishida et al [29] also discuss the inversion of treeless languages. Since treelessness essentially characterizes transducer-like computation, where the inputs and outputs are separated, we believe a similar technique would be applicable to invertible transducers [15,22].…”

Section: Discussion and Related Workmentioning

confidence: 99%

Embedding invertible languages with binders: a case of the FliPpr language

Matsuda

Wang

2018

Proceedings of the 11th ACM SIGPLAN International Symposium on Haskell

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This paper describes a new embedding technique of invertible programming languages, through the case of the FliPpr language. Embedded languages have the advantage of inheriting host languages' features and supports; and one of the influential methods of embedding is the tagless-final style, which enables a high level of programmability and extensibility. However, it is not straightforward to apply the method to the family of invertible/reversible/bidirectional languages, due to the different ways functions in such domains are represented. We consider FliPpr, an invertible pretty-printing system, as a representative of such languages, and show that Atkey et al.'s unembedding technique can be used to address the problem. Together with a reformulation of FliPpr, our embedding achieves a high level of interoperability with the host language Haskell, which is not found in any other invertible languages. We implement the idea and demonstrate the benefits of the approach with examples.

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Section: Discussion and Related Workmentioning

confidence: 99%

Embedding invertible languages with binders: a case of the FliPpr language

Matsuda

Wang

2018

Proceedings of the 11th ACM SIGPLAN International Symposium on Haskell

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“…The first step is tupling [Chin 1993;Hu et al 1997] which eliminates multiple data traversals. The elimination of multiple data traversals is known to be useful for program inversion [Eppstein 1985;Matsuda et al 2012].…”

Section: Rebuilding Trees From a Pre-order And An In-order Traversalsmentioning

confidence: 99%

Sparcl: a language for partially-invertible computation

Matsuda

Wang

2020

Proc. ACM Program. Lang.

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Invertibility is a fundamental concept in computer science, with various manifestations in software development (serializer/deserializer, parser/printer, redo/undo, compressor/decompressor, and so on). Full invertibility necessarily requires bijectivity, but the direct approach of composing bijective functions to develop invertible programs is too restrictive to be useful. In this paper, we take a different approach by focusing on partiallyinvertible functionsÐfunctions that become invertible if some of their arguments are fixed. The simplest example of such is addition, which becomes invertible when fixing one of the operands. More involved examples include entropy-based compression methods (e.g., Huffman coding), which carry the occurrence frequency of input symbols (in certain formats such as Huffman tree), and fixing this frequency information makes the compression methods invertible. We develop a language Sparcl for programming such functions in a natural way, where partial-invertibility is the norm and bijectivity is a special case, hence gaining significant expressiveness without compromising correctness. The challenge in designing such a language is to allow ordinary programming (the łpartiallyž part) to interact with the invertible part freely, and yet guarantee invertibility by construction. The language Sparcl is linear-typed, and has a type constructor to distinguish data that are subject to invertible computation and those that are not. We present the syntax, type system, and semantics of the language, and prove that Sparcl correctly guarantees invertibility for its programs. We demonstrate the expressiveness of Sparcl with examples including tree rebuilding from preorder and inorder traversals and Huffman coding. CCS Concepts: • Software and its engineering → Functional languages; Domain specific languages.

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

Cited by 12 publications

References 35 publications

Embedding invertible languages with binders: a case of the FliPpr language

Embedding invertible languages with binders: a case of the FliPpr language

Sparcl: a language for partially-invertible computation

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

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