Tree Transducers Research Articles

Higher-order multi-parameter tree transducers and recursion schemes for program verification

We introduce higher-order, multi-parameter, tree transducers (HMTTs, for short), which are kinds of higher-order tree transducers that take input trees and output a (possibly infinite) tree. We study the problem of checking whether the tree generated by a given HMTT conforms to a given output specification, provided that the input trees conform to input specifications (where both input/output specifications are regular tree languages). HMTTs subsume higher-order recursion schemes and ordinary tree transducers, so that their verification has a number of potential applications to verification of functional programs using recursive data structures, including resource usage verification, string analysis, and exact type-checking of XML-processing programs. We propose a sound but incomplete verification algorithm for the HMTT verification problem: the algorithm reduces the verification problem to a model-checking problem for higher-order recursion schemes extended with finite data domains, and then uses (an extension of)Kobayashi's algorithm for model-checking recursion schemes. While the algorithm is incomplete (indeed, as we show in the paper, the verification problem is undecidable in general), it is sound and complete for a subclass of HMTTs called linear HMTTs . We have applied our HMTT verification algorithm to various program verification problems and obtained promising results.

Binarization of Synchronous Context-Free Grammars

Systems based on synchronous grammars and tree transducers promise to improve the quality of statistical machine translation output, but are often very computationally intensive. The complexity is exponential in the size of individual grammar rules due to arbitrary re-orderings between the two languages. We develop a theory of binarization for synchronous context-free grammars and present a linear-time algorithm for binarizing synchronous rules when possible. In our large-scale experiments, we found that almost all rules are binarizable and the resulting binarized rule set significantly improves the speed and accuracy of a state-of-the-art syntax-based machine translation system. We also discuss the more general, and computationally more difficult, problem of finding good parsing strategies for non-binarizable rules, and present an approximate polynomial-time algorithm for this problem.

Extended multi bottom–up tree transducers

Extended multi bottom–up tree transducers are defined and investigated. They are an extension of multi bottom–up tree transducers by arbitrary, not just shallow, left-hand sides of rules; this includes rules that do not consume input. It is shown that such transducers, even linear ones, can compute all transformations that are computed by linear extended top–down tree transducers, which are a theoretical model for syntax-based machine translation. Moreover, the classical composition results for bottom–up tree transducers are generalized to extended multi bottom–up tree transducers. Finally, characterizations in terms of extended top–down tree transducers and tree bimorphisms are presented.

Sentence Compression as Tree Transduction

This paper presents a tree-to-tree transduction method for sentence compression. Our model is based on synchronous tree substitution grammar, a formalism that allows local distortion of the tree topology and can thus naturally capture structural mismatches. We describe an algorithm for decoding in this framework and show how the model can be trained discriminatively within a large margin framework. Experimental results on sentence compression bring significant improvements over a state-of-the-art model.

Sentence compression as tree transduction

This paper presents a tree-to-tree transduction method for sentence compression. Our model is based on synchronous tree substitution grammar, a formalism that allows local distortion of the tree to...

Deterministic bottom-up tree transducers and ground term rewrite systems

We show that it is decidable for any deterministic bottom-up tree transducer A and ground term rewrite system R , which one of the following conditions holds: (i) τ ( A ) ⊂ ↔ R ∗ , (ii) ↔ R ∗ ⊂ τ ( A ) , (iii) τ ( A ) = ↔ R ∗ , (iv) τ ( A ) and ↔ R ∗ are incomparable. Here τ ( A ) is the tree transformation induced by A , and ↔ R ∗ is the congruence generated by R .

Deciding equivalence of top–down XML transformations in polynomial time

Many useful XML transformations can be expressed by deterministic top–down tree transducers. A normal form is presented for such transducers (extended with the facility to inspect their input trees). A transducer in normal form has a unique canonical form which can be obtained by a minimization procedure, in polynomial time. Thus, equivalence of transducers in normal form can be decided in polynomial time. If the transducer is total, the normal form can be obtained in polynomial time as well.

Computing the Cost of Typechecking of Composition of Macro Tree Transducers

Macro tree transducers are a classical formal model for structural-recursive tree transformation with accumulative parameters. They have recently been applied to model XML transformations and queries. Typechecking a tree transformation means checking whether all valid input trees are transformed into valid output trees, for the given regular tree languages of input and output trees. Typechecking macro tree transducers is generally based on inverse type inference, because of the advantageous property that inverse transformations effectively preserve regular tree languages. It is known that the time complexity of typechecking an n-fold composition of macro tree transducers is non-elementary. The cost of typechecking can be reduced if transducers in the composition have special properties, such as being deterministic or total, or having no accumulative parameters. In this paper, the impact of such properties on the cost of typechecking is investigated. Reductions in cost are achieved by applying composition and decomposition constructions to tree transducers. Even though these constructions are well-known, they have not yet been analyzed with respect to the precise sizes of the transducers involved. The results can directly be applied to typechecking XML transformations, because type formalisms for XML are captured by regular tree languages.

Head-Needed Strategy of Higher-Order Rewrite Systems and Its Decidable Classes

The present paper discusses a head-needed strategy and its decidable classes of higher-order rewrite systems (HRSs), which is an extension of the head-needed strategy of term rewriting systems (TRSs). We discuss strong sequential and NV-sequential classes having the following three properties, which are mandatory for practical use: (1) the strategy reducing a head-needed redex is head normalizing (2) whether a redex is head-needed is decidable, and (3) whether an HRS belongs to the class is decidable. The main difficulty in realizing (1) is caused by the β-reductions induced from the higher-order reductions. Since β-reduction changes the structure of higher-order terms, the definition of descendants for HRSs becomes complicated. In order to overcome this difficulty, we introduce a function, PV, to follow occurrences moved by β-reductions. We present a concrete definition of descendants for HRSs by using PV and then show property (1) for orthogonal systems. We also show properties (2) and (3) using tree automata techniques, a ground tree transducer (GTT), and recognizability of redexes.

The Power of Extended Top-Down Tree Transducers

Extended top-down tree transducers (transducteurs généralisés descendants; see [A. Arnold and M. Dauchet, Bi-transductions de forêts, in Proceedings of the 3rd International Colloquium on Automata, Languages and Programming, Edinburgh University Press, Edinburgh, 1976, pp. 74–86]) received renewed interest in the field of natural language processing. Here those transducers are extensively and systematically studied. Their main properties are identified and their relation to classical top-down tree transducers is exactly characterized. The obtained properties completely explain the Hasse diagram of the induced classes of tree transformations. In addition, it is shown that most interesting classes of transformations computed by extended top-down tree transducers are not closed under composition.

Computing the Cost of Typechecking of Composition of Macro Tree Transducers

Computing the Cost of Typechecking of Composition of Macro Tree Transducers

The time complexity of typechecking tree-walking tree transducers

Tree-walking tree transducers can be typechecked in double exponential time. More generally, compositions of k tree-walking tree transducers can be typechecked in (k + 1)-fold exponential time. Consequently k-pebble tree transducers, which form a model of XML transformations and XML queries, can be typechecked in (k + 2)-fold exponential time. The results hold for both ranked and unranked trees.

Training Tree Transducers

Many probabilistic models for natural language are now written in terms of hierarchical tree structure. Tree-based modeling still lacks many of the standard tools taken for granted in (finite-state) string-based modeling. The theory of tree transducer automata provides a possible framework to draw on, as it has been worked out in an extensive literature. We motivate the use of tree transducers for natural language and address the training problem for probabilistic tree-to-tree and tree-to-string transducers.

Composing Stack-Attributed Tree Transducers

Stack-attributed tree transducers extend attributed tree transducers with a pushdown stack device for attribute values, which make them strictly more powerful. This paper presents an algorithm for the composition of stack-attributed tree transducers with attributed tree transducers. The algorithm is an extension of the existing method to compose attributed tree transducers. It leads to some natural closure properties of the corresponding classes of tree transformations.

Compositions of extended top-down tree transducers

Unfortunately, the class of transformations computed by linear extended top-down tree transducers with regular look-ahead is not closed under composition. It is shown that the class of transformations computed by certain linear bimorphisms coincides with the previously mentioned class. Moreover, it is demonstrated that every linear epsilon-free extended top-down tree transducer with regular look-ahead can be implemented by a linear multi bottom-up tree transducer. The class of transformations computed by the latter device is shown to be closed under composition, and to be included in the composition of the class of transformations computed by top-down tree transducers with itself. More precisely, it constitutes the composition closure of the class of transformations computed by finite-copying top-down tree transducers.

POWER CLONES AND NON-DETERMINISTIC HYPERSUBSTITUTIONS

Hypersubstitutions map operation symbols to terms of the corresponding arity. Any hypersubstitution can be extended to a mapping defined on the set Wτ(X) of all terms of type τ. If σ : {fi | i ∈ I} → Wτ(X) is a hypersubstitution and [Formula: see text] its canonical extension, then the set [Formula: see text] is a tree transformation where the original language and the image language are of the same type. Tree transformations of the type Tσ can be produced by tree transducers. Here Tσ is the graph of the function [Formula: see text]. Since the set of all hypersubstitutions of type r forms a semigroup with respect to the multiplication [Formula: see text], semigroup properties influence the properties of tree transformations of the form Tσ. For instance, if σ is idempotent, the relation Tσ is transitive (see [1]). Non-deterministic tree transducers produce tree transformations which are not graphs of some functions. If such tree transformations have the form Tσ, then σ is no longer a function. Therefore, there is some interest to study non-deterministic hypersubstitutions. That means, there are operation symbols which have not only one term of the corresponding arity as image, but a set of such terms. To define the extensions of non-deterministic hypersubstitutions, we have to extent the superposition operations for terms to a superposition defined on sets of terms. Let [Formula: see text] be the power set of the set of all n-ary terms of type τ. Then we define a superposition operation [Formula: see text] and get a heterogeneous algebra [Formula: see text] (ℕ+ is the set of all positive natural numbers), which is called the power clone of type τ. We prove that the algebra [Formula: see text] satisfies the well-known clone axioms (C1), (C2), (C3), where (C1) is the superassociative law (see e.g. [5], [4]). It turns out that the extensions of non-deterministic hypersubstitutions are precisely those endomorphisms of the heterogeneous algebra [Formula: see text] which preserve unions of families of sets. As a consequence, to study tree transformations of the form Tσ, where σ is a non-deterministic hypersubstitution, one can use the structural properties of non-deterministic hypersubstitutions. Sets of terms of type τ are tree languages in the sense of [3] and the operations [Formula: see text] are operations on tree languages. In [3] also another kind of superposition of tree languages is introduced which generalizes the usual complex product of subsets of the universe of a semigroup. We show that the extensions of non-deterministic hypersubstitutions are not endomorphisms with respect to this kind of superposition.

Circularity, composition, and decomposition results for pebble macro tree transducers

The pebble tree transducer is a model for XML query languages while the macro tree transducer is a well-known model for syntax-directed semantics. We consider their combination, called pebble macro...

Transducer-based analysis of cryptographic protocols

Cryptographic protocols can be divided into (1) protocols where the protocol steps are simple from a computational point of view and can thus be modeled by simple means, for instance, by single rewrite rules—we call these protocols non-looping—and (2) protocols, such as group protocols, where the protocol steps are complex and typically involve an iterative or recursive computation—we call them recursive. While much is known on the decidability of security for non-looping protocols, only little is known for recursive protocols. In this paper, we prove decidability of security (with respect to the standard Dolev–Yao intruder) for a core class of recursive protocols and undecidability for several extensions. The key ingredient of our protocol model is specifically designed tree transducers which work over infinite signatures and have the ability to generate new constants (which allow us to mimic key generation). The decidability result is based on an automata-theoretic construction which involves a new notion of regularity, designed to work well with the infinite signatures we use.

Capturing practical natural language transformations

We study automata for capturing the transformations in practical natural language processing (NLP) systems, especially those that translate between human languages. For several variations of finite-state string and tree transducers, we survey answers to formal questions about their expressiveness, modularity, teachability, and generalization. We conclude that no formal device yet captures everything that is desirable, and we point to future research.

Formal Efficiency Analysis for Tree Transducer Composition

We study the question of efficiency improvement or deterioration for a semantics-preserving program transformation technique for (lazy) functional languages, based on composition of restricted macro tree transducers. By annotating programs to reflect the intensional property "computation time" explicitly in the computed output and by manipulating such annotations, we formally prove syntactic conditions under which the composed program is guaranteed to be no less efficient than the original program with respect to the number of call-by-need reduction steps required to reach normal form. The criteria developed can be checked automatically and efficiently, and thus they are suitable for integration into an optimizing compiler.