Macro Tree Transducers Research Articles

Characterizing attributed tree translations in terms of macro tree transducers

It is well known that attributed tree transducers can be equipped with “regular look-around” in order to obtain a more robust class of translations. We present two characterizations of this class in terms of macro tree transducers (MTTs): the first one is a static restriction on the rules of the MTTs, where the MTTs need to be equipped with regular look-around. The second characterization is a dynamic one, where the MTTs only need regular look-ahead.

Deciding origin equivalence of weakly self-nesting macro tree transducers

We consider a notion of origin for deterministic macro tree transducers with look-ahead which records for each output node, the corresponding input node for which a rule-application generated that output node. With respect to this natural notion, we show that “origin equivalence” is decidable — whenever the transducers are weakly self-nesting. The latter means that whenever two nested calls on the same input node occur, then there must be at least one other node (a terminal output node or a call on another input node) in between these nested calls. Besides origin equivalence we are also able to decide “origin injectivity” for such transducers.

Linear-bounded composition of tree-walking tree transducers: linear size increase and complexity

Compositions of tree-walking tree transducers form a hierarchy with respect to the number of transducers in the composition. As main technical result it is proved that any such composition can be realized as a linear-bounded composition, which means that the sizes of the intermediate results can be chosen to be at most linear in the size of the output tree. This has consequences for the expressiveness and complexity of the translations in the hierarchy. First, if the computed translation is a function of linear size increase, i.e., the size of the output tree is at most linear in the size of the input tree, then it can be realized by just one, deterministic, tree-walking tree transducer. For compositions of deterministic transducers it is decidable whether or not the translation is of linear size increase. Second, every composition of deterministic transducers can be computed in deterministic linear time on a RAM and in deterministic linear space on a Turing machine, measured in the sum of the sizes of the input and output tree. Similarly, every composition of nondeterministic transducers can be computed in simultaneous polynomial time and linear space on a nondeterministic Turing machine. Their output tree languages are deterministic context-sensitive, i.e., can be recognized in deterministic linear space on a Turing machine. The membership problem for compositions of nondeterministic translations is nondeterministic polynomial time and deterministic linear space. All the above results also hold for compositions of macro tree transducers. The membership problem for the composition of a nondeterministic and a deterministic tree-walking tree translation (for a nondeterministic IO macro tree translation) is log-space reducible to a context-free language, whereas the membership problem for the composition of a deterministic and a nondeterministic tree-walking tree translation (for a nondeterministic OI macro tree translation) is possibly NP-complete.

Multiple context-free tree grammars: Lexicalization and characterization

Multiple (simple) context-free tree grammars are investigated, where “simple” means “linear and nondeleting”. Every multiple context-free tree grammar that is finitely ambiguous can be lexicalized; i.e., it can be transformed into an equivalent one (generating the same tree language) in which each rule of the grammar contains a lexical symbol. Due to this transformation, the rank of the nonterminals increases at most by 1, and the multiplicity (or fan-out) of the grammar increases at most by the maximal rank of the lexical symbols; in particular, the multiplicity does not increase when all lexical symbols have rank 0. Multiple context-free tree grammars have the same tree generating power as multi-component tree adjoining grammars (provided the latter can use a root-marker). Moreover, every multi-component tree adjoining grammar that is finitely ambiguous can be lexicalized. Multiple context-free tree grammars have the same string generating power as multiple context-free (string) grammars and polynomial time parsing algorithms. A tree language can be generated by a multiple context-free tree grammar if and only if it is the image of a regular tree language under a deterministic finite-copying macro tree transducer. Multiple context-free tree grammars can be used as a synchronous translation device.

Decision problems of tree transducers with origin

A tree transducer with origin translates an input tree into a pair of output tree and origin information. The origin information maps each node in the output tree to the unique node in the input tree 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 information is added. For instance, equivalence of nondeterministic top-down and MSO transducers with origin becomes decidable. Both problems are undecidable without origin. The equivalence of deterministic top-down tree-to-string transducers is decidable with origin, while without origin it has (until very recently) been 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.

A Survey on Decidable Equivalence Problems for Tree Transducers

The decidability of equivalence for three important classes of tree transducers is discussed. Each class can be obtained as a natural restriction of deterministic macro tree transducers (MTTs): (1) no context parameters, i.e., top-down tree transducers, (2) linear size increase, i.e., MSO definable tree transducers, and (3) monadic input and output ranked alphabets. For the full class of mtts, decidability of equivalence remains a long-standing open problem.

Equivalence Problems for Tree Transducers: A Brief Survey

The decidability of equivalence for three important classes of tree transducers is discussed. Each class can be obtained as a natural restriction of deterministic macro tree transducers (MTTs): (1) no context parameters, i.e., top-down tree transducers, (2) linear size increase, i.e., MSO definable tree transducers, and (3) monadic input and output ranked alphabets. For the full class of MTTs, decidability of equivalence remains a long-standing open problem.

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

The problem of inverse computation has many potential applications such as serialization/deserialization, providing support for undo, and test-case generation for software testing. In this paper, we propose an inverse computation method that always terminates for a class of functions known as parameter-linear macro tree transducers, which involve multiple data traversals and the use of accumulations. The key to our method is the observation that a function in the class can be regarded as a non-accumulative context-generating transformation without multiple data traversals. Accordingly, we demonstrate that it is easy to achieve terminating inverse computation for the class by context-wise memoization of the inverse computation results. We also show that when we use a tree automaton to express the inverse computation results, the inverse computation runs in time polynomial to the size of the original output and the textual program size.

Implementation of a deterministic partial E-unification algorithm for macro tree transducers

Implementation of a deterministic partial E-unification algorithm for macro tree transducers

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.

Computing the Cost of Typechecking of Composition of Macro Tree Transducers

Computing the Cost of Typechecking of Composition of Macro Tree Transducers

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...

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.

Composition of functions with accumulating parameters

Many functional programs with accumulating parameters are contained in the class of macro tree transducers. We present a program transformation technique that can be used to solve the efficiency problems due to creation and consumption of intermediate data structures in compositions of such functions, where classical deforestation techniques fail. To do so, given two macro tree transducers under appropriate restrictions, we construct a single macro tree transducer that implements the composition of the two original ones. The imposed restrictions are more liberal than those in the literature on macro tree transducer composition, thus generalising previous results.

Macro forest transducers

Xml documents conceptually are not trees, but forests. Therefore, we extend the concept of macro tree transducers (mtt's) to a transformation formalism of forests, macro forest transducers (mft's). We show that mft's form a strict superset of mtt's operating on forests (represented as binary trees). On the other hand, the transformation of every mft can be simulated by the composition of two mtt's. Although macro forest transducers are more powerful, the complexity of inverse type inference, i.e., computing the pre-image of a recognizable language, is almost the same as for tree transducers.

A comparison of pebble tree transducers with macro tree transducers

The n-pebble tree transducer was recently proposed as a model for XML query languages. The four main results on deterministic transducers are: First, (1) the translation $\tau$ of an n-pebble tree transducer can be realized by a composition of n+1 0-pebble tree transducers. Next, the pebble tree transducer is compared with the macro tree transducer, a well-known model for syntax-directed semantics, with decidable type checking. The -pebble tree transducer can be simulated by the macro tree transducer, which, by the first result, implies that (2) $\tau$ can be realized by an (n+1)-fold composition of macro tree transducers. Conversely, every macro tree transducer can be simulated by a composition of 0-pebble tree transducers. Together these simulations prove that (3) the composition closure of n-pebble tree transducers equals that of macro tree transducers (and that of 0-pebble tree transducers). Similar results hold in the nondeterministic case. Finally, (4) the output languages of deterministic n-pebble tree transducers form a hierarchy with respect to the number n of pebbles.

An operational and denotational approach to non-context-freeness

The main result of this paper is a description of linguistically motivated non-context-free phenomena equivalently in terms of regular tree languages (to express the recursive properties) and both a logical and an operational perspective (to establish the intended linguistic relations). The result is exemplified with a particular non-context-free phenomenon, namely cross-serial dependencies in natural languages such as Swiss German or Dutch. The logical description is specified in terms of binary monadic second-order (MSO) formulas and the operational description is achieved by means of a linear and non-deleting macro tree transducer. Besides giving a grammatical presentation for the regular tree language we shall also specify an implementation in the form of a finite-state (tree) automaton to emphasize the effectivity of our approach.

Macro Tree Translations of Linear Size Increase are MSO Definable

The first main result is that if a macro tree translation is of linear size increase, i.e., if the size of every output tree is linearly bounded by the size of the corresponding input tree, then the translation is MSO definable (i.e., definable in monadic second-order logic). This gives a new characterization of the MSO definable tree translations in terms of macro tree transducers: they are exactly the macro tree translations of linear size increase. The second main result is that given a macro tree transducer, it can be decided whether or not its translation is MSO definable, and if it is, then an equivalent MSO transducer can be constructed. Similar results hold for attribute grammars, which define a subclass of the macro tree translations.

Output String Languages of Compositions of Deterministic Macro Tree Transducers

The composition of total deterministic macro tree transducers gives rise to a proper hierarchy with respect to their output string languages (these are the languages obtained by taking the yields of the output trees). There is a language not in this hierarchy which can be generated by a (quite restricted) nondeterministic string transducer, namely, a two-way generalized sequential machine. Similar results hold for attributed tree transducers, for controlled EDT0L systems, and for YIELD mappings (which proves properness of the IO-hierarchy). Witnesses for the properness of the macro tree transducer hierarchy can already be found in the latter three hierarchies.

A Characterization of Attributed Tree Transformations by a Subclass of Macro Tree Transducers

We present a characterization of the class of tree transformations computed by (noncircular) attributed tree transducers, in terms of a subclass of macro tree transducers, called attributed-like macro tree transducers. In fact, we prove formally that attributed tree transducers and attributed-like macro tree transducers generate the same class of tree transformations. Moreover, we prove that it is decidable whether a macro tree transducer is attributed-like.