Composition Of Macro Tree Transducers Research Articles

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.

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

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.

Modular tree transducers

A new tree transducer, called a modular tree transducer, is introduced. This device specifies operations on trees and can be considered as a formalization of the concept of nested simultaneous primitive recursion on trees. Roughly speaking, a modular tree transducer is a special left-linear and non-overlapping term rewriting system of which the set of rules is partitioned into modules, each module being equipped with a non-negative integer: the number of the module. Then, a module with number i may call modules with numbers not less than i. Three properties are proved: (1) modular tree transducers compute exactly the (inductively defined) class of primitive recursive functions on trees; (2) the number of modules in modular tree transducers induces a strict hierarchy on the class of all modular tree transductions; and (3) by appropriately restricting the calling structure between modules, modular tree transducers characterize the compositions of macro tree transducers where the number of modules and the number of compositions coincide.