Modular tree transducers

Abstract

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.