Tree Homomorphisms Research Articles

Stochastic tree functions

We introduce and study stochastic treeautomata generalizing in the framework of trees the well known stochastic string automata. We prove that events induced by such automata constitute a convex subset of the space of all tree events as well as that stochastic events are closed under composition with tree homomor-phisms. Moreover, it is shown that a stochastic treelanguage determined by an isolated cutpoint is actually recognizable.

The congruence theory of closure properties of regular tree languages

Boolean operations, tree homomorphisms and their converses, and forest product, in special cases σ-catenation, x-product, x-quotient and x-iteration, preserve the regularity of forests. These closure properties are proved algebraically by using congruences of term algebras which saturate the forests operated on and constructing, by means of them, a congruence which saturates the product forest. The index of the constructed congruence is finite, if the congruences saturating the forests to operate are of finite indexes. The cardinalities of ranked and frontier alphabets are arbitrary. The preservation of recognizability is a straightforward consequence of those congruence constructions and the Nerode type of congruence characterization for recognizable forests. Furthermore, the constructed congruences can also be applied directly to construct explictly tree automatas to recognize the product forests.

The copying power of one-state tree transducers

One-state deterministic top-down tree transducers (or, tree homomorphisms) cannot handle “prime copying,” i.e., their class of output (string) languages is not closed under the operation L → {$( w$) f( n) ∥ w ϵ L, f( n) ⩾ 1}, where f is any integer function whose range contains numbers with arbitrarily large prime factors (such as a polynomial). The exact amount of nonclosure under these copying operations is established for several classes of input (tree) languages. These results are relevant to the extended definable (or, restricted parallel level) languages, to the syntax-directed translation of context-free languages, and to the tree transducer hierarchy.

The Copying Power of One-State Tree Transducers

One-state deterministic top-down transducers (or tree homomorphisms) cannot handle ''prime copying'', i.e. their class of output (string) languages is not closed under the operation L ( -> { (w)^f(n) | w in L, f(n) >= 1 } ) , where f is any integer function whose range contains numbers with arbitrary large prime factors (such as a polynomial). The exact amount of nonclosure under these copying operations is established for several classes of input (tree) languages. These results are relevant to the extended definable (or restricted parallel level) languages, to the syntax-directed translation of context-free languages and to the tree transducer hierarchy.