Bottom Up Quotients and Residuals for Tree Languages

TL;DR

This paper introduces a novel bottom-up quotient approach for tree languages, enabling the computation of residuals as sets of k-ary trees and leading to the development of minimal deterministic automata for regular tree languages.

Contribution

It extends tree language quotients to bottom-up computation, defining quotient formulas for various language operations and constructing minimal automata for regular tree languages.

Findings

Defined bottom-up quotient formulas for union, products, and compositions.

Introduced the bottom-up quotient tree automaton.

Proved minimality of the automaton for regular tree languages.

Abstract

In this paper, we extend the notion of tree language quotients to bottom-up quotients. Instead of computing the residual of a tree language from top to bottom and producing a list of tree languages, we show how to compute a set of k-ary trees, where k is an arbitrary integer. We define the quotient formula for different combinations of tree languages: union, symbol products, compositions, iterated symbol products and iterated composition. These computations lead to the definition of the bottom-up quotient tree automaton, that turns out to be the minimal deterministic tree automaton associated with a regular tree language in the case of the 0-ary trees.