In machine translation, there is a classical tradeoff between the expressivity of a grammar formalism and its computational complexity. The tree languages induced by context-free tree grammars (CFTGs) subsume the formal languages of many important grammar formalisms that are used in the area of machine translation. However, the computational cost required to work with CFTGs is high and the formalism might be unfeasible for certain applications. Hence, it is worthwhile to investigate the expressivity of CFTGs and find ways to reduce the complexity. In the thesis, three methods are explored to express context-free tree languages, i.e., languages that are induced by CFTGs, by regular tree grammars (RTGs). A RTG is a CFTG with strong syntactic restrictions and thus, it can be efficiently used for computations. Expressing context-free tree languages by RTGs unveils details about the complexity of CFTGs and provides knowledge about the two fundamental grammar formalisms. The first method finds restrictions for CFTGs such that each CFTG which complies with these restrictions can be characterized by a RTG. The restrictions are given in form of syntactic and decidable properties. Furthermore, it is shown how to construct a RTG which is equivalent to a given restricted CFTG. The properties are inspired by non-self-embedding context-free grammars [3, 10]. The second method approximates the tree language induced by an arbitrary CFTG. An approximation in this sense is a RTG that induces a superset of the tree language generated by the CFTG. In this way, non-regular dependencies within a tree are broken. This method uses an already known result that shows how context-free tree languages can be characterized by RTG extended by a pushdown storage [9, 6]. This characterization is turned into an approximation by restricting the pushdown storage and only allow a finite amount of storage configurations. To obtain the approximation, the finite information from the pushdown storage is incorporated into the nonterminals of the RTG. Whenever information is lost by the restriction of the pushdown, the missing data is guessed. Hence, a superset approximation is obtained (cf. [1, 2, 15, 8, 10] for similar results for strings). The third method relates weighted variants of CFTG and RTG. Weighted CFTG induce a weighted tree grammar, i.e., each tree from the tree language induced by the grammar is associated with a weight from the reals. The third method describes how the weights for a weighted RTG can be

Expressing Context-Free Tree Languages by Regular Tree Grammars

For the class of linear context-free tree grammars, we define a decidable property called self-embedding. We prove that each non-self-embedding grammar in this class generates a regular tree language and show how to construct the equivalent regular tree grammar.

Non-self-embedding linear context-free tree grammars generate regular tree languages

A practical tool for natural language modeling and development of human-machine interaction is developed in the context of formal grammars and languages. A new type of formal grammars, called grammars with prohibition, is introduced. Grammars with prohibition provide more powerful tools for natural language generation and better describe processes of language learning than the conventional formal grammars. Here we study relations between languages generated by different grammars with prohibition based on conventional types of formal grammars such as context-free or context sensitive grammars. Besides, we compare languages generated by different grammars with prohibition and languages generated by conventional formal grammars. In particular, it is demonstrated that they have essentially higher computational power and expressive possibilities in comparison with the conventional formal grammars. Thus, while conventional formal grammars are recursive and subrecursive algorithms, many classes of grammars with prohibition are superrecursive algorithms. Results presented in this work are aimed at the development of human-machine interaction, modeling natural languages, empowerment of programming languages, computer simulation, better software systems, and theory of recursion.

R Parchmann

Papers

Self-embedding indexed grammars