Enumerating syntactical graphs and lattices of derivations
Derivations in phrase-structure grammars are by now well understood, and it is generally considered convenient to study equivalence classes of derivations rather than individual derivations themselves. It has been established that classes can be represented by canonical derivations, syntactical graphs, or derivation words, and the categorical algebra of derivations provides the framework for their study. Regarded in this way, it is known that each derivation induces a distributive lattice of subderivations. In this paper a simple algorithm is given for enumerating this lattice for any derivation. The simplicity of this algorithm depends on the nature of the topological sort which allows a canonical derivation (or derivation word) to be constructed uniquely from a syntactical graph. The enumeration algorithm constructs the members of the lattice directly. In the process a new characterization of the syntactical graphs is given using the concept of a “doubly ordered graph.” This characterization greatly simplifies some of the previous work in this field. A direct correspondence between these graphs and the symmetric group (set of permutations) is shown.
Key wordsPhrase-structure grammars derivation languages syntactical graphs lattice of derivations doubly ordered graphs permutations
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