Ultrametric Distance in Syntax

Ultrametric Distance in Syntax

Mark D. Roberts,

Physikalisches Institut, Albert-Ludwigs Universitat Freiburg,
Herman-Herder Straβe 3, Freiburg im Breisgau, Germany, D-79104
[email protected].

October 26, 2006

Abstract

Phrase structure trees have a hierarchical structure. In many subjects,
most notably in taxonomy such tree structures have been studied using
ultrametrics. Here syntactical hierarchical phrase trees are subject to a
similar analysis, which is much simpler as the branching structure is more
readily discernible and switched. The occurrence of hierarchical structure
elsewhere in linguistics is mentioned. The phrase tree can be represented
by a matrix and the elements of the matrix can be represented by trian-
gles. The height at which branching occurs is not prescribed in previous
syntactic models, but it is by using the ultrametric matrix. In other words
the ultrametric approach gives a complete description of phrase trees, un-
like previous approaches. The ambiguity of which branching height to
choose, is resolved by postulating that branching occurs at the lowest
height available. An ultrametric produces a measure of the complexity
of sentences: presumably the complexity of sentences increases as a lan-
guage is acquired so that this can be tested. All ultrametric triangles
are equilateral or isosceles. Here it is shown that XX structure implies that
there are no equilateral triangles. Restricting attention to simple syntax a
minimum ultrametric distance between lexical categories is calculated. A
matrix constructed from this ultrametric distance is shown to be different
than the matrix obtained from features. It is shown that the definition
of c-command can be replaced by an equivalent ultrametric definition.
The new definition invokes a minimum distance between nodes and this is
more aesthetically satisfying than previous varieties of definitions. From
the new definition of c-command follows a new definition of of the central
notion in syntax namely government.

1 Introduction

1.1 Ultrametric Literature

Ultrametrics are used to model any system that can be represented by a bi-
furcating hierarchical tree. To briefly list some areas where ultrametrics have

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