Section 1 of this paper presents the basic mathematical definitions for our work. Section 2 defines the notion of a weighted phrase-structure grammar over either a semiring or zero monoid coefficient structure. The notion of canonical derivations (from Griffiths [1968]) and top-down derivations is defined in section 3, along with some of their basic properties. Equivalence relations over weighted phrase-structure grammars are defined in section 4 as well as power series representations of the associated languages.
Section 5 shows that every weighted type 0 grammar is equivalent to a grammar in standard form.
Section 6 describes the equations associated with a grammar in standard form and describes a process of iteratively obtaining approximations to the language.
[1]
Noam Chomsky,et al.
The Algebraic Theory of Context-Free Languages*
,
1963
.
[2]
H. W. Buttelmann.
Syntax-semantics systems as structure manipulation systems: phrase structure grammars and generalized finite automata
,
1971,
SIGA.
[3]
Timothy V. Griffiths.
Some Remarks on Derivations in General Rewriting Systems
,
1968,
Inf. Control..
[4]
H. W. Buttelmann,et al.
On generalized finite automata and unrestricted generative grammars
,
1971,
STOC.
[5]
Eliahu Shamir,et al.
A Representation Theorem for Algebraic and Context-Free Power Series in Noncommuting Variables
,
1967,
Inf. Control..