Left-Associative Grammar: The Algebraic Definitions

LEFT-ASSOCIATIVEGRAMMAR: The Algebraic

Definitions

ROLAND HAUSSER Institut fur Deutsche Philologie West

Munchen,

Germany

ABSTRACT:

This in

informally only

are

parsers

examples.

Finally,

sition

Networks,

of

the grammar

to be

is shown between

is proven

system

described

to generate

all

and

and context-sensitive context-free, regular, The relation between LA-grammar

of

in LA-grammar.

'type

transparent', and Finite

LA-grammar

and

State

Automata,

and with

illustrated

RTNs,

are explained.

Analyzers

Recursive

Power,

Context-Sensitive

Languages,

Input-Output

classes

generators

Generative

KEYWORDS:

account

of CaT.1 LA-grammar

The

differences

and Predictive

ATNs,

issue

languages.

and

a formal

presents

reconstructed

associated

Free

paper

the previous

the recursive

languages

Miinchen

Universitdt

Ludwig-Maximilians

Languages,

Languages,

Finite-State

Networks,

Augmented-Transition

Parsers,

Regular

Languages,

Automata, Generators,

Context

Recursive-Tran Type-Transparency,

Equivalence.

1. THE FORMAL

DEFINITION

Left-associative grammars constitute a new class of formal objects for which we are going to give an algebraic definition. Let us recall some notation from set theory needed for this purpose. If X is a set, then X+ is the 'positive closure', i.e., the set of all concatenations of elements of X. X* is the Kleene closure of X, defined as X+ u e, where c is the 'empty sequence'. The power set of X is denoted by 2X . If X and Y are sets, then (X x Y) is the Cartesian product of X and Y, i.e., the set of ordered pairs consisting of an element of X and an element of Y. For convenience, we also identify integers with sets, i.e., n = {i | 0 < i for all w? w, GW, and all c* G C+, is an instance of lexical look-up only if w, = w,. (w, c*), (wt ci), (w, cm), etc., are called the 'lexical readings' of the word surface wt G W. For example, the word surface gave is mapped into the lexical readings [gave (NDA V)] and [gave (N A TO

Thus, a categorized word a category. For example,

V)].

2Since

LX

is finite,

each w 6 W

is related by LX

to a non-empty

finite

set of elements

of C+.

LEFT-ASSOCIATIVE GRAMMAR 123 of a Sentence

1.4 Definition A

A

start

sentence

is an

Start of

element

x C*).

(W+

start is an ordered pair consisting of a sequence of word surfaces of the sentence start) and a category. For example, if John, 'surface' the (called are and the word surfaces inW, and if GQ, V are category segments in read, then read C, [{John the) (GQ V)] is a sentence start consisting of the surface sentence

{John read the) ? W+ 1.5 Definition

that

(GQ V) e C+.

Concatenation

of Surface

is the function SC from (W* x W)

concatenation

Surface such

and the category

...

SC((w-l

w-k),

=

w-k+1)

... w-k

(w-1

into W+

w-k+1),

for

all

w-j e W. The

function SC concatenates the surfaces of the input expressions3 into the surface of the resulting sentence start. This completely regular operation gives rise to the name of Left-Associative Grammar. 1.6 Definition The

of a Left-Associative

i-th left-associative

Rule

rule r, is the triple (co/, SC, rpt).

A left-associative rule r, takes a sentence start ss and a next word nw as input, and applies the categorial operation co, to the sentence category cat-1 and the next word category cat-2. If the input condition of the categorial operation is satisfied by (cat-1, cat-2), the application of r, is successful and an output is derived. The output consists of the pair (rp,, ss')> where rp, is a rule package and ss'

is a resulting

sentence

start.

If the

is not satisfied by (cat-1, cat-2), no output is derived.

input

condition

the application

of

the

categorial

operation

of rule r, is not successful

and

The rule package rp, provided by the rule r, contains all rules which may apply after rule r, was successful. A rule package is defined as a set of rule names, where the name of a rule rg is the place number g of its categorial operation cog in the sequence CO.4 In practice, the rules are called by more mnemonic

nw

=

such

names,

The resulting (wn+i

as

sentence

cat-2),

then

or

'rule-g'

'Fverb+main\

start ss' is derived as follows. ss'

=

(

cat-3),

where

If ss = (A? cat-1) and

is derived

from An and wrt+1 by SC, and cat-3 is derived from cat-1 and cat-2 by co,-. The categorial operation co, specifies the input categories and the output category by 3I.e., a pair consisting

of a sentence

start and a next word.

4To give computational recursion a form that is not impredicative in the mathematical Dana Scott suggested the use of rule names in the rule packages.

Professor

sense,

124

ROLAND HAUSSER

of category expressions. Category expressions may contain variables for category segments (written as segl, seg2, etc.), and for sequences of category segments of length > 0 (written as X, Y, etc.).

means

1.7 Definition

of a Category

Expression

is a list consisting of zero or more cate expression more zero or segment variables, and zero or more gory segments, The variables. empty list is represented as e. segment-sequence

A

category

Examples of category expressions are (a), (a X), where a is a category segment eCA category cat-i, with a category expression CAT-i, if the structure of pattern specified by the expression. For example, the is matched by categories like (a), (a a), (a b c), etc.

(X), (segl a X), (X a Y), is compatible cat-i e C, the the category matches (a X)

category expression

A category expression CAT-m is in the subset relation with the category expression CAT-n (CAT-m C CAT-n) if every category compatible with CAT m is compatible with CAT-n, and similarly for C, =, ? ?, and ?. A list of category segments (without any variables), e.g., (a), (a a), or e, is regarded as it is a Otherwise a category if it occurs as part of an analyzed expression. as CAT-i, category expression. Category expressions are represented abstractly while compatible categories are represented as cat-i. rules are notated by expressions of the form [(ss nw) => (rp, Left-associative x and are ss')], where ss, ss' expressions representing sentence starts G (W+ C), nw

is an

expression

representing

a

'next

word'

G

(W

x

C+).

Because

surface

composition SC is the same in all left-associative rules, itmay be omitted in the definitions. This results in the following simplified notation of left-associative rules:

1.8 Notation The

of a Left-Associative

Rule

i-th rule of an LA-grammar

r,: [CAT-1 CAT-2]

has the form

=? [rp, CAT-3],

where CAT-1, CAT-2, and CAT-3 are the category expressions ss, nw, and ss', respectively, and rp, is the rule package of r,.

of

is assumed implicitly in 1.8. If the categorial that surface concatenation cannot be cot expressed directly by the structure of the category ex operation pressions CAT-1, CAT-2, and CAT-3, a rule may be augmented with additional and the relationships among clauses which explain the category expressions them. If the categorial operation of a rule does not express a function, the rule is not considered well-formed.

Note

LEFT-ASSOCIATIVE GRAMMAR 125 2.

THE DERIVATIONAL

STRUCTURE

OF LA-GRAMMAR

In LA-grammar a complete well-formed expression is derived in a sequence of transitions from a start state via a number of rule states to a final state. These notions have the following definitions. 2.1 Definition

of the Set of Rule

States

A rule state in LA-grammar is defined as a pair [rp? CAT-3] where a is rule and is the category expression specified CAT-3 rp, package in the output of co,. The state associated with rule r, is called st,. The set of all rule states is called ST/?. 2.2 Definition

of the Set of Start States

If rps is the start rule package of an LA-grammar, nw = (surf cat-i), then [rps cat-i] is a start state.

and nw e LX,

For the sake of efficiency and precision, the set of start states is usually not defined with all lexical categories, but with a set of initial category expressions IC, defined as the set of all the category expressions which appear as CAT-1 in any of the rules in rp5, i.e., STS =d*f {[rps, CAT-1] |CAT-1 e IC}. 2.3 Definition

of the Set of Final

States

Given the set of 'final rule packages' RPf C RP, and the associated set of states {[rp, CAT-3,-] | rp, e RP/r}, the set of final states ST/r |CAT-3,/7 C CAT-3,}. =def {[rp, CAT-3^1 In other words, expressions

final states are like certain rule states, except be more

may

2.4 Definition

that their category

restricted.

of a State

A state is an element of the set STS U ST/? u STF. The set of states in an LA-grammar and

n rules,

2.5 Definition

it has

at most

2n

+

is finite.

If an LA-grammar

has / start states

/ states.

of a State Token

A

state (rpt CAT-3) represents a possibly infinite number of state tokens (rp, cat-3), where CAT-3 is a category expression and cat-3 represents categories compatible with CAT-3.

The

set of state tokens is unbounded because there is no upper limit on the length of categories generated by the rules. Two state tokens are equivalent only if they consist of the same rule package and the same category token.

ROLAND HAUSSER

126 2.6 Definition

of an Application

Set

An application set in LA-grammar is defined as a pair [rp, (cat-1 a where is rule rp, cat-2)] package and (cat-1 cat-2) is a pair of categories. Application

sets are derived from

2.7 Definition

of the nw-Intake

states by means

of the following

function:

Function

The function nw-intake takes a state token st, (0 < i < n) of the form [rp, cat-1] and a nw of the form (surf cat-2), nw e LX, and renders an application set of the form [rp, (cat-1 cat-2)] as output State function:

tokens are derived from

2.8 Definition

of the Application

application

sets by means

of the following

Function

The function application takes an application set [rp, (cat-1 cat-2)] as input, applies each rule j e RP, to (cat-1 cat-2), and renders a (possibly empty) set of state tokens as output. The interaction of states, nw-intake, applications sets, and application left-associative derivation is illustrated in the following schema: 2.9 The Recursion

of a Left-Associative

Derivation

STATE TOKENS [rps cat-1]

|

[rp/cat-1'] .

im

I

[rp/cat-r]-. |-m^ [rp/c cat-1'"]

f-^

APPLICATION

NW-INTAKE

L?[rpy

(cat-1" cat-2")]

^

I-[rp/ (cat-1' cat-2')]

^I

[rps (cat-1 cat-2)]

^

APPLICATION SETS

I

in a

GRAMMAR127 LEFT-ASSOCIATIVE 2.10 Definition

of a Left-Associative

Transition

A left-associative transition is a function from a state into a set of transition is a composite function, rule states. A left-associative of and nw-intake application. consisting transition can result in two different types of ambiguity, left-associative lexical ambiguity and syntactic ambiguity. Each reading of a lexically ambiguous next word is represented by a category; if an nw has i readings in nw-intake will create i application sets. Syntactic ambiguity arises when, a given application set, more than one state is generated because more than A

called

one rule in the rule package accepts the input pair. Since lexical ambiguity is associated with nw-intake and syntactic ambiguity is associated with application, in a transition. both kinds of ambiguity may occur simultaneously 2.11 Definition

of the Set of Well-Formed

The setWE of well-formed is defined as follows:

expressions

Expressions generated by an LA-grammar

1. If rps is the start rule package, and nw = (surf cat-i) is in LX, then [rp5 ((surf) cat-i)] G WE. 2. If (rpi, ss) G WE, j G rp,, and nw, nw G LX, is accepted by r;: [(ss nw) => (rp; ss')], then (rp, ss') G WE. 3. Nothing is inWE unless it so follows from (1) and (2). WE

is also called the reflexive-transitive closure of an LA-grammar. an is if it can be continued Intuitively, expression regarded as well-formed into a complete expression. For example, [rp-2 (aaab (bbccc))] is a well-formed expression of the language akbkck defined below because it can be continued into the complete well-formed expression [rp-3 (aaabbbccc, e)]. The set of complete well-formed expressions of a language is characterized by the final states of its grammar. Because [rp-3, e] is the final state of akbkck, all well formed expressions of that language with rp-3 as their rule package and e as their category are considered complete. 2.12 Definition The {s

of the Set of Surfaces

set of 'phrases' or surfaces S, S C W+, s | is the surface of we G WE}.

is

Special subsets of the well-formed expressions and the surfaces are the com plete well-formed expressions and the complete surfaces, respectively, which are defined as follows. 2.13 Definition

of the Set of Complete

Well-Formed

Expressions

The set of complete well-formed expressions CWE C WE of pairs [rp/ (s/ C/)], where [rp/ c/] G STF.

is the set

128 2.14 Definition The {s/

3.

ROLAND HAUSSER

of the Set of Complete

Surfaces

or complete surfaces CS, CS C W+, | S/ is the surface of ewe G CWE}.

set of 'sentences'

THE FORMAL

AN EXAMPLE:

LANGUAGE

is

akbkck

An LA-grammar is usually specified by (i) a lexicon LX, (ii) a set of start states STs, (iii) a sequence of rules, and (iv) a set of final states STF. Let us illustrate this general format of LA-grammars with a simple example of a formal language, namely the context-sensitive language akbkck. 3.1 The Definition

LX=^

of akb*ck5

{[a(bc)],[b(b)],[c(c)]}

STs=def {({r-l,r-2}(bc))} r-1: [(X)(bc)] =>[{r-l,r-2}(bXc)], r-2: [(bXc) (b)] => [{r-2, r-3} (Xc)],

r-3: [(cX) (c)] => [{r-3} (X)] STF=^

{[rp-3e]}.

Given example 3.1, let us consider the relation between the definition of as a 6-tuple <W, C, LX, CO, RP, rp,s>, and the specification of LA-grammar an LA-grammar in terms of LX, STs, a list of rules, and STf. The sets of in the word surfaces W and category segments C are implicitly characterized definition of LX: W -d^ {a, b, c} and C =