10. Left-associative grammar (LAG)

10.1 Rule types and derivation order

10.1.1 The notion left-associative

When we combine operators to form expressions, the order in which the operators are to be applied may not be obvious. For example, a + b + c can be interpreted as ((a + b) + c) or as (a + (b + c)). We say that + is left-associative if operands are grouped left to right as in ((a + b) + c). We say it is right-associative if it groups operands in the opposite direction, as in (a + (b + c)).

A.V. Aho & J.D. Ullman 1977, p. 47

10.1.2 Incremental left- and right-associative derivation

left-associative: right-associative:

    a                                                      a
   (a + b)                                          (b + a)
  ((a + b) + c)                               (c + (b + a))
 (((a + b) + c) + d)                   (d + (c + (b + a)))
      ...                                             ...

⟹ ⟸ 10.1.3 Left-associative derivation order

Derivation is based on the principle of possible continuations Used to model the time-linear structure of language

10.1.4 Irregular bracketing structures corresponding to the trees of C- and PS-grammar

   (((a + b) + (c +d)) + e)
   ((a + b) + ((c +d)) + e)
   (a + ((b + c)) + (d + e))
   ((a + (b + c)) + (d + e))
   (((a + b) + c) + (d +e))
             ...

The number of these irregular bracketings grows exponentially with the length of the string and is infinite, if bracketings like (a), ((a)), (((a))), etc., are permitted.

10.1.5 Irregular bracketing structure

Derivation is based on the principle of possible substitutions Used to model constituent structure

FoCL, Chapter 10: Left-associative grammar (LAG) 152

10.1.6 The principle of possible continuations Beginning with the first word of the sentence, the grammar describes the possible continuations for each sentence start by specifying the rules which may perform the next grammatical composition (i.e., add the next word).

10.1.7 Schema of left-associative rule in LA-grammar

r_{i}: cat_{1} cat_{2} ⇒ cat_{3} rp_{i} 10.1.8 Schema of a canceling rule in C-grammar

α_{(Y|X)} ∘ β_{(Y)} ⇒ αβ_{(X)} 10.1.9 Schema of a rewrite rule in PS-grammar

A → B C 10.1.10 Three conceptual derivation orders

[figure: LA-grammar; C-grammar; PS-grammar; bot.-up left-associative; bottom-up amalgamating; top-down expanding]

10.2 Formalism of LA-grammar

10.2.1 Algebraic definition of LA-grammar

A left-associative grammar (or LA-grammar) is defined as a 7-tuple <W, C, LX, CO, RP, ST_{S}, ST_{F} >, where

1. W is a finite set of word surfaces.

2. C is a finite set of category segments.

3. LX ⊂ (W × C^{+}) is a finite set comprising the lexicon.

4. CO = (co_{0} ... co_{n−1}) is a finite sequence of total recursive functions from (C^{∗} × C^{+}) into C^{∗} ∪ {⊥}, called categorial operations.

5. RP = (rp_{0} ... rp_{n−1}) is an equally long sequence of subsets of n, called rule packages.

6. ST_{S} = {(cat_{s} rp_{s}), ...} is a finite set of initial states, whereby each rp_{s} is a subset of n called start rule package and each cat_{s} ϵ C^{+}.

7. ST_{F} = {( cat_{f} rp_{f}), ...} is a finite set of final states, whereby each cat_{f} ϵ C^{∗} and each rp_{f} ϵ RP.

10.2.2 A concrete LA-grammar is specified by

1. a lexicon LX (cf. 3),

2. a set of initial states ST_{S} (cf. 6),

3. a sequence of rules r_{i}, each defined as an ordered pair (co_{i}, rp_{i}), and

4. a set of final states ST_{F}.

10.2.3 LA-grammar for a^{k}b^{k}

LX =_{def} {[a (a)], [b (b)]} ST_{S} =_{def} {[(a) {r_{1}, r_{2}}]} r_{1}: (X) (a) ⇒ (aX) {r_{1}, r_{2}} r_{2}: (aX) (b) ⇒ (X) {r_{2}} ST_{F} =_{def} {[ε rp_{2}]}.

10.2.4 LA-grammar for a^{k}b^{k}c^{k}

LX =_{def} {[a (a)], [b (b)], [c (c)]} ST_{S} =_{def} {[(a) {r_{1}, r_{2}}]} r_{1}: (X) (a) ⇒ (aX) {r_{1}, r_{2}} r_{2}: (aX) (b) ⇒ (Xb) {r_{2}, r_{3}} r_{3}: (bX) (c) ⇒ (X) {r_{3}} ST_{F} =_{def} {[ε rp_{3}]}.

10.2.5 The finite state backbone of the LA-grammar for a^{k}b^{k}c^{k}

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10.2.6 Recursion of left-associative algorithm

[figure: [rp_{k} cat-1′′′]; [rp_{j} cat-1′′]; [rp_{i} cat-1′]; [rp_{s} cat-1]; [rp_{s} (cat-1 cat-2)]; [rp_{i} (cat-1′ cat-2′)]; [rp_{j} (cat-1′′ cat-2′′)]; STATES; APPLICATION; NW-INTAKE; APPLICATION SETS]

10.3 Time-linear analysis

10.3.1 LA-trees as structured lists

(A (AB (ABC ABCD B) C) D) (ii)  ABCD ABC) AB) A) (D (C  (iii) (B (i)^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} A  B AB ABC C D ABCD

10.3.2 LA-grammar derivation of a^{k}b^{k} for k =3

NEWCAT>  a a a b b b
*START-0
1
   (A) A
   (A) A
*RULE-1
2
   (A A) A A
   (A) A
*RULE-1
3
   (A A A) A A A
   (B) B
*RULE-2
4
   (A A) A A A B
   (B) B
*RULE-2
5
   (A) A A A B B
   (B) B
*RULE-2
6
   (NIL) A A A B B B

10.3.3 Interpretation of a history section

active rule package:          *START-0
composition number:           1
sentence start:                (A) A
next word:                      (A) A
successful rule:              *RULE-1
next composition number:      2
result:                         (A A) A A

10.3.4 Overlap between history sections

active rule package:           *RULE-1
composition number:             2
sentence start :                (A A) A A
next word:                       (A) A
successful rule :               *RULE-1
next composition number:       3
result:                          (A A A) A A A

10.4 Absolute type transparency of LA-grammar

10.4.1 Parsing aaabbbccc with active rule counter

NEWCAT>  a a a b b b c c c
;  1: Applying rules (RULE-1 RULE-2)                         (A A B) A A A B
;  2: Applying rules (RULE-1 RULE-2)                         (B) B
;  3: Applying rules (RULE-1 RULE-2)                      *RULE-2
;  4: Applying rules (RULE-2 RULE-3)                      5
;  5: Applying rules (RULE-2 RULE-3)                         (A B B) A A A B B
;  6: Applying rules (RULE-2 RULE-3)                         (B) B
;  7: Applying rules (RULE-3)                              *RULE-2
;  8: Applying rules (RULE-3)                              6
; Number of rule applications: 14.                             (B B B) A A A B B B
                                                                   (C) C
     *START-0                                                 *RULE-3
     1                                                         7
        (A) A                                                      (C C) A A A B B B C
        (A) A                                                      (C) C
     *RULE-1                                                  *RULE-3
     2                                                         8
        (A A) A A                                                 (C) A A A B B B C C
        (A) A                                                      (C) C
     *RULE-1                                                  *RULE-3
     3                                                         9
        (A A A) A A A                                             (NIL) A A A B B B C C C
        (B) B
     *RULE-2
     4

10.4.2 Generating a representative sample in a^{k}b^{k}c^{k}

NEWCAT> (gram-gen 3 '(a b c))
Parses of length 2:                                                  1 1 2 2   (A B B)
A B                                                                 A A A A B
  2   (B)                                                             1 1 1 2   (A A A B)
A A
  1   (A A)                                                        Parses of length 6:
                                                                     A A B B C C
Parses of length 3:                                                  1 2 2 3 3   (NIL)
A B C                                                               A A A B B B
  2 3   (NIL)                                                        1 1 2 2 2   (B B B)
A A B                                                               A A A A B B
  1 2   (A B)                                                        1 1 1 2 2   (A A B B)
A A A
  1 1   (A A A)                                                   Parses of length 7:
                                                                     A A A B B B C
Parses of length 4:                                                  1 1 2 2 2 3   (B B)
A A B B                                                             A A A A B B B
  1 2 2   (B B)                                                      1 1 1 2 2 2   (A B B B)
A A A B
  1 1 2   (A A B)                                                 Parses of length 8:
A A A A                                                             A A A B B B C C
  1 1 1   (A A A A)                                                 1 1 2 2 2 3 3   (C)
                                                                     A A A A B B B B
Parses of length 5:                                                  1 1 1 2 2 2 2   (B B B B)
A A B B C
  1 2 2 3   (B)                                                   Parses of length 9:
A A A B B                                                          A A A B B B C C C

1 1 2 2 2 3 3 3   (NIL)
A A A A B B B B C
1 1 1 2 2 2 2 3   (B B B)
Parses of length 10:
A A A A B B B B C C
1 1 1 2 2 2 2 3 3   (B B)
Parses of length 11:
A A A A B B B B C C C
1 1 1 2 2 2 2 3 3 3   (B)
Parses of length 12:
A A A A B B B B C C C C
1 1 1 2 2 2 2 3 3 3 3   (NIL)

10.4.3 Complete well-formed expression in a^{k}b^{k}c^{k}

 A A A B B B C C C
   1 1 2 2 2 3 3 3   (NIL)

10.5 LA-grammar for natural language

10.5.1 Constituent structure analysis in C-grammar

[figure: Mary gives Fido a bone; (V); gives Fido a bone; (S3 V); gives Fido; (S3 A V); a bone; (SNP); Mary; (SNP); gives; (S3 D A V); Fido; (SNP); a; (SN SNP); bone; (SN)]

10.5.2 Time-linear analysis in LA-grammar

[figure: Mary gives Fido a bone; (V); Mary gives Fido a; (SN V); Mary gives Fido; (A V); Mary gives; (D A V); Mary; (SNP); gives; (S3 D A V); Fido; (SNP); a; (SN SNP); bone; (SN)]

FoCL, Chapter 10: Left-associative grammar (LAG) 166

10.5.3 Categorial operation combining Mary and gives (SNP) (N D A V) ⇒ (D A V) 10.5.4 Categorial operation combining Mary gives and Fido (D A V) (SNP) ⇒ (A V) 10.5.5 Categorial operation combining Mary gives Fido and a (A V) (SN SNP) ⇒ (SN V) 10.5.6 Categorial operation combining Mary gives Fido a and book (SN V) (SN) ⇒ (V)

10.5.7 Left-associative parsing of example 10.5.2

NEWCAT>  Mary gives Fido a bone \.
*START
1
   (SNP) MARY
   (S3 D A V) GIVES
*NOM+FVERB
2
   (D A V) MARY GIVES
   (SNP) FIDO
*FVERB+MAIN
3
   (A V) MARY GIVES FIDO
   (SN SNP) A
*FVERB+MAIN
4
   (SN V) MARY GIVES FIDO A
   (SN) BONE
*DET+NOUN
5
   (V) MARY GIVES FIDO A BONE
   (V DECL) .
*CMPLT
6
   (DECL) MARY GIVES FIDO A BONE .

10.5.8 Analysis of a discontinuous element

NEWCAT>  Fido dug the bone up \.
*START
1
   (SNP) FIDO
   (N A UP V) DUG
*NOM+FVERB
2
   (A UP V) FIDO DUG
   (SN SNP) THE
*FVERB+MAIN
3
   (SN UP V) FIDO DUG THE
   (SN) BONE
*DET+NOUN
4
   (UP V) FIDO DUG THE BONE
   (UP) UP
*FVERB+MAIN
5
   (V) FIDO DUG THE BONE UP
   (V DECL) .
*CMPLT
6
   (DECL) FIDO DUG THE BONE UP .

10.5.9 LA-analysis of ungrammatical input

NEWCAT>  the young girl give Fido the bone \.
ERROR
Ungrammatical continuation at: "GIVE"
*START
1
   (SN SNP) THE
   (ADJ) YOUNG
*DET+ADJ
2
   (SN SNP) THE YOUNG
   (SN) GIRL
 *DET+NOUN
3
   (SNP) THE YOUNG GIRL