A Grammar Tool

Applied by Dennis J. Darland

June 15, 2007

See

http://smlweb.cpsc.ucalgary.ca/start.html

I’ve made my grammar acceptable to it.

But you will have to copy & paste it.

This is a context free grammar that I think would appear to be propositions of Principia Mathematica.

I think the grammar for what would include only valid abbreviations would not be context free and I have never studied that.

 S -> A .

S -> A eq A.

S -> A equ A.

S -> A and A.

S -> not A.

A -> F ( V ).

A -> V e ( C ).

C -> P excl  z.

A -> P excl x.

A -> F ( w ( F w ) ).

A -> F ( P excl z ).

S -> ( ex V ) F V.

S -> ( ex F ) F V.

S -> ( ex P ) P excl V.

A ->  ( all F ) F V.

A ->  ( all V ) P excl V.

A -> ( all V ) F V.

F ->  f.

F ->  g.

P ->  p.

P ->  q.

C  ->  c.

C ->  d.

V ->  x.

V ->  y.

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Originally the grammar is not LL(1).

After 3 attempts to transform it you get:

Result of Analysis

Grammar
S →	F FF FA
|	V e ( C ) FA
|	P excl x FA
|	( F(3
|	not A .
F(3 →	F(1 FA
|	F( .
FA →
|	eq A
|	equ A
|	and A .
F( →	ex Fex .
Fex →	V ) F V
|	F ) F V
|	P ) P excl V .
A →	F FF
|	V e ( C )
|	P excl x
|	( F(1 .
FF →	( F(2 .
F(2 →	V )
|	w ( F w ) )
|	P excl z ) .
F(1 →	all Fall .
Fall →	F ) F V
|	V FV .
FV →	) F) .
F) →	P excl V
|	F V .
C →	P excl z
|	c
|	d .
F →	f
|	g .
P →	p
|	q .
V →	x
|	y .

Some sentences generated by this grammar: {q excl x, p excl x, x e ( d ), y e ( c ), x e ( c ), y e ( d ), not q excl x, not p excl x, not y e ( c ), not x e ( d ), not y e ( d ), not x e ( c ), not y e ( p excl z ), not y e ( q excl z ), not x e ( q excl z ), p excl x eq p excl x, not x e ( p excl z ), p excl x eq q excl x, p excl x equ p excl x, p excl x equ q excl x}

• All nonterminals are reachable and realizable.
• The nullable nonterminals are: FA.
• The endable nonterminals are: F) FV Fall F(₂ V Fex F( F(₁ S A F(₃ FA FF.
• No cycles.

nonterminal	first set	follow set	nullable	endable
S	( not f g x y p q	∅	no	yes
F(3	all ex	∅	no	yes
FA	eq equ and	∅	yes	yes
F(	Ex	∅	no	yes
Fex	x y f g p q	∅	no	yes
A	( f g x y p q	∅	no	yes
FF	(	eq equ and	no	yes
F(2	w x y p q	eq equ and	no	yes
F(1	All	eq equ and	no	yes
Fall	f g x y	eq equ and	no	yes
FV	)	eq equ and	no	yes
F)	p q f g	eq equ and	no	yes
C	c d p q	)	no	no
F	f g	( w ) x y	no	no
P	p q	) excl	no	no
V	x y	eq equ and e )	no	yes

The grammar is LL(1).