1. Chapter 4. Syntax Analysis (1)

2. Application of a production A in a derivation step i  i+1

3. Formal grammars (1/3)
Example : Let G1 have N = {A, B, C}, T = {a, b, c} and the set of productions
  A CB  BC
A  aABC bB  bb
A  abC bC  bc
cC  cc
The reader should convince himself that the word akbkck is in L(G1) for all k  1 and that only these words are in L(G1). That is,
L(G1) = { akbkck | k  1}.

4. Formal grammars (2/3) Example : Grammar G2 is a modification of G1:
G2:   A CB  BC
A  aABC bB  bb
A  abC bC  b
The reader may verify that L(G2) = { akbk | k  1}. Note that the last rule, bC  b, erases all the C's from the derivation, and that only this production removes the nonterminal C from sentential forms.

5.   S  aSb  aaSbb  aaabbb
Formal grammars (3/3)
Example : A simpler grammar that generates { akbk | k  1} is the grammar G3 :
G3:   S
S  aSb
S  ab
A derivation of a3b3 is
  S  aSb  aaSbb  aaabbb
The reader may verify that L(G3) = { akbk | k  1}.

6. The four types of formal grammars
Format of Productions
Remarks
φAψ→ φω ψ
Unrestricted
Substitution
Rules 1
φAψ→ φω ψ, ω≠λ
∑→λ
Context
Sensitive
Free
Right
Linear
Left 2
A →ω, ω≠λ 3
A→aB
A→a
A→Ba
A →a
Contracting
Noncon-
tracting
Regular
The four types of formal grammars

7. Context-Sensitive Grammars(Type1)
Unrestricted Grammars(Type0)
Context-Sensitive Grammars(Type1)
Definition : A context-sensitive grammar G = (N,T,P,) is a formal grammar in which all productions are of the form
φAψ→φωψ, ω≠ 
The grammar may also contain the production  →, if G is a context-sensitive (type1) grammar, then L(G) is a context-sensitive (type1) language.

8. Context-Free Grammars (Type2)
Definition : A context-free grammar G=(N,T,P,) is a formal grammar in which all productions are of the form
A→ω
The grammar may also contain the production  →λ. If G is a context-free (type2) grammar, then L(G) is a context-free (type2) language.
A∈N∪{}
ω∈(N∪T)*-{λ}

9. Regular Grammars (Type3) (1/2)
Definition : A production of the form
A→aB or A→a
is called a right linear production. A production of the form
A→Ba or A→a
is a left linear production. A formal grammar is right linear if it contains only right linear productions, and is left linear if it contains only left linear production  →λ. Left and right linear grammars are also known as regular grammars. If G is a regular (type3) grammar, then L(G) is a regular (type3) language.
A∈N∪{∑}
B∈N
a∈T
A∈N∪{∑}
B∈N
a∈T

10. Regular Grammars (Type3) (2/2)
Example: A left linear grammar G1 and a right linear grammar G2 have productions as follows:
G1 : G2 :
The reader may verify that
L(G1) = (10)*1=1(01)*=L(G2)
∑ → 1B
∑ → 1
A → 1B
B → 0A
A → 1
∑ → B1
∑ → 1
A → B1
B → A0
A → 1

11. Fig. 4.23. Operator-precedence relations.id + * $
·>
<·
Fig Operator-precedence relations.

12. Operator-Precedence Relations from Associativity and Precedence (1/2)
1. If operator θ1 has higher precedence than operator θ2, make θ1 ·> θ2 and θ2 <· θ1 . For example, if * has higher precedence than +, make * ·> + and + <· *. These relations ensure that, in an expression of the form E+E*E+E, the central E*E is the handle that will be reduced first.
2. If θ1 and θ2 are operators of equal precedence (they may in fact be the same operator), then make θ1 ·> θ2 and θ2 ·> θ1 if the operators are left-associative, or make θ1 <· θ2 and θ2 <· θ1 if they are right-associative. For example, if + and – are left-associative, then make + ·> +, + ·> -, - ·> - and - ·> +. If  is right associative, then make  <· . These relations ensure that E-E+E will have handle E-E selected and EEE will have the last EE selected.

13. Operator-Precedence Relations from Associativity and Precedence (2/2)
3. Make θ <· id, id ·> θ, θ <· (, ( <· θ , ) ·> θ, θ ·> ), θ ·> $, and $ <· θ for all operators θ. Also, let
These rules ensure that both id and (E) will be reduced to E. Also, $ serves as both the left and right endmarker, causing handles to be found between $’s wherever possible.
( = )
$ <· (
$ <· id
( <· (
id ·> $
) ·> $
( <· id
id ·> )
) ·> )

14. Fig. 4.25. Operator-precedence relations.+ - * / ↑ id ( ) $
·>
<· =
Fig Operator-precedence relations.

15. Precedence Functions
Example 4.29 The Precedence table of Fig has the following pair of precedence functions,
For example, * <· id and, f(*) < g(id). Note that f(id) > g(id) suggests that id ·> id; but, in fact, no precedence relation holds between id and id. Other error entries in Fig are similarly replaced by one or another precedence relation. + - * / ↑ ( ) id $ f 2 4 6 g 1 3 5

16. Fig. 4.26. Graph representing precedence functions.

17. Fig. 4.28. Operator-precedence matrix with error entriesid ( ) $ e3
·>
<· = e4 e2 e1
Fig Operator-precedence matrix with error entries