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 G 1 have N = {A, B, C}, T = {a, b, c} and the set of productions   ACB  BC A  aABCbB  bb A  abCbC  bc cC  cc The reader should convince himself that the word a k b k c k is in L(G 1 ) for all k  1 and that only these words are in L(G 1 ). That is, L(G 1 ) = { a k b k c k | k  1}.

4 Formal grammars (2/3)  Example : Grammar G 2 is a modification of G 1 : G 2 :   ACB  BC A  aABCbB  bb A  abCbC  b The reader may verify that L(G 2 ) = { a k b k | 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 Formal grammars (3/3)  Example : A simpler grammar that generates { a k b k | k  1} is the grammar G 3 : G 3 :   S S  aSb S  ab A derivation of a 3 b 3 is   S  aSb  aaSbb  aaabbb The reader may verify that L(G 3 ) = { a k b k | k  1}.

6 TypeFormat of ProductionsRemarks 0φAψ→ φω ψUnrestricted Substitution Rules 1φAψ→ φω ψ, ω≠λ ∑→λ Context Sensitive Context Free Right Linear Left Linear 2A →ω, ω≠λ ∑→λ 3A→aB A→a ∑→λ A→Ba A →a ∑→λ Regular Noncon- tracting Contracting The four types of formal grammars

7 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. Unrestricted Grammars(Type0 )

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 G 1 and a right linear grammar G 2 have productions as follows: G 1 : G 2 : The reader may verify that L(G 1 ) = (10)*1=1(01)*=L(G 2 ) ∑ → 1B ∑ → 1 A → 1B B → 0A A → 1 ∑ → B1 ∑ → 1 A → B1 B → A0 A → 1

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

12 Operator-Precedence Relations from Associativity and Precedence (1/2) 1. If operator θ 1 has higher precedence than operator θ 2, make θ 1 ·> θ 2 and θ 2 + 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 θ 1 and θ 2 are operators of equal precedence (they may in fact be the same operator), then make θ 1 ·> θ 2 and θ 2 ·> θ 1 if the operators are left- associative, or make θ 1 +, + ·> -, - ·> - 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 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 ·> ) ) ·> ) ·

*/↑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, * 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 g

16 Fig Graph representing precedence functions.

17 id()$ e3 ·>·> ·>·> ( <·<· <·<· =e4 )e3 ·>·> ·>·> $ <·<· <·<· e2e1 · Fig Operator-precedence matrix with error entries