Chapter 4. Syntax Analysis (1)

Chapter 4.Syntax Analysis (1)

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

Formal grammars (1/3) • Example : Let G1have 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}.

Formal grammars (2/3) • Example : Grammar G2is 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.

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}.

Contracting Noncon- tracting Regular The four types of formal grammars

Context-Sensitive Grammars(Type1) Unrestricted Grammars(Type0) • Definition : A context-sensitive grammarG = (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.

Context-Free Grammars (Type2) • Definition : A context-free grammarG=(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}

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

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

Fig. 4.23. Operator-precedence relations.

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.

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. ·

· Fig. 4.25. Operator-precedence relations.

Precedence Functions Example 4.29 The Precedence table of Fig. 4.25 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. 4.25 are similarly replaced by one or another precedence relation.

Fig. 4.26. Graph representing precedence functions.

Fig. 4.28. Operator-precedence matrix with error entries ·

Chapter 4. Syntax Analysis (1)