This lecture covers Chapter 5 of HMU: Context-free Grammars

Transcription

1 This lecture covers Chapter 5 of HMU: Context-free rammars (Context-free) rammars (Leftmost and Rightmost) Derivations Parse Trees An quivalence between Derivations and Parse Trees Ambiguity in rammars Additional Reading: Chapter 5 of HMU.

2 rammars Introduction to rammars We have so far seen machine-like means (e.g., DFAs) and declarative means (e.g., regular expressions) of defining languages rammars are a generative means of defining languages. rammars can be used to create a strictly larger class of languages. They are especially useful in compiler and parser design; they can be used to check if: parantheses are balanced in a program, else occurrences have a matching if, etc. 2 / 14

3 rammars rammars: Formal Definition A context-free grammar (CF) = (V, T, P, ), where V is a finite set whose elements are called variables or non-terminal symbols. Notation: upper case letters, e.g., A, B,.... T is a finite set whose elements are called terminal symbols; T is precisely the alphabet of the language generated by the grammar. Notation: lower case letters, e.g., s 1, s 2,.... P V (V T ) is a finite set of production rules. ach production rule (A, α) is also written as A α. Terminology: A, α are called the head and body of the production rule, resp. T is the unique variable/non-terminal that generates the language. Notation trings consisting of non-terminals and/or terminals will be denoted by greek symbols, e.g., α, β,.... trings of terminals will be denoted by lower case letters, e.g., w, u, v 3 / 14

4 Derivations How do rammars enerate Languages? A string w T is in the language L() generated by = (V, T, P, ) iff we can derive w from, i.e., start from and use production rule(s) repeatedly to replace heads of the rules by their bodies until a string in T is obtained. xample Let = ({}, {0, 1}, P, ) be a CF with P given by { } (, ɛ), (, 0), (, 1) (1) (, 00), (, 11) (2) ɛ (3) ɛ (tart) / 14

5 Derivations Derivation: Formal Definition Definition iven = (V, T, P, ) and α, β (V T ), a derivation of β from α is a finite sequence of strings γ 1 γ 2 γ k for some k N where 1. γ 1 = α and γ k = β; 2. γ 1,..., γ k (V T ) 3. For each i = 1,..., k 1, either γ i = γ i+1 or γ i+1 is obtained from γ i by replacing the head of a production rule of P by its body. The following phrases are used interchangeably. xample β is derived from α there exists a derivation of β from α α For the grammar = ({}, {0, 1}, P, )with P given by ɛ , the following is a derivation of from β / 14

6 Derivations entential Forms and Language enerated by a rammar: Definitions Definition iven = (V, T, P, ), any string in (V T ) derived from is a sentential form. The set of all sentential forms of (denoted by F()) is defined inductively: Basis: F() Induction: if αaγ F() for some α, γ (V T ) and A V, and A β is a production rule, then αβγ F(). Only those strings that are generated by the above induction are sentential forms. Definition iven CF = (V, T, P, ), the language L() generated by are the sentential forms that are in T, i.e., L() = F() T. xample For the CF = ({}, {0, 1}, P, )with P given by ɛ , (1), ɛ, 0, 1 00, 00, 000, 010, 11, 11, 101, 111,... are all sentential forms. (2), ɛ, 0, 1 00, 00, 000, 010, 11, 11, 101, 111,... are in L(). 6 / 14

7 Derivations Other entential Forms At each step of a derivation, one can replace any variable by a suitable production. If at each non-trivial step of the derivation the leftmost (or rightmost) variable is replaced by a production rule, then the derivation is said to be a leftmost (or rightmost) derivation, respectively. We let α β (or α β) to denote the RM existence of a leftmost (or rightmost) derivation of β from α, respectively. entential forms derived via leftmost (or rightmost) derivations are known as leftmost (or rightmost) sentential forms, respectively. Balanced Parantheses xample Consider the CF = ({}, {(, )}, P, ) with P given by () (). [Derivation] [Leftmost Derivation] [Rightmost Derivation () ( ) () ( )() (()) ( )() (())() (())() (())() In the above, indicates the variable that is replaced in the following step 7 / 14

8 Parse Trees Parse Trees Parse trees are a graphical method of representing derivations. They are used in compilers to represent the source program. Definition iven a CF = (V, T, P, ), a parse tree for is any directed labelled tree that meets the following three conditions: every interior node is labelled by a non-terminal (i.e., variable); every leaf node is labelled by a non-terminal, or a terminal or ɛ; however if it is labelled by ɛ, it is the sole child of its parent. if an interior node is labelled by A V, and it s children are labelled s 1,..., s k V T {ɛ}, then A s 1 s k is a production rule in P. The yield of a parse tree is the string formed from the labels of the tree leaves read from left to right. Note: The yield is not necessarily a string of terminals. =({}; {(; )}; P;) P :! () ( ) ( ) ( ) yield = (())() 8 / 14

9 An quivalence between Parse Trees and Derivations Derivations and Parse Trees Parse trees, derivations, leftmost derivations, and rightmost derivations are equivalent means of generating the language L() of a CF. The proof for equivalence of rightmost derivations mirrors that of leftmost derivations. (o we ll not delve into rightmost derivations). Theorem Let CF = (V, T, P, ) be given. Let A V and w T. Then, A Proof Idea w A w there exists a parse tree with root A and yield w A w. RM We ll show the following implications. xistence of a parse tree with root A and yield w (b) (a) A ) By Definition w A ) w 9 / 14

10 An quivalence between Parse Trees and Derivations Part (a) of Proof of Theorem 5.5.1: A w Parse Tree We prove the following generalization of Part (a) by induction on the length of the derivation. Lemma Let CF = (V, T, P, ) be given. Let A V and α F() with α A. Then, A α there exists a parse tree with root A and yield α Proof of Lemma (Induction on the length of derivation) ince α A the minimum length of the derivation is at least 1. Basis: Let A α be a one-step derivation. ince α A, this derivation has to be the production rule A α. Hence, the parse tree is trivially the one on the right. Basis: A s 1 s 2 s = s 1 s (A; ) (A! ) 2 P 10 / 14

11 An quivalence between Parse Trees and Derivations Part (a) of Proof of Theorem 5.5.1: A w Parse Tree Proof of Lemma (Induction on the length of derivation) Induction: uppose that the claim is true for all derivations of length k 1 or lesser for some k 2. uppose a derivation of α from A in k steps exists. A = γ 1 γ 2 γ 3 γ k 1 γ k = α We may assume γ k 1 A. o by the induction hypothesis, there exists a parse tree with root A and yield γ k 1. [If γ k 1 = A, the derivation contains one step, and the basis case applies.] We may assume that γ k 1 γ k or else the derivation of γ k 1 from A, which has a corresponding parse tree is also a parse tree with yield α and root label A. Thus, the last step involves the application of a production rule. Hence, γ k 1 = βbω and α = βλω where (a) β, ω (V T ), (b) B V, and (b) B λ is a production rule. Parse tree for A )! = A Parse tree for A ) k 1 B {z } {z }! B! {z } 11 / 14

12 An quivalence between Parse Trees and Derivations Part (b) of Proof of Theorem 5.5.1: Parse Tree A w Proof of Theorem (Induction on the depth of the tree) ince A w, the parse tree has a depth of at least one. Basis: Let the parse tree with root A and yield α be of depth one. Then, by definition A α is a production rule, since the root is an internal node. Hence, A α. Induction: Let the claim be true for all parse trees of up to depth l 1. Consider the root and its (say k) children. This corresponds to a production rule A X 1 X k. If X i is a leaf, then the yield of the sub-tree rooted at X i is w i = X i itself. Then trivially X i w i. If X i is not a leaf, let w i be the yield of the parse (sub-)tree rooted at X i of depth l 1 or less. Then, by induction hypothesis, X i w i. Then, the following is a leftmost derivation for α from A A X 1X 2 X k w 1X 2 X k w 1w 2X 3 X k Basis: Induction: A s 1 s 2 s = s 1 s (A; ) (A! ) 2 P A X 1 X 2 X k Depth 1 w 1 w k 12 / 14

13 Ambiguous rammars Ambiguity in CFs Definition A given CF is ambiguous if a string w L() is the yield of two different parse trees. quivalently, a CF is ambiguous if a string w L() has two different leftmost (or rightmost) derivations. Ambiguity is a property of a grammar, and not the language it generates. An xample CF = ({}, {0, 1,..., 9, +, }, P, ) with P : Consider the parse trees for ince there are two distinct parse trees, a compiler will not know to reduce this to 13 or to This ambiguity is addressed by precedence rules for operators. 13 / 14

14 Ambiguous rammars Ambiguity in CFs ome languages are generated by unambiguous as well as ambiguous grammars. Balanced Parantheses xample CF 1 = ({}, {(, )}, P, ) with P : () () CF 2 = ({B, R}, {(, )}, Q, B) with Q : B (RB ɛ and R ) (RR 1 is ambiguous for there are two leftmost derivations for ()()(). () () ()() ()()() () ()() ()()() 2 is not ambiguous, since there is precisely only one rule at any stage of derivation. B (RB ()B ()(RB ()()B ()()()B ()()()ɛ ome languages are intrinsically ambiguous, e.g., {0 i 1 j 2 k : i = j or j = k}. All grammars for such languages are ambiguous. In general, there is no way to tell if a grammar is ambiguous. 14 / 14