Context-Free Grammars

Grammar, which knows how to control even kings. (Moliere)

I am the Roman Emperor, and am above grammar. (Emperor Sigismund)

CS 4313/5353 Theory of Computation Context-Free Grammars – 1 Examples 1

⊲ Examples 1 Example 1: Balanced Parentheses Example 2 Example 3 Definition More Defs. S → SS Derivations Parse Trees S → (S) Ambiguity Chomsky Normal Form S → ǫ CYK Algorithm CYK Pseudocode

CS 4313/5353 Theory of Computation Context-Free Grammars – 2 Example 2

Examples 1 ⊲ Example 2 Example 2: Arithmetic Expressions (start E) Example 3 Definition More Defs. E → EOE Derivations Parse Trees E → (E) Ambiguity Chomsky Normal a b c Form E → | | CYK Algorithm CYK Pseudocode O → + |−|∗| /

CS 4313/5353 Theory of Computation Context-Free Grammars – 3 Example 3

Examples 1 0n1n 1n0n Example 2 Example 3: or ⊲ Example 3 Definition More Defs. S → A | B Derivations Parse Trees A → 0A1 Ambiguity Chomsky Normal 1 0 Form B → B CYK Algorithm CYK Pseudocode A → ǫ B → ǫ

CS 4313/5353 Theory of Computation Context-Free Grammars – 4 Definition of CFGs

Examples 1 Example 2 A context-free grammar is: Example 3 ⊲ Definition • V , a finite set of variables More Defs. Derivations • Σ, a finite set of terminals, Σ ∩ V = ∅ Parse Trees Ambiguity • R, a finite set of rules, each rule has the form Chomsky Normal ∗ Form A → w, where A ∈ V and w ∈ (V ∪ Σ) CYK Algorithm CYK Pseudocode • S ∈ V , a start variable

CS 4313/5353 Theory of Computation Context-Free Grammars – 5 More Definitions

Examples 1 ∗ Example 2 If u, v, and w are strings in (V ∪ Σ) , and A → w Example 3 Definition is a rule, then uAv ⇒ uwv (uAv yields uwv). ⊲ More Defs. Derivations Parse Trees ∗ Ambiguity u ⇒ v (u derives v) Chomsky Normal Form if u = v or u ⇒ v or CYK Algorithm ∗ CYK Pseudocode there exists an x such that u ⇒ x ⇒ v

The language of a grammar is {w | w ∈ Σ∗ ∧ S ⇒∗ w}.

The context-free languages are those generated by CFGs.

CS 4313/5353 Theory of Computation Context-Free Grammars – 6 Derivations

Examples 1 Example 2 Derivation of ()(()()) Example 3 Definition More Defs. S ⇒ SS ⇒ (S)S ⇒ ()S ⇒ ()(S) ⇒ ⊲ Derivations Parse Trees ()(SS) ⇒ ()((S)S) ⇒ ()(()S) ⇒ Ambiguity Chomsky Normal ()(()(S)) ⇒ ()(()()) Form CYK Algorithm CYK Pseudocode Derivation of a−b/c E ⇒ EOE ⇒ EOEOE ⇒ aOEOE ⇒ a−EOE ⇒ a−bOE ⇒ a−b/E ⇒ a−b/c

CS 4313/5353 Theory of Computation Context-Free Grammars – 7 Parse Trees

Examples 1 Example 2 Parse Tree for ()(()()) Example 3 Definition More Defs. S Derivations ⊲ Parse Trees Ambiguity Chomsky Normal SS Form CYK Algorithm CYK Pseudocode ( SS))( ε SS ( SS) ( ) ε ε

CS 4313/5353 Theory of Computation Context-Free Grammars – 8 Ambiguity

Examples 1 a b c Example 2 Parse Trees for − / Example 3 Definition More Defs. E E Derivations Parse Trees ⊲ Ambiguity E O E E O E Chomsky Normal Form CYK Algorithm c a _ CYK Pseudocode E OE / E OE a_ b b/ c

A CFG is ambiguous if there is some string w that has more than one parse tree.

CS 4313/5353 Theory of Computation Context-Free Grammars – 9 Chomsky Normal Form

Examples 1 Example 2 A CFG is in Chomsky normal form if every rule has Example 3 Definition the form A → BC or A → a, where A, B, and C More Defs. Derivations are variables and a is a terminal. Also, S → ǫ is Parse Trees Ambiguity allowed, where S is the start variable. Chomsky Normal ⊲ Form CYK Algorithm CYK Pseudocode Theorem: Any CFG can be converted to Chomsky normal form.

CS 4313/5353 Theory of Computation Context-Free Grammars – 10 CYK Algorithm

Examples 1 Example 2 The CYK (Cocke-Younger-Kasami) algorithm is a Example 3 Definition general algorithm for parsing CFGs. It is an More Defs. Derivations example of dynamic programming. Parse Trees Ambiguity Chomsky Normal Form For a grammar G in Chomsky normal form, the ⊲ CYK Algorithm 2 3 CYK Pseudocode CYK algorithm is O(l + m n ), where l = |G|, m = |V |, n = |w|.

The entry V [i,j] will contain the variable symbols that can generate the substring w[i] ...w[j].

CS 4313/5353 Theory of Computation Context-Free Grammars – 11 CYK Pseudocode

Examples 1 Example 2 procedure CYK(G, w) Example 3 Definition Initialize all entries V [i,j] to ∅ More Defs. Derivations for i ← 1 to |w| Parse Trees Ambiguity V [i,i] ← {A | A → w[i]} Chomsky Normal Form end for CYK Algorithm CYK for d ← 1 to |w|− 1 ⊲ Pseudocode for i ← 1 to |w|− d for m ← i to i + d − 1 for B in V [i,m] for C in V [m +1,i + d] V [i,i + d] ← V [i,i + d] ∪ {A | A → BC} return V

CS 4313/5353 Theory of Computation Context-Free Grammars – 12