Theory of Computer Science C5. Context-free Languages: Normal Forms, - PowerPoint PPT Presentation

Theory of Computer Science C5. Context-free Languages: Normal Forms, Closure, Decidability Malte Helmert University of Basel April 6, 2016

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary Agenda for this Chapter repetition of context-free grammars ε -rules normal form for context-free grammars closure properties decidability

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary Context-free Grammars and ε -Rules

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary Repetition: Context-free Grammars Definition (Context-free Grammar) A context-free grammar is a 4-tuple � Σ , V , P , S � with 1 Σ finite alphabet of terminal symbols, 2 V finite set of variables (with V ∩ Σ = ∅ ), 3 P ⊆ ( V × ( V ∪ Σ) + ) ∪ {� S , ε �} finite set of rules, 4 If S → ε ∈ P , then all other rules in V × (( V \ { S } ) ∪ Σ) + . 5 S ∈ V start variable.

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary Repetition: Context-free Grammars Definition (Context-free Grammar) A context-free grammar is a 4-tuple � Σ , V , P , S � with 1 Σ finite alphabet of terminal symbols, 2 V finite set of variables (with V ∩ Σ = ∅ ), 3 P ⊆ ( V × ( V ∪ Σ) + ) ∪ {� S , ε �} finite set of rules, 4 If S → ε ∈ P , then all other rules in V × (( V \ { S } ) ∪ Σ) + . 5 S ∈ V start variable. Rule X → ε is only allowed if X = S and S never occurs on a right-hand side.

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary Repetition: Context-free Grammars Definition (Context-free Grammar) A context-free grammar is a 4-tuple � Σ , V , P , S � with 1 Σ finite alphabet of terminal symbols, 2 V finite set of variables (with V ∩ Σ = ∅ ), 3 P ⊆ ( V × ( V ∪ Σ) + ) ∪ {� S , ε �} finite set of rules, 4 If S → ε ∈ P , then all other rules in V × (( V \ { S } ) ∪ Σ) + . 5 S ∈ V start variable. Rule X → ε is only allowed if X = S and S never occurs on a right-hand side. With regular grammars, this restriction could be lifted. How about context-free grammars?

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary ε -Rules Theorem For every grammar G with rules P ⊆ V × ( V ∪ Σ) ∗ there is a context-free grammar G ′ with L ( G ) = L ( G ′ ) .

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary ε -Rules Theorem For every grammar G with rules P ⊆ V × ( V ∪ Σ) ∗ there is a context-free grammar G ′ with L ( G ) = L ( G ′ ) . Proof. Let G = � Σ , V , P , S � be a grammar with P ⊆ V × ( V ∪ Σ) ∗ . Let V ε = { A ∈ V | A ⇒ ∗ ε } . We can find this set V ε by first collecting all variables A with rule A → ε ∈ P and then successively adding additional variables B if there is a rule B → A 1 A 2 . . . A k ∈ P and the variables A i are already in the set for all 1 ≤ i ≤ k . . . .

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary ε -Rules Theorem For every grammar G with rules P ⊆ V × ( V ∪ Σ) ∗ there is a context-free grammar G ′ with L ( G ) = L ( G ′ ) . Proof (continued). Let P ′ be the rule set that is constructed from P by adding rules that obviate the need for A → ε rules: for every existing rule B → w with B ∈ V , w ∈ ( V ∪ Σ) + , let I ε be the set of positions where w contains a variable A ∈ V ε . For every non-empty set I ′ ⊆ I ε , add a new rule B → w ′ , where w ′ is constructed from w by removing the variables at all positions in I ′ . removing all rules of the form A → ε (after the previous step). . . .

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary ε -Rules Theorem For every grammar G with rules P ⊆ V × ( V ∪ Σ) ∗ there is a context-free grammar G ′ with L ( G ) = L ( G ′ ) . Proof (continued). Then L ( G ) \ { ε } = L ( � Σ , V , P ′ , S � ) and P ′ contains no rule A → ε . If the start variable S of G is not in V ε , we are done.

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary ε -Rules Theorem For every grammar G with rules P ⊆ V × ( V ∪ Σ) ∗ there is a context-free grammar G ′ with L ( G ) = L ( G ′ ) . Proof (continued). Then L ( G ) \ { ε } = L ( � Σ , V , P ′ , S � ) and P ′ contains no rule A → ε . If the start variable S of G is not in V ε , we are done. Otherwise, let S ′ be a new variable and construct P ′′ from P ′ by 1 replacing all occurrences of S on the right-hand side of rules with S ′ , 2 adding the rule S ′ → w for every rule S → w , and 3 adding the rule S → ε . Then L ( G ) = L ( � Σ , V ∪ { S ′ } , P ′′ , S � ).

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary Questions Questions?

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary Chomsky Normal Form

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary Chomsky Normal Form: Motivation As in logical formulas (and other kinds of structured objects), normal forms for grammars are useful: they show which aspects are critical for defining grammars and which ones are just syntactic sugar they allow proofs and algorithms to be restricted to a limited set of grammars (inputs): those in normal form Hence we now consider a normal form for context-free grammars.

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary Chomsky Normal Form: Definition Definition (Chomsky Normal Form) A context-free grammar G is in Chomsky normal form (CNF) if all rules have one of the following three forms: A → BC with variables A , B , C , or A → a with variable A , terminal symbol a , or S → ε with start variable S . German: Chomsky-Normalform in short: rule set P ⊆ ( V × ( VV ∪ Σ)) ∪ {� S , ε �}

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary Chomsky Normal Form: Theorem Theorem For every context-free grammar G there is a context-free grammar G ′ in Chomsky normal form with L ( G ) = L ( G ′ ) .

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary Chomsky Normal Form: Theorem Theorem For every context-free grammar G there is a context-free grammar G ′ in Chomsky normal form with L ( G ) = L ( G ′ ) . Proof. The following algorithm converts the rule set of G into CNF: Step 1: Eliminate rules of the form A → B with variables A , B . If there are sets of variables { B 1 , . . . , B k } with rules B 1 → B 2 , B 2 → B 3 , . . . , B k − 1 → B k , B k → B 1 , then replace these variables by a new variable B . Then rename all variables to V = { A 1 , . . . , A n } in a way that A i → A j ∈ P implies that i < j . For k = n − 1 , . . . , 1: Eliminate all rules of the form A k → A k ′ with k ′ > k and add a rule A k → w for every rule A k ′ → w with w ∈ ( V ∪ Σ) + . . . .

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary Chomsky Normal Form: Theorem Theorem For every context-free grammar G there is a context-free grammar G ′ in Chomsky normal form with L ( G ) = L ( G ′ ) . Proof (continued). Step 2: Eliminate rules with terminal symbols on the right-hand side that do not have the form A → a . For every terminal symbol a ∈ Σ add a new variable A a and the rule A a → a . Replace all terminal symbols in all rules that do not have the form A → a with the corresponding newly added variables. . . .

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary Chomsky Normal Form: Theorem Theorem For every context-free grammar G there is a context-free grammar G ′ in Chomsky normal form with L ( G ) = L ( G ′ ) . Proof (continued). Step 3: Eliminate rules of the form A → B 1 B 2 . . . B k with k > 2 For every rule of the form A → B 1 B 2 . . . B k with k > 2, add new variables C 2 , . . . , C k − 1 and replace the rule with A → B 1 C 2 C 2 → B 2 C 3 . . . C k − 1 → B k − 1 B k

Context-free Grammars and ε -Rules Chomsky Normal Form Closure Properties Decidability Summary Chomsky Normal Form: Length of Derivations Observation Let G be a grammar in Chomsky normal form, and let w ∈ L ( G ) be a non-empty word generated by G . Then all derivations of w have exactly 2 | w | − 1 derivation steps. Proof. � Exercises