Fundamentele Informatica 1 (I&E) najaar 2011 http://www.liacs.nl/home/rvvliet/fi1ie/ Rudy van Vliet rvvliet(at)liacs.nl college 9, 21 november 2011 4. Context-Free Languages 4.5. Simplified Forms and Normal Forms 5. Pushdown Automata 5.1. Definitions and Examples 1
4.5. Simplified Forms and Normal Forms 2
Definition 4.13. Regular Grammars. A context-free grammar G = ( V, Σ , S, P ) is regular if every production is of the form A → σB or A → Λ , where A, B ∈ V and σ ∈ Σ. 3
Definition 4.29. Chomsky Normal Form A context-free grammar is said to be in Chomsky normal form if every production is of one of these two types: A → BC (where B and C are variables) A → σ (were σ is a terminal symbol) 4
Exercise. For each grammar G below, try to find a grammar G ′ in Chomsky normal form which generates the same language. a. S → TB T → aTb | ab B → bB | b b. S → aSb | B B → bB | b c. S → aSb | aBb B → bB | Λ B → bB | Λ d. S → aSb | B 5
Theorem 4.30 For every context-free grammar G , there is another CFG G 1 in Chomsky normal form such that L ( G 1 ) = L ( G ) − { Λ } . The proof of this result does not have to be known for the exam. What if Λ / ∈ L ( G ) ? 6
reg. languages reg. grammar FA reg. expression cf. languages cf. grammar PDA 7
5. Pushdown Automata 8
just like FA, PDA accepts strings / language just like FA, PDA has states just like FA, PDA reads input one letter at a time unlike FA, PDA has auxiliary memory: a stack unlike FA, by default PDA is nondeterministic unlike FA, by default Λ-transitions are allowed in PDA 9
Stack in PDA contains symbols from certain alphabet. Usual stack operations: pop, top, push Extra possiblity: replace top element X by string α α = Λ pop α = X top α = Y X push push ∗ α = βX α = . . . Top element X is required to do a move! 10
Example 5.3. A PDA Accepting the Language AnBn 11
Definition 5.1. A Pushdown Automaton A pushdown automaton (PDA) is a 7-tuple M = ( Q, Σ , Γ , q 0 , Z 0 , A, δ ), where Q is a finite set of states. Σ and Γ are finite sets, the input and stack alphabet. q 0 , the initial state, is an element of Q . Z 0 , the initial stack symbol, is an element of Γ. A , the set of accepting states, is a subset of Q . δ , the transition function, is a function from . . . to . . . 12
Definition 5.1. A Pushdown Automaton A pushdown automaton (PDA) is a 7-tuple M = ( Q, Σ , Γ , q 0 , Z 0 , A, δ ), where Q is a finite set of states. Σ and Γ are finite sets, the input and stack alphabet. q 0 , the initial state, is an element of Q . Z 0 , the initial stack symbol, is an element of Γ. A , the set of accepting states, is a subset of Q . δ , the transition function, is a function from Q × (Σ ∪{ Λ } ) × Γ to the set of finite subsets of Q × Γ ∗ . In principle, Z 0 may be removed from the stack, but often it isn’t. 13
Example 5.3. A PDA Accepting the Language AnBn 14
Notation: configuration ( q, x, α ) ( p, x, α ) ⊢ M ( q, y, β ) ( p, x, α ) ⊢ n ( p, x, α ) ⊢ ∗ M ( q, y, β ) M ( q, y, β ) ( p, x, α ) ⊢ n ( q, y, β ) ( p, x, α ) ⊢ ∗ ( q, y, β ) ( p, x, α ) ⊢ ( q, y, β ) 15
Definition 5.2. Acceptance by a PDA If M = ( Q, Σ , Γ , q 0 , Z 0 , A, δ ) and x ∈ Σ ∗ , the string x is accepted by M if ( q 0 , x, Z 0 ) ⊢ ∗ M ( q, Λ , α ) for some α ∈ Γ ∗ and some q ∈ A . A language L ⊆ Σ ∗ is said to be accepted by M , if L is precisely the set of strings accepted by M ; in this case, we write L = L ( M ). Sometimes a string accepted by M , or a language accepted by M , is said to be accepted by final state . 16
Example 5.3. PDAs Accepting the Languages AnBn and SimplePal 17
Example 5.7. A Pushdown Automaton Accepting Pal 18
Exercise 5.4. Consider the PDA in Table 5.8, and for each of the following languages over { a, b } ∗ , modify it to obtain a PDA accepting the language. a. The language of even-length palindromes. b. The language of odd-length palindromes. 19
Exercise 5.5. Give transition diagrams for PDAs accepting each of the follow- ing languages. a. The language of all odd-length strings over { a, b } with middle symbol a . b. { a n x | n ≥ 0 , x ∈ { a, b } ∗ and | x | ≤ n } . c. { a i b j c k | i, j, k ≥ 0 and j = i or j = k } . 20
Exercise 5.6. In both cases below (on the whiteboard), a transition diagram is given for a PDA with intial state q 0 and accepting state q 2 . Describe in each case the language that is accepted. (first case treated) 21
Exercise 5.10. Show that every regular language can be accepted by a (deterministic) PDA M with only two states in which there are no Λ-transitions and no symbols are ever removed from the stack. 22
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