Context Free and Non Context Free Languages

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1 Context Free and Non Context Free Languages

2 Part I: Pumping Theorem for Context Free Languages

3 Context Free Languages and Push Down Automata Theorems For Every Context Free Grammar there Exists an Equivalent Push Down Automata. For Every Push Down Automata There Exist an Equivalent Context Free Grammar. BIG QUESTION Given a Language L, Determine Whether or Not L is Context Free. Examples L = {a n b n n 0} -- Context Free L = {a n b n c n n 0} -- Not Context Free

4 Observations on Context Free Languages Every Regular Language is Context Free -- 2 Proofs Context Free Languages are Countable -- 2 Proofs

5 Pumping Theorem for Regular Languages Pumping Theorem: Let M = {Q, Σ, δ, q 0, F} be a DFSA and let x L(M ) with x Q. Then there are strings u,v, w such that x = uvw u v Q v 1 u v k w L(M ) k = 0,1, 2,

6 Pumping Theorem for Context Free Languages Pumping Theorem: Let L be a Context Free Grammar. There is an integer k 1 such that for all strings w L with w k, there exist strings u, v, x, y, z such that w = uvxyz ª v ε and y ε vxy k uv q xy q z L for all q 0

7 Example Proposition: L = {a n b n c n n 0} is Not Context Free Proof: Suppose L is Context Free. Let w = a n b n c n for n k. By the Pumping Theorem w = uvxyz ª v ε and y ε vxy k uv q xy q z L for all q 0 Case 1: v or y contain a single character -- Impossible. Pump with q = 2. Not all symbols appear equally often. Case 2: v or y contain two different characters -- Impossible. Pump with q = 2. A symbol will appear out of order.

8 Trees Definitions Height -- Longest Path in a Tree Branching Factor = Maximum Number of Children for any Node Yield = Number of Leaf Nodes Theorem: Yield (Branching Factor) Height Proof: By Induction on Height.

9 Pumping Theorem for Context Free Languages Pumping Theorem: Let L be a Context Free Grammar. There is an integer k 1 such that for all strings w L with w k, there exist strings u, v, x, y, z such that w = uvxyz ª v ε or y ε vxy k uv q xy q z L for all q 0

10 Proof of Pumping Theorem for Context Free Languages Let G = { N, Σ,R,S} be a context free grammar, and let w L(G). If no non-terminal appears twice along a path in the parse tree for w, then NonTerminal Symbols w = Yield (Branching Factor) w > (Branching Factor) NonTerminal Symbols a non-terminal X must appear at least twice on the same path in the parse tree for w. Thus S u X z uv X yz uvxyz = w X vx y and X x Consequences S u X z ux z, so ux z L(G) S u X z uv X yz uv p X y p z uv p x y p z, so uv p x y p z L(G) v ε or y ε (Take the smallest parse tree for w.) NonTerminal Symbols vxy (Branching Factor) -- X vx y v x y and no non-terminal appears twice.

11 Closure Properties of Regular Languages 1. The Union, Concatenation, and Star of Regular Languages is Regular Build the Corresponding Non-Deterministic Finite State Automata 2. Every Finite Language is Regular L = s 1 s n 3. The Complement of a Regular Language is Regular Build the Complementary Deterministic Finite State Automata Exchange Accepting and Non-Accepting States 4. The Intersection of Two Regular Languages is Regular L 1 L 2 = ( c c L 1 L2 ) c 5. The Difference of Two Regular Languages is Regular L 1 L 2 = L 1 L 2 c

12 Closure Properties of Context Free Languages 1. The Union of Two Context Free Languages is Context Free Build Union of Grammars S S 1 and S S 2 2. The Concatenation of Two Context Free Languages is Context Free Concatenate Two Grammars S S 1 S 2 3. The Kleene Star of Two Context Free Languages is Context Free Star Two Grammars S ε and S S S 1 4. The Reverse of Two Context Free Languages is Context Free Use Chomsky Normal Form Replace X BC by X C B

13 Non Closure Properties of Context Free Languages 1. The Intersection of Two Context Free Languages Need NOT be Context Free L 1 = {a n b n c m } and L 2 = {a n b m c m } -- Context Free L 1 L 2 = {a p b p c p } -- Not Context Free 2. The Complement of Two Context Free Languages Need NOT be Context Free L 1 L 2 = ( c c L 1 L2 ) c -- Contradicts 1 3. The Difference of Two Context Free Languages Need NOT be Context Free L c = Σ L -- Contradicts 2

14 Closure Properties of Context Free and Regular Languages 1. The Intersection of a Context Free Language and a Regular Language is Context Free L(M ) = L 1 and L(N) = L 2, then L(M N ) = L 1 L 2 where F = F 1 F 2 Only One Stack 2. The Difference Between a Context Free Language and a Regular Language is Context Free c L 1 L 2 = L 1 L 2 Complement of a Regular Language is Regular

15 Examples 1. Removing a Finite Number of Strings F from a Context Free Language L Generates a Context Free Language L F is Context Free because F is Regular 2. L = {w {a,b,c} Number of a's = Number of b 's = Number c' s} is NOT Context Free L {a b c } = {an b n c n } Context Free Regular Not Context Free

16 Part II: Deterministic Context Free Languages

17 Deterministic Push Down Automata Deterministic PDA M = {Q, Σ, Γ,δ, q 0, F} -- Σ = Input Symbols Γ = Stack Symbols -- Q = States -- q 0 = Initial State -- F = Final (Accepting) States Q -- δ :Q Σ {ε} Γ = Q Γ = Deterministic Transition Functions pop push -- Transition to Next State and Next Stack is Unique -- No Transitions Out of an Accepting State Must Consume either an Input Symbol or Symbols from the Stack Must Accept when Can Accept

18 Deterministic Context Free Languages Deterministic Context Free Language Language Accepted by a Deterministic Push Down Automata End of String Symbol $ Required Example L = a {a n b n n 0} Need End of String Symbol

19 Closure Properties of Regular Languages 1. Union 2. Concatenation 3. Star 4. Complement 5. Intersection 6. Difference

20 Closure Properties of Deterministic Context Free Languages Closure Complement Proof: See Appendix D.2 -- Tedious and Unenlightening Non Closure Union Intersection

21 CounterExample: Intersection L 1 = {a i b j c k i = j} -- Deterministic Context Free L 2 = {a i b j c k j = k} -- Deterministic Context Free L 1 L 2 = {a n b n c n } -- Not Context Free

22 CounterExample: Union L 1 = {a i b j c k i j} -- Deterministic Context Free L 2 = {a i b j c k j k} -- Deterministic Context Free L = L 1 L 2 = {a i b j c k i j or j k} -- Union L c = {a i b j c k i = j = k} {(a,b,c) out of order} -- Complement L = L c a b c = {a n b n c n n 0} -- Intersection with Regular Expression -- But Not Context Free L Deterministic Context Free L c Deterministic Context Free L Context Free Contradiction

23 CounterExample: Union and Ambiguity Inherently Ambiguous Language -- NO Unambiguous Grammar. {a i b j c k i = j or j = k} = {a i b i c k } {a i b k c k } Ambiguous Grammar from Union of Two Unambiguous Grammars S S 1 S S 2 S 1 S 1 c S 2 as 2 S 1 A S 2 B A a Ab B b Bc A ε B ε {a i b i c k } {ai b j c j } {a i b i c i }

24 Part III: Hierarchy of Languages

25 Hierarchy of Languages Theorem: The following Languages are Proper Subsets of Each Other: Regular Deterministic Context Free Inherently Unambiguous Context Free Context Free Proof: See the Following Theorems.

26 Theorems 1. There Exist Context Free Languages that are Not Deterministic Context Free. 1a. There Exist Non Deterministic Push Down Automata for which there is no Equivalent Deterministic Push Down Automata. 2. Every Determinist Context Free Language is a Context Free Language. 2a. Every Deterministic Push Down Automata can be Simulated by a Non Deterministic Push Down Automata.

27 Theorem: Every Determinist Context Free Language L is a Context Free Language. Proof: Let M be a Deterministic PDA that recognize L$. Build a PDA M = M 1 M 2 to recognize L: M 1 = M without $ transitions -- Do NOT mark any state of M 1 as Accepting M 2 M with only the ε -transitions (Reads no input symbols) For each $ transition in M, insert an ε -transition from M 1 to M 2 For each accepting state A in M, make A an accepting state of M 2 M recognizes L$ M recognize L.

28 Theorem: There Exist Context Free Languages that are Not Deterministic Context Free Languages. Proof: Consider L = {a i b j c k i j or j k} -- Context Free L c = {a i b j c k i = j = k} {(a,b,c) out of order} -- Complement L = L c a b c = {a n b n c n n 0} -- Intersection of L c with Regular Language L Deterministic Context Free L c Deterministic Context Free L c Context Free L Context Free. Contradiction

29 More Theorems 1. Every Regular Language is Deterministic Context Free. 1a. Every Finite State Automaton is a Deterministic Push Down Automaton 2. There Exist Deterministic Context Free Languages that are NOT Regular. 2a. There Exist Deterministic Push Down Automata for which there is no Equivalent Finite State Automaton.

30 More Theorems 1. Every Deterministic Context Free Grammar has an Unambiguous Grammar. 2. There Exist Inherently Unambiguous Context Free Languages that are NOT Deterministic.

31 Theorem: Every Deterministic Context Free Grammar L has an Unambiguous Grammar. Proof: Recall: For Every Push Down Automata M There Exist an Equivalent Context Free Grammar G. Therefore M Recognizes L$ G Generates L$ But M Deterministic G is Unambiguous (Tedious -- See Appendix D.2) Hence Replacing $ by ε in G G generates L

32 Hierarchy of Languages Theorem: The following Languages are Proper Subsets of Each Other: Regular Deterministic Context Free Inherently Unambiguous Context Free Context Free Proof: See the Previous Theorems.

33 Hierarchy of Languages -- Examples Deterministic Context Free, but NOT Regular L = {a n b n n 0} Inherently Unambiguous but NOT Deterministic Context Free L = {a n b n c m d m,n 0} {a n b m c m e m, n 0} Context Free but NOT Inherently Unambiguous L = {a i b j c k (i, j,k 0) and (i = j or j = k)} = {a i b i c k } {a i b k c k } Not Context Free L = {a n b n c n n 0}

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