1 A Non Context-Free Language (We will prove it at the next class)

2 Regular languages Context-free languages Non Context-free languages

3 Deterministic PDAs DPDAs

4 DPDAs Allowed:

5 Not allowed:

6 Allowed: Something must be matched from the stack

7 Not allowed:

8 DPDA example

9 The language is deterministic context-free

10 Definition: A language is deterministic context-free if some DPDA accepts it

11 Example of Non-DPDA (NPDA)

12 Not allowed in DPDAs

13 NPDAs Have More Power than DPDAs

14 We will show: there is a context-free language which is not deterministic context-free (accepted by a NPDA) (not accepted by a DPDA)

15 The language is:

16 The language is context-free Context-free grammar for : there is an NPDA that accepts

17 is not deterministic context-free Theorem: The language (there is no DPDA that accepts )

18 Proof: Assume for contradiction that is deterministic context free Therefore: there is a DPDA that accepts

19 DPDA with accepts

20 The language is not context-free (we will prove it at the next class) Fact 1:

21 The language is not context-free (we will prove later today that the union of two context-free languages is context-free) Fact 2:

22 We will construct a NPDA that accepts: Contradiction!

23 We modify Modified Replace with

24 The NPDA that accepts Modified Original

25 Since is accepted by a NPDA it is context-free Contradiction! (since is not context-free)

26 Therefore: There is no DPDA that accepts End of Proof Not deterministic context free

27 Positive Properties of Context-Free languages

28 Context-free languages are closed under: Union is context free is context-free Union

29 Example Union LanguageGrammar

30 In general: The grammar of the union has new start variable and additional production For context-free languages with context-free grammars and start variables

31 Context-free languages are closed under: Concatenation is context free is context-free Concatenation

32 Example Concatenation LanguageGrammar

33 In general: The grammar of the concatenation has new start variable and additional production For context-free languages with context-free grammars and start variables

34 Context-free languages are closed under: Star-operation is context freeis context-free Star Operation

35 Example Language Grammar Star Operation

36 In general: The grammar of the star operation has new start variable and additional production For context-free language with context-free grammar and start variable

37 Negative Properties of Context-Free Languages

38 Context-free languages are not closed under: intersection is context free not necessarily context-free Intersection

39 Example Context-free: NOT context-free Intersection

40 Context-free languages are not closed under: complement is context freenot necessarily context-free Complement

41 NOT context-free Example Context-free: Complement

42 Intersection of Context-free languages and Regular Languages

43 The intersection of a context-free language and a regular language is a context-free language context free regular context-free

44 for NPDA DFA Construct a new NPDA machine that accepts Machine context-free regular simulates in parallel and

45 transition NPDADFA transition NPDA

46 initial state NPDADFA Initial state NPDA

47 final state final states NPDADFA final states NPDA

48 simulates in parallel and accepts stringif and only if accepts string and accepts string

49 Therefore: (since is NPDA) is context-free