Pushdown Automata. Courtesy: Costas Busch RPI

Transcription

1 Pushdown Automt Courtesy: Costs Busch RPI

2 Pushdown Automt

3 Pushdown Automt

4 Pushdown Automt

5 Pushdown Automt

6 Pushdown Automt

7 Pushdown Automt Non-Determinism:NPDA PDAs re non-deterministic: non-deterministic trnsitions re llowed

8 Exmple: Pushdown Automt

9 Pushdown Automt

10 Pushing trings: Pushdown Automt Exmple:

11 Pushdown Automt

12 Pushdown Automt

13 Pushdown Automt

14 Pushdown Automt

15 Pushdown Automt

16 Pushdown Automt

17 Pushdown Automt Lnguges o PDA: Acceptnce y Finl tte

18 Pushdown Automt Lnguges o PDA: Acceptnce y Empty tck It is possile to rech the inl stte rom the empty stck position nd vice vers.

19 Pushdown Automt

20 Constructing NPDA M rom grmmr G Given ny grmmr G = V P We cn construct NPDA M = { } V ${ } With L G = L M δ

21 Constructing NPDA M rom grmmr G For ny production A w For ny terminl A w $ $ 1 2

22 Constructing NPDA M rom grmmr G An exmple grmmr: Wht is the euivlent NPDA?

23 1 2 $ $ NPDA: Constructing NPDA M rom grmmr G

24 Constructing NPDA M rom grmmr G Check whether the NPDA ccepts the string

25 Grmmr: A letmost derivtion:

26 Derivtion: Input ime $ tck $ $ 1 2

27 Derivtion: Input ime $ tck $ $ 1 2

28 Derivtion: Input ime 1 $ tck $ $ 1 2

29 Derivtion: Input ime 2 $ tck $ $ 1 2

30 Derivtion: Input ime 3 $ tck $ $ 1 2

31 Derivtion: Input ime 4 $ tck $ $ 1 2

32 Derivtion: Input ime 5 $ tck $ $ 1 2

33 Derivtion: Input ime 6 $ tck $ $ 1 2

34 Derivtion: Input ime 7 $ tck $ $ 1 2

35 Derivtion: Input ime 8 $ tck $ $ 1 2

36 Derivtion: Input ime 9 $ tck ccept $ $ 1 2

37 NPDA to CFG For ny NPDA M we cn construct CFG G with LM=LG Modiy i necessry the NPDA ccepting y reching inl stte so tht: 1 It hs single inl stte nd empties the stck when it ccepts string 2 Hs trnsitions in specil orm

38 NPDA NPDA to CFG 1 Modiy the NPDA so tht it empties the stck nd hs uniue inl stte Empty the stck x x Γ {$} $ Old inl sttes

39 NPDA to CFG 2 modiy the NPDA so tht trnsitions hve the ollowing orms: i σ B j OR i σ B CD j B C D :stck symols

40 NPDA to CFG Convert: i σ y j i σ τ yτ j τ Γ {$}

41 Convert: i NPDA to CFG σ A B j symols i σ A XB σ X j X Γ {$}

42 y 2 NPDA to CFG symols Convert: i σ A By j i Convert recursively σ A y σ X BX X Γ {$} j

43 NPDA to CFG L M = { w: n = n} $ :initil stck symol $ $ $ 1$ $

44 NPDA to CFG In grmmr G : Vriles: tck symol i B j sttes erminls: Input symols o NPDA

45 NPDA to CFG For ech trnsition i B j We dd production i B j

46 For ech trnsition i B NPDA to CFG CD j We dd productions ibk jcl l D k For ll possile sttes k l in the utomton

47 NPDA to CFG tck ottom symol $ trt Vrile: o trt stte inl stte

48 NPDA to CFG $ $ $ 1$ $ Grmmr production: 1

49 $ $ $ 1 $ 1$ 1 11 $ 1 $ 1 $ $ 1 $ 1 $ Grmmr productions: NPDA to CFG

50 NPDA to CFG $ $ $ 1$ $ Grmmr production: $

51 $ 1 $ 1 $ $ 1 $ 1 $ $ $ $ $ $ $ Resulting Grmmr: le :strt vri $ NPDA to CFG

52 1 $ NPDA to CFG

53 NPDA to CFG Derivtion o string $ $ $ 1 $ $

54 NPDA to CFG Consider n NPDA where cceptnce is reched y emptying the stck.

55 NPDA to CFG Q s r rys pxr Zs Z XY p Q r pyr Zr Z Y p Zp Z p ]: ][ [ ] [ ]: [ ] [ ] [ δ δ δ ε

56 NPDA to CFG

57 NPDA to CFG 57

58 NPDA to CFG V = {[ X][ Xp][ Z][ Z p][ pxp][ px][ pz p][ pz] } 58

59 NPDA to CFG 59

60 Deterministic PDA: DPDA

61 Deterministic PDA: DPDA Allowed trnsitions: w 1 2 w c w 2 3 c w 2 3 deterministic choices

62 Deterministic PDA: DPDA Not llowed: w 1 2 w w 2 3 w 2 3 non deterministic choices

63 DPDA exmple n n L M = { : n } $ $ 1 2 3

64 he lnguge DCF n n L M = { : n } is deterministic context-ree Deinition: A lnguge is deterministic context-ree i there exists some DPDA tht ccepts it

65 Exmple o Non-DPDA NPDA L M = R { ww } $ $ 1 2

66 Exmple o Non-DPDA NPDA Not llowed in DPDAs $ $ 1 2

67 It is importnt to note tht tht there exist context-ree lnguges which re not ccepted y ny DPDA ut ccepted y NPDA