Recursively Enumerable and Recursive. Languages

Transcription

1 Recursively Enumerble nd Recursive nguges 1

2 Recll Definition (clss 19.pdf) Definition 10.4, inz, 6 th, pge 279 et S be set of strings. An enumertion procedure for Turing Mchine tht genertes ll strings of S one by one S is nd ech string is generted in finite time. 2

3 Definition: A lnguge is recursively enumerble if some Turing mchine ccepts it 3

4 et nd M be recursively enumerble lnguge the Turing Mchine tht ccepts it w For string : if w then M hlts in finl stte w if then hlts in non-finl stte M or loops forever 4

5 Definition (11.2, pge 287): A lnguge is recursive if some Turing mchine ccepts it nd hlts on ny input string In other words: A recursive lnguge hs membership lgorithm. 5

6 et nd M be recursive lnguge the Turing Mchine tht ccepts it is recursive lnguge if there is Turing Mchine M such tht w For ny string : if w then M hlts in finl stte w if then hlts in non-finl stte M 6

7 We will prove: 1. There is specific lnguge which is not recursively enumerble (not ccepted by ny Turing Mchine) 2. There is specific lnguge which is recursively enumerble but not recursive 7

8 Non Recursively Enumerble Recursively Enumerble Recursive 8

9 We will first prove: If lnguge is recursive then there is n enumertion procedure for it A lnguge is recursively enumerble if nd only if there is n enumertion procedure for it 9

10 Theorem: if lnguge is recursive then there is n enumertion procedure for it 10

11 Proof: Enumertion Mchine M ~ M Enumertes ll strings of input lphbet Accepts 11

12 If the lphbet is M ~ then {, b} cn enumerte strings s follows: b b b bb b... 12

13 Enumertion procedure Repet: M ~ genertes string w M checks if w YES: print to output w NO: ignore w End of Proof 13

14 Exmple: { b, b, bb,,...} M ~ (M ) Enumertion Output b b b b b b b bb b... bb... bb... 14

15 Theorem: if lnguge is recursively enumerble then there is n enumertion procedure for it 15

16 Proof: Enumertion Mchine M ~ M Enumertes ll strings of input lphbet Accepts 16

17 If the lphbet is M ~ then {, b} cn enumerte strings s follows: b b b bb b 17

18 NAIVE APPROACH Enumertion procedure Repet: M ~ genertes string w M checks if w YES: print to output w NO: ignore w Problem: If w mchine M my loop forever 18

19 BETTER APPROACH M ~ Genertes first string w 1 M executes first step on w 1 M ~ Genertes second string w2 M executes first step on second step on w 2 w 1 19

20 M ~ Genertes third string w3 M executes first step on w3 second step on third step on w 2 w 1 And so on... 20

21 w1 w2 w3 w Step in string

22 If for ny string w i the mchine M hlts in finl stte, then it prints w i on the output End of Proof 22

23 Theorem: If for lnguge there is n enumertion procedure, then is recursively enumerble 23

24 Proof: Input Tpe w Mchine tht ccepts Enumertor for Compre 24

25 Turing mchine tht ccepts For input string w Repet: Using the enumertor, generte the next string of Compre generted string with w If sme, ccept nd exit loop End of Proof 25

26 We hve proven: A lnguge is recursively enumerble if nd only if there is n enumertion procedure for it 26

27 A nguge which is not Recursively Enumerble 27

28 We wnt to find lnguge tht is not Recursively Enumerble This lnguge is not ccepted by ny Turing Mchine 28

29 Consider lphbet {} Strings:,,,,

30 Consider Turing Mchines tht ccept lnguges over lphbet {} They re countble: M1, M 2, M3, M 4, 30

31 Exmple lnguge ccepted by M i ( Mi ) {,, } ( Mi ) { 2, 4, 6 } Alterntive representtion ( Mi )

32 ( M1 ) ( M2 ) ( M3 ) ( M4 )

33 Consider the lnguge i i { : ( Mi )} consists from the 1 s in the digonl 33

34 ( M1 ) ( M2 ) ( M3 ) ( M4 ) { 3, 4, } 34

35 Consider the lnguge i i { : ( Mi )} i i { : ( Mi )} consists from the 0 s in the digonl 35

36 ( M1 ) ( M2 ) ( M3 ) ( M4 ) { 1, 2, } 36

37 Theorem: nguge is not recursively enumerble 37

38 Proof: Assume for contrdiction tht is recursively enumerble There must exist some mchine tht ccepts M k ( M k ) 38

39 ( M1 ) ( M2 ) ( M3 ) ( M4 ) Question: M k M 1? 39

40 ( M1 ) ( M2 ) ( M3 ) ( M4 ) Answer: M k M1 1 1 ( M ( M k 1 ) ) 40

41 ( M1 ) ( M2 ) ( M3 ) ( M4 ) Question: M k M 2? 41

42 ( M1 ) ( M2 ) ( M3 ) ( M4 ) Answer: M k M2 2 2 ( M ( M k 2 ) ) 42

43 ( M1 ) ( M2 ) ( M3 ) ( M4 ) Question: M k M 3? 43

44 ( M1 ) ( M2 ) ( M3 ) ( M4 ) Answer: M k M3 3 3 ( M ( M k 3 ) ) 44

45 Similrly: M k M i for ny i Becuse either: i i ( M ( M k i ) ) or i i ( M ( M k i ) ) 45

46 Therefore, the mchine M k cnnot exist Therefore, the lnguge recursively enumerble is not End of Proof 46

47 Observtion: There is no lgorithm tht describes (otherwise some Turing Mchine) would be ccepted by 47

48 Non Recursively Enumerble Recursively Enumerble Recursive 48

49 A nguge which is Recursively Enumerble nd not Recursive 49

50 We wnt to find lnguge which Is recursively enumerble But not recursive There is Turing Mchine tht ccepts the lnguge The mchine doesn t hlt on some input 50

51 We will prove tht the lnguge i i { : ( Mi )} Is recursively enumerble but not recursive 51

52 ( M1 ) ( M2 ) ( M3 ) ( M4 ) { 3, 4, } 52

53 Theorem: The lnguge i i { : ( Mi )} is recursively enumerble 53

54 Proof: We will give Turing Mchine tht ccepts 54

55 Turing Mchine tht ccepts For ny input string w Compute, for which i i w Find Turing mchine M (using the enumertion procedure for Turing Mchines) i Simulte M i on input i If M i ccepts, then ccept w End of Proof 55

56 Observtion: Recursively enumerble i i { : ( Mi )} Not recursively enumerble i i { : ( Mi )} (Thus, lso not recursive) 56

57 Theorem: i i { i The lnguge : ( M )} is not recursive 57

58 Proof: Assume for contrdiction tht is recursive Then is recursive: Tke the Turing Mchine M tht ccepts M hlts on ny input: If If M M ccepts then reject rejects then ccept 58

59 Therefore: is recursive But we know: is not recursively enumerble thus, not recursive CONTRADICTION!!!! 59

60 Therefore, is not recursive End of Proof 60

61 Non Recursively Enumerble Recursively Enumerble Recursive 61