There are two main techniques for showing that problems are undecidable: diagonalization and reduction

Transcription

1 Reducibility 1

2 There are two main techniques for showing that problems are undecidable: diagonalization and reduction 2

3 We say that a problem A is reduced to a problem B if the decidability of A follows from the decidability of B. Linz, 6 th, page 315 3

4 Problem A is reduced to problem B If we can solve problem we can solve problem A B then A B 4

5 A is reducible to B if we can use B as a subroutine to solve A. Use a halting T for B in the construction of a halting T which decides problem A. 5

6 Problem A is reduced to problem B If B is decidable then A is decidable If is undecidable then B is undecidable A 6

7 Example: the halting problem is reduced to the state-entry problem So, if the halting problem is undecidable, then the state-entry problem is undecidable. 7

8 The state-entry problem Inputs: Turing achine State q String w Question: Does on input? w enter state q 8

9 Theorem: The state-entry problem is undecidable Proof: Reduce the halting problem to the state-entry problem. 9

10 Suppose we have an algorithm (Turing achine) that solves the state-entry problem We will construct an algorithm that solves the halting problem 10

11 Assume we have the state-entry algorithm: Algorithm for YES enters q w state-entry q problem NO doesn t enter q 11

12 We want to design the halting algorithm: Algorithm for YES halts on w Halting problem w NO doesn t halt on w 12

13 odify any machine to construct : Add new state q From any halting state add transitions to q halting states q Single halt state 13

14 halts if and only if halts on state q 14

15 Algorithm for halting problem: Inputs: machine and string w 1. Construct machine with state q 2. Run algorithm for state-entry problem,, with inputs: q w 15

16 Halting problem algorithm Generate q w State-entry algorithm YES NO YES NO w 16

17 We reduced the halting problem to the state-entry problem Since the halting problem is undecidable, it must be that the state-entry problem is also undecidable END OF PROOF 17

18 Another example: the halting problem is reduced to the blank-tape halting problem 18

19 The blank-tape halting problem Input: Turing achine Question: Does halt when started with a blank tape? 19

20 Theorem: The blank-tape halting problem is undecidable Proof: Reduce the halting problem to the blank-tape halting problem 20

21 Suppose we have an algorithm for the blank-tape halting problem We will construct an algorithm for the halting problem 21

22 Assume we have the blank-tape halting algorithm/machine. It looks like this: Algorithm for YES halts on blank tape blank-tape halting problem NO doesn t halt on blank tape 22

23 w We want to design a machine that decides the halting problem. The machine looks like this: YES Algorithm for halting problem NO halts on doesn t halt on w w 23

24 Construct a new machine w On blank tape writes w Then continues execution like w step 1 if blank tape then write w step2 execute with input w 24

25 halts on input string w if and only if w halts when started with blank tape 25

26 Algorithm for halting problem: Inputs: machine and string w 1. Construct w 2. Run algorithm for blank-tape halting problem, with input w 26

27 Halting problem algorithm w Generate w w blank-tape halting algorithm YES NO YES NO 27

28 We reduced the halting problem to the blank-tape halting problem Since the halting problem is undecidable, the blank-tape halting problem is also undecidable END OF PROOF 28

29 Summary of Undecidable Problems Halting Problem: Does machine halt on input? w embership problem: Does machine accept string w? In other words: Is a string w member of a recursively enumerable language L? 29

30 Blank-tape halting problem: Does machine on blank tape? halt when starting State-entry Problem: Does machine enter state q w on input? 30

31 Uncomputable Functions 31

32 Uncomputable Functions Domain f Values region A function is uncomputable if it cannot be computed for all of its domain 32

33 An uncomputable function: maximum number of moves until f (n) any Turing machine with n states halts when started with the blank tape 33

34 Theorem: Function f (n) is uncomputable Proof: Assume for contradiction that f (n) is computable Then the blank-tape halting problem is decidable 34

35 Algorithm for blank-tape halting problem: Input: machine 1. Count states of : m 2. Compute f (m) f (m) 3. Simulate for steps starting with empty tape If halts then return YES otherwise return NO 35

36 Therefore, the blank-tape halting problem is decidable However, the blank-tape halting problem is undecidable Contradiction!!! 36

37 Therefore, function f (n) in uncomputable END OF PROOF 37

38 Rice s Theorem 38

39 Definition: Non-trivial properties of recursively enumerable languages: any property possessed by some (not all) recursively enumerable languages 39

40 Some non-trivial properties of recursively enumerable languages: L is empty L is finite L contains two different strings of the same length 40

41 Rice s Theorem: Any non-trivial property of a recursively enumerable language is undecidable 41

42 Rice s Theorem In exactly the same manner, we can substitute other questions such as Does L() contain any string of length five? or Is L() regular? without affecting the argument essentially. These questions, as well as similar questions, are all undecidable. A general result fromalizng this is known as Rice s theorem. 42

43 Rice s Theorem This theorem states that any nontrivial property of a recursively enumerable language is undecidable. The adject nontrivial refers to a property possed by some but not all recursively enumerable languages. A precide statement and a proof of Rice s theory can be found in Hopcroft and Ullman (1979). Linz, 6 th, pages

44 A property of the RE languages is simply a set of RE languages. Thus, the property of being context-free is formally the set of all CFL s. The property of being empty is the set { } consisting of only the empty language. Theorem 9.11: (Rice s Theorem) Every nontrivial property of the RE languages is undecidable. Hopcroft, otwani, Ullman, 3 rd, pages

45 We will prove some non-trivial properties without using Rice s theorem 45

46 Theorem: For any recursively enumerable language L it is undecidable to determine whether L is empty Proof: We will reduce the membership problem to this problem 46

47 Let be the machine that accepts L L( ) L Assume we have the empty language algorithm: Algorithm for YES L( ) empty empty language problem NO L( ) not empty 47

48 We will design the membership algorithm: Algorithm for YES accepts w membership w problem NO rejects w 48

49 First construct machine w : When enters a final state, compare original input string with w Accept if original input is the same with w 49

50 w L if and only if L( w ) is not empty L( w) { w} 50

51 Algorithm for membership problem: Inputs: machine and string w 1. Construct w 2. Determine if L( w ) is empty YES: then w L( ) NO: then w L( ) 51

52 embership algorithm w construct w w Check if L( w ) is empty YES NO NO YES END OF PROOF 52