# CSE 135: Introduction to Theory of Computation Decidability and Recognizability

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1 CSE 135: Introduction to Theory of Computation Decidability and Recognizability Sungjin Im University of California, Merced 04-28,

2 High-Level Descriptions of Computation Instead of giving a Turing Machine, we shall often describe a program as code in some programming language (or often pseudo-code ) Possibly using high level data structures and subroutines (Recall that TM and RAM are equivalent (even polynomially)) Inputs and outputs are complex objects, encoded as strings Examples of objects: Matrices, graphs, geometric shapes, images, videos,... DFAs, NFAs, Turing Machines, Algorithms, other machines...

3 High-Level Descriptions of Computation Encoding Complex Objects Everything finite can be encoded as a (finite) string of symbols from a finite alphabet (e.g. ASCII) Can in turn be encoded in binary (as modern day computers do). No special symbol: use self-terminating representations Example: encoding a graph. (1,2,3,4)((1,2)(2,3)(3,1)(1,4)) encodes the graph

4 High-Level Descriptions of Computation We have already seen several algorithms, for problems involving complex objects like DFAs, NFAs, regular expressions, and Turing Machines For example, convert a NFA to DFA; Given a NFA N and a word w, decide if w L(N);... All these inputs can be encoded as strings and all these algorithms can be implemented as Turing Machines Some of these algorithms are for decision problems, while others are for computing more general functions All these algorithms terminate on all inputs

5 High-Level Descriptions of Computation Examples: Problems regarding Computation Some more decision problems that have algorithms that always halt (sketched in the textbook) On input B, w where B is a DFA and w is a string, decide if B accepts w. Algorithm: simulate B on w and accept iff simulated B accepts On input B where B is a DFA, decide if L(B) =. Algorithm: Use a fixed point algorithm to find all reachable states. See if any final state is reachable. Code is just data: A TM can take the code of a program (DFA, NFA or TM) as part of its input and analyze or even execute this code

6 High-Level Descriptions of Computation Examples: Problems regarding Computation Some more decision problems that have algorithms that always halt (sketched in the textbook) On input B, w where B is a DFA and w is a string, decide if B accepts w. Algorithm: simulate B on w and accept iff simulated B accepts On input B where B is a DFA, decide if L(B) =. Algorithm: Use a fixed point algorithm to find all reachable states. See if any final state is reachable. Code is just data: A TM can take the code of a program (DFA, NFA or TM) as part of its input and analyze or even execute this code Universal Turing Machine (a simple Operating System ): Takes a TM M and a string w and simulates the execution of M on w

7 Decidable and Recognizable Languages Recall: Definition A Turing machine M is said to recognize a language L if L = L(M). A Turing machine M is said to decide a language L if L = L(M) and M halts on every input.

8 Decidable and Recognizable Languages Recall: Definition A Turing machine M is said to recognize a language L if L = L(M). A Turing machine M is said to decide a language L if L = L(M) and M halts on every input. L is said to be Turing-recognizable (Recursively Enumerable (R.E.) or simply recognizable) if there exists a TM M which recognizes L. L is said to be Turing-decidable (Recursive or simply decidable) if there exists a TM M which decides L.

9 Decidable and Recognizable Languages Recall: Definition A Turing machine M is said to recognize a language L if L = L(M). A Turing machine M is said to decide a language L if L = L(M) and M halts on every input. L is said to be Turing-recognizable (Recursively Enumerable (R.E.) or simply recognizable) if there exists a TM M which recognizes L. L is said to be Turing-decidable (Recursive or simply decidable) if there exists a TM M which decides L. Every finite language is decidable

10 Decidable and Recognizable Languages Recall: Definition A Turing machine M is said to recognize a language L if L = L(M). A Turing machine M is said to decide a language L if L = L(M) and M halts on every input. L is said to be Turing-recognizable (Recursively Enumerable (R.E.) or simply recognizable) if there exists a TM M which recognizes L. L is said to be Turing-decidable (Recursive or simply decidable) if there exists a TM M which decides L. Every finite language is decidable: For example, by a TM that has all the strings in the language hard-coded into it We just saw some example algorithms all of which terminate in a finite number of steps, and output yes or no (accept or reject). i.e., They decide the corresponding languages.

11 Decidable and Recognizable Languages But not all languages are decidable! We will show: A tm = { M, w M is a TM and M accepts w} is undecidable

12 Decidable and Recognizable Languages But not all languages are decidable! We will show: A tm = { M, w M is a TM and M accepts w} is undecidable However A tm is Turing-recognizable!

13 Decidable and Recognizable Languages But not all languages are decidable! We will show: A tm = { M, w M is a TM and M accepts w} is undecidable However A tm is Turing-recognizable! Proposition There are languages which are recognizable, but not decidable

14 Recognizing A tm Program U for recognizing A tm : On input M, w simulate M on w if simulated M accepts w, then accept else reject (by moving to q rej )

15 Recognizing A tm Program U for recognizing A tm : On input M, w simulate M on w if simulated M accepts w, then accept else reject (by moving to q rej ) U (the Universal TM) accepts M, w iff M accepts w. i.e., L(U) = A tm

16 Recognizing A tm Program U for recognizing A tm : On input M, w simulate M on w if simulated M accepts w, then accept else reject (by moving to q rej ) U (the Universal TM) accepts M, w iff M accepts w. i.e., But U does not decide A tm L(U) = A tm

17 Recognizing A tm Program U for recognizing A tm : On input M, w simulate M on w if simulated M accepts w, then accept else reject (by moving to q rej ) U (the Universal TM) accepts M, w iff M accepts w. i.e., L(U) = A tm But U does not decide A tm : If M rejects w by not halting, U rejects M, w by not halting.

18 Recognizing A tm Program U for recognizing A tm : On input M, w simulate M on w if simulated M accepts w, then accept else reject (by moving to q rej ) U (the Universal TM) accepts M, w iff M accepts w. i.e., L(U) = A tm But U does not decide A tm : If M rejects w by not halting, U rejects M, w by not halting. Indeed (as we shall see) no TM decides A tm.

19 Deciding vs. Recognizing Proposition If L and L are recognizable, then L is decidable Proof. Program P for deciding L, given programs P L and P L for recognizing L and L:

20 Deciding vs. Recognizing Proposition If L and L are recognizable, then L is decidable Proof. Program P for deciding L, given programs P L and P L for recognizing L and L: On input x, simulate P L and P L on input x.

21 Deciding vs. Recognizing Proposition If L and L are recognizable, then L is decidable Proof. Program P for deciding L, given programs P L and P L for recognizing L and L: On input x, simulate P L and P L on input x. Whether x L or x L, one of P L and P L will halt in finite number of steps. Which one to simulate first?

22 Deciding vs. Recognizing Proposition If L and L are recognizable, then L is decidable Proof. Program P for deciding L, given programs P L and P L for recognizing L and L: On input x, simulate P L and P L on input x. Whether x L or x L, one of P L and P L will halt in finite number of steps. Which one to simulate first? Either could go on forever.

23 Deciding vs. Recognizing Proposition If L and L are recognizable, then L is decidable Proof. Program P for deciding L, given programs P L and P L for recognizing L and L: On input x, simulate P L and P L on input x. Whether x L or x L, one of P L and P L will halt in finite number of steps. Which one to simulate first? Either could go on forever. On input x, simulate in parallel P L and P L on input x until either P L or P L accepts

24 Deciding vs. Recognizing Proposition If L and L are recognizable, then L is decidable Proof. Program P for deciding L, given programs P L and P L for recognizing L and L: On input x, simulate P L and P L on input x. Whether x L or x L, one of P L and P L will halt in finite number of steps. Which one to simulate first? Either could go on forever. On input x, simulate in parallel P L and P L on input x until either P L or P L accepts If P L accepts, accept x and halt. If P L accepts, reject x and halt.

25 Deciding vs. Recognizing Proof (contd). In more detail, P works as follows: On input x for i = 1, 2, 3,... simulate P L on input x for i steps simulate P L on input x for i steps if either simulation accepts, break if P L accepted, accept x (and halt) if P L accepted, reject x (and halt)

26 Deciding vs. Recognizing Proof (contd). In more detail, P works as follows: On input x for i = 1, 2, 3,... simulate P L on input x for i steps simulate P L on input x for i steps if either simulation accepts, break if P L accepted, accept x (and halt) if P L accepted, reject x (and halt) (Alternately, maintain configurations of P L and P L, and in each iteration of the loop advance both their simulations by one step.)

27 Deciding vs. Recognizing So far: A tm is undecidable (will learn soon) But it is recognizable

28 Deciding vs. Recognizing So far: A tm is undecidable (will learn soon) But it is recognizable Is every language recognizable?

29 Deciding vs. Recognizing So far: A tm is undecidable (will learn soon) But it is recognizable Is every language recognizable? No!

30 Deciding vs. Recognizing So far: A tm is undecidable (will learn soon) But it is recognizable Is every language recognizable? No! Proposition A tm is unrecognizable

31 Deciding vs. Recognizing So far: A tm is undecidable (will learn soon) But it is recognizable Is every language recognizable? No! Proposition A tm is unrecognizable Proof. If A tm is recognizable, since A tm is recognizable, the two languages will be decidable too!

32 Deciding vs. Recognizing So far: A tm is undecidable (will learn soon) But it is recognizable Is every language recognizable? No! Proposition A tm is unrecognizable Proof. If A tm is recognizable, since A tm is recognizable, the two languages will be decidable too! Note: Decidable languages are closed under complementation, but recognizable languages are not.

33 Decision Problems and Languages A decision problem requires checking if an input (string) has some property. Thus, a decision problem is a function from strings to boolean. A decision problem is represented as a formal language consisting of those strings (inputs) on which the answer is yes.

34 Recursive Enumerability A Turing Machine on an input w either (halts and) accepts, or (halts and) rejects, or never halts.

35 Recursive Enumerability A Turing Machine on an input w either (halts and) accepts, or (halts and) rejects, or never halts. The language of a Turing Machine M, denoted as L(M), is the set of all strings w on which M accepts.

36 Recursive Enumerability A Turing Machine on an input w either (halts and) accepts, or (halts and) rejects, or never halts. The language of a Turing Machine M, denoted as L(M), is the set of all strings w on which M accepts. A language L is recursively enumerable/turing recognizable if there is a Turing Machine M such that L(M) = L.

37 Decidability A language L is decidable if there is a Turing machine M such that L(M) = L and M halts on every input.

38 Decidability A language L is decidable if there is a Turing machine M such that L(M) = L and M halts on every input. Thus, if L is decidable then L is recursively enumerable.

39 Undecidability Definition A language L is undecidable if L is not decidable.

40 Undecidability Definition A language L is undecidable if L is not decidable. Thus, there is no Turing machine M that halts on every input and L(M) = L. This means that either L is not recursively enumerable. That is there is no turing machine M such that L(M) = L, or L is recursively enumerable but not decidable. That is, any Turing machine M such that L(M) = L, M does not halt on some inputs.

41 Big Picture Languages Recursively Enumerable (Recognizable) Decidable (Recursive) Lanbncn CFL Regular L 0n1n Relationship between classes of Languages

42 Machines as Strings For the rest of this lecture, let us fix the input alphabet to be {0, 1}

43 Machines as Strings For the rest of this lecture, let us fix the input alphabet to be {0, 1}; a string over any alphabet can be encoded in binary.

44 Machines as Strings For the rest of this lecture, let us fix the input alphabet to be {0, 1}; a string over any alphabet can be encoded in binary. Any Turing Machine/program M can itself be encoded as a binary string.

45 Machines as Strings For the rest of this lecture, let us fix the input alphabet to be {0, 1}; a string over any alphabet can be encoded in binary. Any Turing Machine/program M can itself be encoded as a binary string. Moreover every binary string can be thought of as encoding a TM/program.

46 Machines as Strings For the rest of this lecture, let us fix the input alphabet to be {0, 1}; a string over any alphabet can be encoded in binary. Any Turing Machine/program M can itself be encoded as a binary string. Moreover every binary string can be thought of as encoding a TM/program. (If not the correct format, considered to be the encoding of a default TM.)

47 Machines as Strings For the rest of this lecture, let us fix the input alphabet to be {0, 1}; a string over any alphabet can be encoded in binary. Any Turing Machine/program M can itself be encoded as a binary string. Moreover every binary string can be thought of as encoding a TM/program. (If not the correct format, considered to be the encoding of a default TM.) We will consider decision problems (language) whose inputs are Turing Machine (encoded as a binary string)

48 The Diagonal Language Definition Define L d = {M M L(M)}.

49 The Diagonal Language Definition Define L d = {M M L(M)}. Thus, L d is the collection of Turing machines (programs) M such that M does not halt and accept (i.e. either reject or never ends) when given itself as input.

50 A non-recursively Enumerable Language Proposition L d is not recursively enumerable.

51 A non-recursively Enumerable Language Proposition L d is not recursively enumerable. Proof. Recall that,

52 A non-recursively Enumerable Language Proposition L d is not recursively enumerable. Proof. Recall that, Inputs are strings over {0, 1}

53 A non-recursively Enumerable Language Proposition L d is not recursively enumerable. Proof. Recall that, Inputs are strings over {0, 1} Every Turing Machine can be described by a binary string and every binary string can be viewed as Turing Machine

54 A non-recursively Enumerable Language Proposition L d is not recursively enumerable. Proof. Recall that, Inputs are strings over {0, 1} Every Turing Machine can be described by a binary string and every binary string can be viewed as Turing Machine In what follows, we will denote the ith binary string (in lexicographic order) as the number i.

55 A non-recursively Enumerable Language Proposition L d is not recursively enumerable. Proof. Recall that, Inputs are strings over {0, 1} Every Turing Machine can be described by a binary string and every binary string can be viewed as Turing Machine In what follows, we will denote the ith binary string (in lexicographic order) as the number i. Thus, we can say j L(i), which means that the Turing machine corresponding to ith binary string accepts the jth binary string.

56 Completing the proof Diagonalization: Cantor Proof (contd). We can organize all programs and inputs as a (infinite) matrix, where the (i, j)th entry is Y if and only if j L(i). Inputs TMs 1 N N N N N N N 2 N N N N N N N 3 Y N Y N Y Y Y 4 N Y N Y Y N N 5 N Y N Y Y N N 6 N N Y N Y N Y

57 Completing the proof Diagonalization: Cantor Proof (contd). We can organize all programs and inputs as a (infinite) matrix, where the (i, j)th entry is Y if and only if j L(i). Inputs TMs 1 N N N N N N N 2 N N N N N N N 3 Y N Y N Y Y Y 4 N Y N Y Y N N 5 N Y N Y Y N N 6 N N Y N Y N Y For the sake of contradiction, suppose L d is recognized by a Turing machine. Say by the jth binary string. i.e., L d = L(j).

58 Completing the proof Diagonalization: Cantor Proof (contd). We can organize all programs and inputs as a (infinite) matrix, where the (i, j)th entry is Y if and only if j L(i). Inputs TMs 1 N N N N N N N 2 N N N N N N N 3 Y N Y N Y Y Y 4 N Y N Y Y N N 5 N Y N Y Y N N 6 N N Y N Y N Y For the sake of contradiction, suppose L d is recognized by a Turing machine. Say by the jth binary string. i.e., L d = L(j). But j L d iff j L(j)! More concretly, suppose j / L(j) note that j can be a string or a TM. Then, by definition, j L d = L(j). The other case j L(j) can be handled similarly.

59 Acceptor for L d? Consider the following program On input i Run program i on i Output yes if i does not accept i Output no if i accepts i

60 Acceptor for L d? Consider the following program On input i Run program i on i Output yes if i does not accept i Output no if i accepts i Does the above program recognize L d?

61 Acceptor for L d? Consider the following program On input i Run program i on i Output yes if i does not accept i Output no if i accepts i Does the above program recognize L d? No, because it may never output yes if i does not halt on i.

62 Recursively Enumerable but not Decidable L d not recursively enumerable, and therefore not decidable.

63 Recursively Enumerable but not Decidable L d not recursively enumerable, and therefore not decidable. Are there languages that are recursively enumerable but not decidable?

64 Recursively Enumerable but not Decidable L d not recursively enumerable, and therefore not decidable. Are there languages that are recursively enumerable but not decidable? Yes, A tm = { M, w M is a TM and M accepts w}

65 The Universal Language Proposition A tm is r.e. but not decidable.

66 The Universal Language Proposition A tm is r.e. but not decidable. Proof. We have already seen that A tm is r.e.

67 The Universal Language Proposition A tm is r.e. but not decidable. Proof. We have already seen that A tm is r.e. Suppose (for contradiction) A tm is decidable. Then there is a TM M that always halts and L(M) = A tm.

68 The Universal Language Proposition A tm is r.e. but not decidable. Proof. We have already seen that A tm is r.e. Suppose (for contradiction) A tm is decidable. Then there is a TM M that always halts and L(M) = A tm. Consider a TM D as follows: On input i Run M on input i, i Output yes if i rejects i Output no if i accepts i

69 The Universal Language Proposition A tm is r.e. but not decidable. Proof. We have already seen that A tm is r.e. Suppose (for contradiction) A tm is decidable. Then there is a TM M that always halts and L(M) = A tm. Consider a TM D as follows: On input i Run M on input i, i Output yes if i rejects i Output no if i accepts i Observe that L(D) = L d!

70 The Universal Language Proposition A tm is r.e. but not decidable. Proof. We have already seen that A tm is r.e. Suppose (for contradiction) A tm is decidable. Then there is a TM M that always halts and L(M) = A tm. Consider a TM D as follows: On input i Run M on input i, i Output yes if i rejects i Output no if i accepts i Observe that L(D) = L d! But, L d is not r.e. which gives us the contradiction.

71 A more complete Big Picture Languages Recursively Enumerable Decidable CFL Regular L anbncn L 0n1n L d, A tm A tm

72 Reductions A reduction is a way of converting one problem into another problem such that a solution to the second problem can be used to solve the first problem. We say the first problem reduces to the second problem.

73 Reductions A reduction is a way of converting one problem into another problem such that a solution to the second problem can be used to solve the first problem. We say the first problem reduces to the second problem. Informal Examples: Measuring the area of rectangle reduces to measuring the length of the sides

74 Reductions A reduction is a way of converting one problem into another problem such that a solution to the second problem can be used to solve the first problem. We say the first problem reduces to the second problem. Informal Examples: Measuring the area of rectangle reduces to measuring the length of the sides; Solving a system of linear equations reduces to inverting a matrix

75 Reductions A reduction is a way of converting one problem into another problem such that a solution to the second problem can be used to solve the first problem. We say the first problem reduces to the second problem. Informal Examples: Measuring the area of rectangle reduces to measuring the length of the sides; Solving a system of linear equations reduces to inverting a matrix The problem L d reduces to the problem A tm as follows: To see if w L d check if w, w A tm.

76 Undecidability using Reductions Proposition Suppose L 1 reduces to L 2 and L 1 is undecidable. Then L 2 is undecidable.

77 Undecidability using Reductions Proposition Suppose L 1 reduces to L 2 and L 1 is undecidable. Then L 2 is undecidable. Proof Sketch. Suppose for contradiction L 2 is decidable. Then there is a M that always halts and decides L 2. Then the following algorithm decides L 1

78 Undecidability using Reductions Proposition Suppose L 1 reduces to L 2 and L 1 is undecidable. Then L 2 is undecidable. Proof Sketch. Suppose for contradiction L 2 is decidable. Then there is a M that always halts and decides L 2. Then the following algorithm decides L 1 On input w, apply reduction to transform w into an input w for problem 2 Run M on w, and use its answer.

79 Schematic View w Reductions schematically

80 Schematic View w Reduction f f (w) Reductions schematically

81 Schematic View w Reduction f f (w) Algorithm for Problem 2 yes no Reductions schematically

82 Schematic View Algorithm for Problem 1 w Reduction f f (w) Algorithm for Problem 2 yes no Reductions schematically

83 The Halting Problem Proposition The language HALT = { M, w M halts on input w} is undecidable.

84 The Halting Problem Proposition The language HALT = { M, w M halts on input w} is undecidable. Proof. We will reduce A tm to HALT. Based on a machine M, let us consider a new machine f (M) as follows:

85 The Halting Problem Proposition The language HALT = { M, w M halts on input w} is undecidable. Proof. We will reduce A tm to HALT. Based on a machine M, let us consider a new machine f (M) as follows: On input x Run M on x If M accepts then halt and accept If M rejects then go into an infinite loop

86 The Halting Problem Proposition The language HALT = { M, w M halts on input w} is undecidable. Proof. We will reduce A tm to HALT. Based on a machine M, let us consider a new machine f (M) as follows: On input x Run M on x If M accepts then halt and accept If M rejects then go into an infinite loop Observe that f (M) halts on input w if and only if M accepts w

87 The Halting Problem Completing the proof Proof (contd). Suppose HALT is decidable. Then there is a Turing machine H that always halts and L(H) = HALT.

88 The Halting Problem Completing the proof Proof (contd). Suppose HALT is decidable. Then there is a Turing machine H that always halts and L(H) = HALT. Consider the following program T On input M, w Construct program f (M) Run H on f (M), w Accept if H accepts and reject if H rejects

89 The Halting Problem Completing the proof Proof (contd). Suppose HALT is decidable. Then there is a Turing machine H that always halts and L(H) = HALT. Consider the following program T On input M, w Construct program f (M) Run H on f (M), w Accept if H accepts and reject if H rejects T decides A tm.

90 The Halting Problem Completing the proof Proof (contd). Suppose HALT is decidable. Then there is a Turing machine H that always halts and L(H) = HALT. Consider the following program T On input M, w Construct program f (M) Run H on f (M), w Accept if H accepts and reject if H rejects T decides A tm. But, A tm is undecidable, which gives us the contradiction.

91 Mapping Reductions Definition A function f : Σ Σ is computable if there is some Turing Machine M that on every input w halts with f (w) on the tape.

92 Mapping Reductions Definition A function f : Σ Σ is computable if there is some Turing Machine M that on every input w halts with f (w) on the tape. Definition A mapping/many-one reduction from A to B is a computable function f : Σ Σ such that w A if and only if f (w) B

93 Mapping Reductions Definition A function f : Σ Σ is computable if there is some Turing Machine M that on every input w halts with f (w) on the tape. Definition A mapping/many-one reduction from A to B is a computable function f : Σ Σ such that w A if and only if f (w) B In this case, we say A is mapping/many-one reducible to B, and we denote it by A m B.

94 Convention In this course, we will drop the adjective mapping or many-one, and simply talk about reductions and reducibility.

95 Reductions and Recursive Enumerability Proposition If A m B and B is recursively enumerable then A is recursively enumerable.

96 Reductions and Recursive Enumerability Proposition If A m B and B is recursively enumerable then A is recursively enumerable. Proof. Let f be the reduction from A to B and let M B be the Turing Machine recognizing B.

97 Reductions and Recursive Enumerability Proposition If A m B and B is recursively enumerable then A is recursively enumerable. Proof. Let f be the reduction from A to B and let M B be the Turing Machine recognizing B. Then the Turing machine recognizing A is On input w Compute f (w) Run M B on f (w) Accept if M B does and reject if M B rejects

98 Reductions and non-r.e. Corollary If A m B and A is not recursively enumerable then B is not recursively enumerable.

99 Reductions and Decidability Proposition If A m B and B is decidable then A is decidable.

100 Reductions and Decidability Proposition If A m B and B is decidable then A is decidable. Proof. Let M B be the Turing machine deciding B and let f be the reduction. Then the algorithm deciding A, on input w, computes f (w) and runs M B on f (w).

101 Reductions and Decidability Proposition If A m B and B is decidable then A is decidable. Proof. Let M B be the Turing machine deciding B and let f be the reduction. Then the algorithm deciding A, on input w, computes f (w) and runs M B on f (w). Corollary If A m B and A is undecidable then B is undecidable.

102 Mapping Reductions Definition A function f : Σ Σ is computable if there is some Turing Machine M that on every input w halts with f (w) on the tape. Definition A reduction (a.k.a. mapping reduction/many-one reduction) from a language A to a language B is a computable function f : Σ Σ such that w A if and only if f (w) B In this case, we say A is reducible to B, and we denote it by A m B.

103 Reductions and Recursive Enumerability Proposition If A m B and B is r.e., then A is r.e. Proof. Let f be a reduction from A to B and let M B be a Turing Machine recognizing B. Then the Turing machine recognizing A is On input w Compute f (w) Run M B on f (w) Accept if M B accepts, and reject if M B rejects Corollary If A m B and A is not r.e., then B is not r.e.

104 Reductions and Decidability Proposition If A m B and B is decidable, then A is decidable. Proof. Let f be a reduction from A to B and let M B be a Turing Machine deciding B. Then a Turing machine that decides A is On input w Compute f (w) Run M B on f (w) Accept if M B accepts, and reject if M B rejects Corollary If A m B and A is undecidable, then B is undecidable.

105 The Halting Problem Proposition The language HALT = { M, w M halts on input w} is undecidable. Proof. Recall A tm = { M, w w L(M)} is undecidable. Will give reduction f to show A tm m HALT = HALT undecidable. Let f ( M, w ) = N, w where N is a TM that behaves as follows: On input x Run M on x If M accepts then halt and accept If M rejects then go into an infinite loop N halts on input w if and only if M accepts w.

106 The Halting Problem Proposition The language HALT = { M, w M halts on input w} is undecidable. Proof. Recall A tm = { M, w w L(M)} is undecidable. Will give reduction f to show A tm m HALT = HALT undecidable. Let f ( M, w ) = N, w where N is a TM that behaves as follows: On input x Run M on x If M accepts then halt and accept If M rejects then go into an infinite loop N halts on input w if and only if M accepts w. i.e., M, w A tm iff f ( M, w ) HALT

107 Emptiness of Turing Machines Proposition The language E tm = {M L(M) = } is not decidable. Note: in fact, E tm is not recognizable.

108 Emptiness of Turing Machines Proposition The language E tm = {M L(M) = } is not decidable. Note: in fact, E tm is not recognizable. Proof. Recall A tm = { M, w w L(M)} is undecidable.

109 Emptiness of Turing Machines Proposition The language E tm = {M L(M) = } is not decidable. Note: in fact, E tm is not recognizable. Proof. Recall A tm = { M, w w L(M)} is undecidable. For the sake of contradiction, suppose there is a decider B for E tm.

110 Emptiness of Turing Machines Proposition The language E tm = {M L(M) = } is not decidable. Note: in fact, E tm is not recognizable. Proof. Recall A tm = { M, w w L(M)} is undecidable. For the sake of contradiction, suppose there is a decider B for E tm. Then we first transform M, w to M 1 which is the following: On input x If x w, reject else run M on w, and accept if M accepts w, and accept if B rejects M 1, and rejects if B accepts M 1.

111 Emptiness of Turing Machines Proposition The language E tm = {M L(M) = } is not decidable. Note: in fact, E tm is not recognizable. Proof. Recall A tm = { M, w w L(M)} is undecidable. For the sake of contradiction, suppose there is a decider B for E tm. Then we first transform M, w to M 1 which is the following: On input x If x w, reject else run M on w, and accept if M accepts w, and accept if B rejects M 1, and rejects if B accepts M 1. Then we show that (1) if M, w A tm, then accept, and (2) M, w A tm, then reject. (how?)

112 Emptiness of Turing Machines Proposition The language E tm = {M L(M) = } is not decidable. Note: in fact, E tm is not recognizable. Proof. Recall A tm = { M, w w L(M)} is undecidable. For the sake of contradiction, suppose there is a decider B for E tm. Then we first transform M, w to M 1 which is the following: On input x If x w, reject else run M on w, and accept if M accepts w, and accept if B rejects M 1, and rejects if B accepts M 1. Then we show that (1) if M, w A tm, then accept, and (2) M, w A tm, then reject. (how?) This implies A tm is decidable, which is a contradiction.

113 Checking Regularity Proposition The language REGULAR = {M L(M) is regular} is undecidable.

114 Checking Regularity Proposition The language REGULAR = {M L(M) is regular} is undecidable. Proof. We give a reduction f from A tm to REGULAR.

115 Checking Regularity Proposition The language REGULAR = {M L(M) is regular} is undecidable. Proof. We give a reduction f from A tm to REGULAR. Let f ( M, w ) = N, where N is a TM that works as follows: On input x If x is of the form 0 n 1 n then accept x else run M on w and accept x only if M does

116 Checking Regularity Proposition The language REGULAR = {M L(M) is regular} is undecidable. Proof. We give a reduction f from A tm to REGULAR. Let f ( M, w ) = N, where N is a TM that works as follows: On input x If x is of the form 0 n 1 n then accept x else run M on w and accept x only if M does If w L(M) then L(N) =

117 Checking Regularity Proposition The language REGULAR = {M L(M) is regular} is undecidable. Proof. We give a reduction f from A tm to REGULAR. Let f ( M, w ) = N, where N is a TM that works as follows: On input x If x is of the form 0 n 1 n then accept x else run M on w and accept x only if M does If w L(M) then L(N) = Σ.

118 Checking Regularity Proposition The language REGULAR = {M L(M) is regular} is undecidable. Proof. We give a reduction f from A tm to REGULAR. Let f ( M, w ) = N, where N is a TM that works as follows: On input x If x is of the form 0 n 1 n then accept x else run M on w and accept x only if M does If w L(M) then L(N) = Σ. If w L(M) then L(N) =

119 Checking Regularity Proposition The language REGULAR = {M L(M) is regular} is undecidable. Proof. We give a reduction f from A tm to REGULAR. Let f ( M, w ) = N, where N is a TM that works as follows: On input x If x is of the form 0 n 1 n then accept x else run M on w and accept x only if M does If w L(M) then L(N) = Σ. If w L(M) then L(N) = {0 n 1 n n 0}.

120 Checking Regularity Proposition The language REGULAR = {M L(M) is regular} is undecidable. Proof. We give a reduction f from A tm to REGULAR. Let f ( M, w ) = N, where N is a TM that works as follows: On input x If x is of the form 0 n 1 n then accept x else run M on w and accept x only if M does If w L(M) then L(N) = Σ. If w L(M) then L(N) = {0 n 1 n n 0}. Thus, N REGULAR if and only if M, w A tm

121 Checking Equality Proposition EQ tm = { M 1, M 2 L(M 1 ) = L(M 2 )} is not r.e.

122 Checking Equality Proposition EQ tm = { M 1, M 2 L(M 1 ) = L(M 2 )} is not r.e. Proof. We will give a reduction f from E tm (assume that we know E tm is R.E.) to EQ tm.

123 Checking Equality Proposition EQ tm = { M 1, M 2 L(M 1 ) = L(M 2 )} is not r.e. Proof. We will give a reduction f from E tm (assume that we know E tm is R.E.) to EQ tm. Let M 1 be the Turing machine that on any input, halts and rejects

124 Checking Equality Proposition EQ tm = { M 1, M 2 L(M 1 ) = L(M 2 )} is not r.e. Proof. We will give a reduction f from E tm (assume that we know E tm is R.E.) to EQ tm. Let M 1 be the Turing machine that on any input, halts and rejects i.e., L(M 1 ) =. Take f (M) = M, M 1.

125 Checking Equality Proposition EQ tm = { M 1, M 2 L(M 1 ) = L(M 2 )} is not r.e. Proof. We will give a reduction f from E tm (assume that we know E tm is R.E.) to EQ tm. Let M 1 be the Turing machine that on any input, halts and rejects i.e., L(M 1 ) =. Take f (M) = M, M 1. Observe M E tm iff L(M) = iff L(M) = L(M 1 ) iff M, M 1 EQ tm.

126 Checking Properties Given M Does L(M) contain M? Is L(M) non-empty? Is L(M) empty? Undecidable Is L(M) infinite? Is L(M) finite? Is L(M) co-finite (i.e., is L(M) finite)? Is L(M) = Σ? Which of these properties can be decided?

127 Checking Properties Given M Does L(M) contain M? Is L(M) non-empty? Is L(M) empty? Undecidable Is L(M) infinite? Is L(M) finite? Is L(M) co-finite (i.e., is L(M) finite)? Is L(M) = Σ? Which of these properties can be decided? None! Undecidable

128 Checking Properties Given M Does L(M) contain M? Is L(M) non-empty? Is L(M) empty? Undecidable Is L(M) infinite? Is L(M) finite? Is L(M) co-finite (i.e., is L(M) finite)? Is L(M) = Σ? Undecidable Which of these properties can be decided? None! By Rice s Theorem

129 Properties Definition A property of languages is simply a set of languages.

130 Properties Definition A property of languages is simply a set of languages. We say L satisfies the property P if L P.

131 Properties Definition A property of languages is simply a set of languages. We say L satisfies the property P if L P. Definition For any property P, define language L P to consist of Turing Machines which accept a language in P: L P = {M L(M) P}

132 Properties Definition A property of languages is simply a set of languages. We say L satisfies the property P if L P. Definition For any property P, define language L P to consist of Turing Machines which accept a language in P: L P = {M L(M) P} Deciding L P : deciding if a language represented as a TM satisfies the property P. Example: {M L(M) is infinite}

133 Properties Definition A property of languages is simply a set of languages. We say L satisfies the property P if L P. Definition For any property P, define language L P to consist of Turing Machines which accept a language in P: L P = {M L(M) P} Deciding L P : deciding if a language represented as a TM satisfies the property P. Example: {M L(M) is infinite}; E tm = {M L(M) = }

134 Properties Definition A property of languages is simply a set of languages. We say L satisfies the property P if L P. Definition For any property P, define language L P to consist of Turing Machines which accept a language in P: L P = {M L(M) P} Deciding L P : deciding if a language represented as a TM satisfies the property P. Example: {M L(M) is infinite}; E tm = {M L(M) = } Non-example: {M M has 15 states}

135 Properties Definition A property of languages is simply a set of languages. We say L satisfies the property P if L P. Definition For any property P, define language L P to consist of Turing Machines which accept a language in P: L P = {M L(M) P} Deciding L P : deciding if a language represented as a TM satisfies the property P. Example: {M L(M) is infinite}; E tm = {M L(M) = } Non-example: {M M has 15 states} of TMs, and not languages! This is a property

136 Trivial Properties Definition A property is trivial if either it is not satisfied by any r.e. language, or if it is satisfied by all r.e. languages.

137 Trivial Properties Definition A property is trivial if either it is not satisfied by any r.e. language, or if it is satisfied by all r.e. languages. Otherwise it is non-trivial.

138 Trivial Properties Definition A property is trivial if either it is not satisfied by any r.e. language, or if it is satisfied by all r.e. languages. Otherwise it is non-trivial. Example Some trivial properties: P all = set of all languages P r.e. = set of all r.e. languages P where P is trivial

139 Trivial Properties Definition A property is trivial if either it is not satisfied by any r.e. language, or if it is satisfied by all r.e. languages. Otherwise it is non-trivial. Example Some trivial properties: P all = set of all languages P r.e. = set of all r.e. languages P where P is trivial P = {L L is recognized by a TM with an even number of states}

140 Trivial Properties Definition A property is trivial if either it is not satisfied by any r.e. language, or if it is satisfied by all r.e. languages. Otherwise it is non-trivial. Example Some trivial properties: P all = set of all languages P r.e. = set of all r.e. languages P where P is trivial P = {L L is recognized by a TM with an even number of states} = P r.e.

141 Trivial Properties Definition A property is trivial if either it is not satisfied by any r.e. language, or if it is satisfied by all r.e. languages. Otherwise it is non-trivial. Example Some trivial properties: P all = set of all languages P r.e. = set of all r.e. languages P where P is trivial P = {L L is recognized by a TM with an even number of states} = P r.e. Observation. For any trivial property P, L P is decidable. (Why?)

142 Trivial Properties Definition A property is trivial if either it is not satisfied by any r.e. language, or if it is satisfied by all r.e. languages. Otherwise it is non-trivial. Example Some trivial properties: P all = set of all languages P r.e. = set of all r.e. languages P where P is trivial P = {L L is recognized by a TM with an even number of states} = P r.e. Observation. For any trivial property P, L P is decidable. (Why?) Then L P = Σ or L P =.

143 Rice s Theorem Proposition If P is a non-trivial property, then L P is undecidable.

144 Rice s Theorem Proposition If P is a non-trivial property, then L P is undecidable. Thus {M L(M) P} is not decidable (unless P is trivial)

145 Rice s Theorem Proposition If P is a non-trivial property, then L P is undecidable. Thus {M L(M) P} is not decidable (unless P is trivial) We cannot algorithmically determine any interesting property of languages represented as Turing Machines!

146 Properties of TMs Note. Properties of TMs, as opposed to those of languages they accept, may or may not be decidable.

147 Properties of TMs Note. Properties of TMs, as opposed to those of languages they accept, may or may not be decidable. Example } { M M has 193 states} Decidable { M M uses at most 32 tape cells on blank input} { M M halts on blank input} { M on input 0011 M at some point writes the Undecidable symbol \$ on its tape}

148 Proof of Rice s Theorem Rice s Theorem If P is a non-trivial property, then L P is undecidable. Proof.

149 Proof of Rice s Theorem Rice s Theorem If P is a non-trivial property, then L P is undecidable. Proof. Suppose P non-trivial and P.

150 Proof of Rice s Theorem Rice s Theorem If P is a non-trivial property, then L P is undecidable. Proof. Suppose P non-trivial and P. (If P, then in the following we will be showing LP is undecidable. Then L P = L P is also undecidable.)

151 Proof of Rice s Theorem Rice s Theorem If P is a non-trivial property, then L P is undecidable. Proof. Suppose P non-trivial and P. (If P, then in the following we will be showing LP is undecidable. Then L P = L P is also undecidable.) Recall L P = { M L(M) satisfies P}. We ll reduce A tm to L P.

152 Proof of Rice s Theorem Rice s Theorem If P is a non-trivial property, then L P is undecidable. Proof. Suppose P non-trivial and P. (If P, then in the following we will be showing LP is undecidable. Then L P = L P is also undecidable.) Recall L P = { M L(M) satisfies P}. We ll reduce A tm to L P. Then, since A tm is undecidable, L P is also undecidable.

153 Proof of Rice s Theorem Rice s Theorem If P is a non-trivial property, then L P is undecidable. Proof. Suppose P non-trivial and P. (If P, then in the following we will be showing LP is undecidable. Then L P = L P is also undecidable.) Recall L P = { M L(M) satisfies P}. We ll reduce A tm to L P. Then, since A tm is undecidable, L P is also undecidable.

154 Proof of Rice s Theorem Proof (contd). Since P is non-trivial, at least one r.e. language satisfies P.

155 Proof of Rice s Theorem Proof (contd). Since P is non-trivial, at least one r.e. language satisfies P. i.e., L(M 0 ) P for some TM M 0.

156 Proof of Rice s Theorem Proof (contd). Since P is non-trivial, at least one r.e. language satisfies P. i.e., L(M 0 ) P for some TM M 0. Will show a reduction f that maps an instance M, w for A tm, to N such that If M accepts w then N accepts the same language as M 0. Then L(N) = L(M 0 ) P If M does not accept w then N accepts. Then L(N) = P

157 Proof of Rice s Theorem Proof (contd). Since P is non-trivial, at least one r.e. language satisfies P. i.e., L(M 0 ) P for some TM M 0. Will show a reduction f that maps an instance M, w for A tm, to N such that If M accepts w then N accepts the same language as M 0. Then L(N) = L(M 0 ) P If M does not accept w then N accepts. Then L(N) = P Thus, M, w A tm iff N L P.

158 Proof of Rice s Theorem Proof (contd). Since P is non-trivial, at least one r.e. language satisfies P. i.e., L(M 0 ) P for some TM M 0. Will show a reduction f that maps an instance M, w for A tm, to N such that If M accepts w then N accepts the same language as M 0. Then L(N) = L(M 0 ) P If M does not accept w then N accepts. Then L(N) = P Thus, M, w A tm iff N L P.

159 Proof of Rice s Theorem Proof (contd). The reduction f maps M, w to N, where N is a TM that behaves as follows: On input x Ignore the input and run M on w If M does not accept (or doesn t halt) then do not accept x (or do not halt) If M does accept w then run M 0 on x and accept x iff M 0 does. Notice that indeed if M accepts w then L(N) = L(M 0 ). Otherwise L(N) =.

160 Rice s Theorem Recap Every non-trivial property of r.e. languages is undecidable

161 Rice s Theorem Recap Every non-trivial property of r.e. languages is undecidable Rice s theorem says nothing about properties of Turing machines

162 Rice s Theorem Recap Every non-trivial property of r.e. languages is undecidable Rice s theorem says nothing about properties of Turing machines Rice s theorem says nothing about whether a property of languages is recurisvely enumerable or not.

163 Big Picture... again Languages L d, A tm, E tm Recursively Enumerable A tm, E tm, HALT Decidable CFL Regular L anbncn L 0n1n

164 Big Picture... again almost all properties! Languages L d, A tm, E tm Recursively Enumerable A tm, E tm, HALT Decidable CFL Regular L anbncn L 0n1n

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