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CS3052 Complexity > 4: The limits of computation > Flashcards

4: The limits of computation Flashcards

1
Q

What is the Church-Turing thesis?

A

That any algorithmic procedure can be implemented and carried out on a Turing machine.

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2
Q

What does it mean for a programming language to be Turing-complete?

A

A programming language is Turing-complete if it is able to simulate every Turing machine (or equivalently, the Universal Turing Machine). Automatons and other machines can be said to be Turing-complete as well as programming languages so long as they too can simulate all Turing machines.

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3
Q

How does being Turing-complete affect the power of programming languages?

A

Languages are more powerful when Turing-complete rather than Turing-incomplete in that they are able to run some number of TMs that they would not be able to run otherwise. Note, however, that despite the Turing machine-based architecture of modern computers, regular languages (that can run on TMs) are a tiny number of languages so there is a large capacity of power for languages outside of Turing-completeness.

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4
Q

What is the Universal Turing Machine?

A

A TM that is able to simulate any other TM; when given the input , an encoding of a Turing machine M and an input word w for M, it simulates the action of M on w.

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5
Q

Prove that the Universal Turing Machine exists (in theory, not as a machine implementing it).

A

??

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6
Q

What is decidability?

A

A true/false decision problem is decidable if there exists an effective method, here a TM, for deriving the correct answer.

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7
Q

What is a decision problem?

A

A decision problem is any well-specified problem whose result is true or false.

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8
Q

True/false: finding whether there exist algorithms to solve decision problems is equivalent to finding whether one can solve these problems using a Turing machine. Why?

A

True, since by the Church-Turing thesis says there is a TM that exists to implement any algorithm (that will accept when it results in a true solution).

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9
Q

What is a decider?

A

A TM that always halts (accepts or rejects) and never infinitely loops.

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10
Q

What does it mean for a language to be Turing-decidable?

A

Whether or not a problem may be solved, here by a decider TM (decider = always halts and never infinitely loops). A language is Turing-decidable if there exists a decider TM whose language is the same as the language.

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11
Q

What does it mean for a language to be Turing-recognisable?

A

A language is Turing-recognisable if and only if there exists a Turing Machine which will halt and accept only the strings in that language and for strings not in the language, the TM either rejects or does not halt at all.

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12
Q

True/False: Not all Turing-decidable languages are Turing-recognisable?

A

False, all Turing-decidable languages are Turing-recognisable.

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13
Q

Prove that all Turing-decidable languages are Turing-recognisable.

A

Consider a Turing-decidable language L. By assumption of the Church-Turing thesis, there is a Turing machine M such that L = L(M) and M is a decider. So, in particular, M is a Turing machine that recognises L.

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14
Q

True/false: all context-free languages are Turing-decidable.

A

True.

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15
Q

Let A = {<b> | B is a DFA that accepts w} .</b>

True/false: any such language A is not decidable.</b>

A

False, any such language will be decidable.

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16
Q

Prove that any language {<b> | B is a DFA that accepts w} will be decidable.</b>

A

??

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17
Q

What is the acceptance problem?

A

To find a decider TM for the acceptance language, AT = { | M is a TM which accepts the word w}.

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18
Q

Is the language { | M is a TM which accepts the word w} Turing-recognisable?

A

Yes, the acceptance language is Turing-recognisable but it is not decidable.

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19
Q

Is the language { | M is a TM which accepts the word w} decidable? What does this signify?

A

No. The acceptance language is Turing-recognisable but not decidable.

20
Q

Prove that the language { | M is a TM which accepts the word w} is not solvable. What does this signify?

A

??

Since this is the acceptance language, from the acceptance problem, the acceptance problem is therefore not solvable.

21
Q

What is the Halting Problem?

A

To find a decider TM for the halting language, AT = { | M is a TM which halts on the word w}.

22
Q

Is the language { | M is a TM that halts on input w} decidable? What does this signify?

A

No, the halting language of the halting problem is not decidable. This proves that the halting problem is therefore unsolvable.

23
Q

Prove that the language { | M is a TM that halts on input w} is not decidable. What does this signify?

A

??

This is the halting language of the halting problem proves that the halting problem is therefore unsolvable.

24
Q

What is a property of a Turing machine?

A

A property P of a TM is a statement that is true or false about the TM.

25
Q

What is a non-trivial property of a Turing machine?

A

A property P of a TM (a statement that is true or false about the TM) is non-trivial when there is at least one TM for which P is true and at least one for which P is not (i.e. P not always true and P not always false).

26
Q

What is Rice’s theorem?

A

Let P be a nontrivial property of the languages recognised by TMs. Then the following language is undecidable:

LP = { | M satisfies P}.

That is, there is no algorithm to decide whether a TM satisfies P.

27
Q

What is the proof of Rice’s theorem?

A

??

28
Q

What is the complement of a language?

A

If a language X is a subset of another language from the alphabet Y, Y, then the complement of X (notated as X̄) is the language of all words in Y (built from alphabet Y) that are not in X.

29
Q

True/false: If a language and its complement are Turing-recognisable, then the language is decidable.

A

True.

30
Q

Prove that if a language and its complement are Turing-recognisable, then the language is decidable.

A

??

31
Q

Prove that the language { | M is a TM which accepts the word w} is not Turing-recognisable>

A

??

32
Q

What does it mean for a set to be countable?

A

A set is countable if it is finite or its elements can be put into an infinite list.

33
Q

True/false: the union of a finite number of countable sets is countable?

A

True. Since the product will also be a countable number of numbers from all the sets.

34
Q

True/false: the union of countably many countable sets is countable?

A

True. Since the product will also be a countable number of numbers from all the sets.

35
Q

True/false: the set of all infinitely-long words from the alphabet {0, 1} is countable.

A

False. The set of all infinitely-long words from the alphabet {0, 1} is uncountable.

36
Q

Prove that the set of all infinitely-long words from the alphabet {0, 1} is uncountable.

A

??

37
Q

True/false: most languages aren’t Turing-recognisable?

A

True: the vast majority of languages are not Turing-recognisable?

38
Q

True/false: there are only countably many languages that are Turing-recognisable?

A

True.

39
Q

True/false: there are countably many languages that are not Turing recognisable?

A

False. There are uncountably many languages that are not Turing recognisable.

40
Q

Prove that there are only countably many languages that are Turing
recognisable.

A

??

41
Q

Prove that there are uncountably many languages that are not Turing recognisable.

A

??

42
Q

True/false: all regular languages are context-free?

A

True.

43
Q

True/false: all context-free languages are decidable?

A

True.

44
Q

True/false: all decidable languages are recognisable?

A

True.

45
Q

Prove that all regular languages are context-free.

A

??

46
Q

Prove that all context-free languages are decidable.

A

??

47
Q

Prove that all decidable languages are recognisable.

A

??

CS3052 Complexity (9 decks)

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