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Recursion Theory

Description: Recursion Theory is a branch of mathematical logic that studies the foundations of computation and the limits of what can be computed. This quiz covers basic concepts in recursion theory, including computability, decidability, and Turing machines.
Number of Questions: 14
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Tags: recursion theory computability decidability turing machines
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What is the Halting Problem?

  1. The problem of determining whether a given program will ever halt.

  2. The problem of determining whether a given program will halt in a finite number of steps.

  3. The problem of determining whether a given program will halt in an infinite number of steps.

  4. The problem of determining whether a given program will halt in a finite or infinite number of steps.

Show answer
Correct Option: A
Explanation:

The Halting Problem is undecidable, meaning that there is no algorithm that can solve it for all possible programs.

What is a Turing machine?

  1. A mathematical model of computation that consists of a tape, a head, and a set of instructions.

  2. A physical device that can be used to perform computations.

  3. A programming language that is used to write programs for Turing machines.

  4. A software program that can be used to simulate Turing machines.

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Correct Option: A
Explanation:

Turing machines are a theoretical model of computation that can be used to define the concept of computability.

What is the Church-Turing thesis?

  1. The thesis that every computable function can be computed by a Turing machine.

  2. The thesis that every Turing machine can be simulated by a computer.

  3. The thesis that every computer program can be written in a Turing-complete programming language.

  4. The thesis that every algorithm can be implemented on a physical computer.

Show answer
Correct Option: A
Explanation:

The Church-Turing thesis is a widely accepted conjecture in computer science that states that Turing machines are a universal model of computation.

What is a decidable problem?

  1. A problem that can be solved by an algorithm that always terminates.

  2. A problem that can be solved by an algorithm that sometimes terminates.

  3. A problem that can be solved by an algorithm that never terminates.

  4. A problem that cannot be solved by any algorithm.

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Correct Option: A
Explanation:

A decidable problem is a problem that can be solved by an algorithm that always terminates.

What is an undecidable problem?

  1. A problem that can be solved by an algorithm that always terminates.

  2. A problem that can be solved by an algorithm that sometimes terminates.

  3. A problem that can be solved by an algorithm that never terminates.

  4. A problem that cannot be solved by any algorithm.

Show answer
Correct Option: D
Explanation:

An undecidable problem is a problem that cannot be solved by any algorithm.

What is the Rice's theorem?

  1. A theorem that states that every non-trivial property of the set of all partial recursive functions is undecidable.

  2. A theorem that states that every non-trivial property of the set of all Turing machines is undecidable.

  3. A theorem that states that every non-trivial property of the set of all computer programs is undecidable.

  4. A theorem that states that every non-trivial property of the set of all algorithms is undecidable.

Show answer
Correct Option: A
Explanation:

Rice's theorem is a fundamental result in recursion theory that shows that many problems about the behavior of programs are undecidable.

What is the Kleene's recursion theorem?

  1. A theorem that states that every partial recursive function can be defined by a Turing machine.

  2. A theorem that states that every Turing machine can be simulated by a computer.

  3. A theorem that states that every computer program can be written in a Turing-complete programming language.

  4. A theorem that states that every algorithm can be implemented on a physical computer.

Show answer
Correct Option: A
Explanation:

Kleene's recursion theorem is a fundamental result in recursion theory that shows that Turing machines are a universal model of computation.

What is the Post's correspondence problem?

  1. A problem that asks whether there is a way to match up two lists of symbols so that the corresponding symbols in each list form a valid word.

  2. A problem that asks whether there is a way to match up two lists of symbols so that the corresponding symbols in each list form a valid sentence.

  3. A problem that asks whether there is a way to match up two lists of symbols so that the corresponding symbols in each list form a valid program.

  4. A problem that asks whether there is a way to match up two lists of symbols so that the corresponding symbols in each list form a valid algorithm.

Show answer
Correct Option: A
Explanation:

The Post's correspondence problem is an undecidable problem that is often used to illustrate the limits of computation.

What is the Busy Beaver problem?

  1. A problem that asks what is the maximum number of steps that a Turing machine with a given number of states can take before it halts.

  2. A problem that asks what is the maximum number of steps that a Turing machine with a given number of symbols can take before it halts.

  3. A problem that asks what is the maximum number of steps that a Turing machine with a given number of instructions can take before it halts.

  4. A problem that asks what is the maximum number of steps that a Turing machine with a given number of tapes can take before it halts.

Show answer
Correct Option: A
Explanation:

The Busy Beaver problem is an undecidable problem that is often used to study the limits of computation.

What is the halting problem?

  1. A problem that asks whether a given Turing machine will ever halt.

  2. A problem that asks whether a given Turing machine will halt in a finite number of steps.

  3. A problem that asks whether a given Turing machine will halt in an infinite number of steps.

  4. A problem that asks whether a given Turing machine will halt in a finite or infinite number of steps.

Show answer
Correct Option: A
Explanation:

The halting problem is an undecidable problem that is often used to illustrate the limits of computation.

What is the diagonalization argument?

  1. An argument that shows that there is no algorithm that can solve the halting problem.

  2. An argument that shows that there is no algorithm that can solve the Post's correspondence problem.

  3. An argument that shows that there is no algorithm that can solve the Busy Beaver problem.

  4. An argument that shows that there is no algorithm that can solve any undecidable problem.

Show answer
Correct Option: A
Explanation:

The diagonalization argument is a powerful technique that can be used to prove that certain problems are undecidable.

What is the incompleteness theorem?

  1. A theorem that states that every consistent axiomatic system is either incomplete or contradictory.

  2. A theorem that states that every axiomatic system is either complete or contradictory.

  3. A theorem that states that every consistent axiomatic system is either complete or undecidable.

  4. A theorem that states that every axiomatic system is either incomplete or undecidable.

Show answer
Correct Option: A
Explanation:

The incompleteness theorem is a fundamental result in mathematical logic that shows that there are certain statements that cannot be proven or disproven within a given axiomatic system.

What is the Gödel's incompleteness theorem?

  1. A theorem that states that every consistent axiomatic system is either incomplete or contradictory.

  2. A theorem that states that every axiomatic system is either complete or contradictory.

  3. A theorem that states that every consistent axiomatic system is either complete or undecidable.

  4. A theorem that states that every axiomatic system is either incomplete or undecidable.

Show answer
Correct Option: A
Explanation:

Gödel's incompleteness theorem is a fundamental result in mathematical logic that shows that there are certain statements that cannot be proven or disproven within a given axiomatic system.

What is the Turing's proof of the halting problem?

  1. A proof that shows that there is no algorithm that can solve the halting problem.

  2. A proof that shows that there is no algorithm that can solve the Post's correspondence problem.

  3. A proof that shows that there is no algorithm that can solve the Busy Beaver problem.

  4. A proof that shows that there is no algorithm that can solve any undecidable problem.

Show answer
Correct Option: A
Explanation:

Turing's proof of the halting problem is a seminal result in computer science that shows that there is no algorithm that can solve the halting problem.

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