Theory of Computation: P vs NP
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Questions and Answers

Which of the following statements about the complexity class P is TRUE?

  • P is considered the class of problems that are computationally intractable.
  • P stands for 'Polynomial time' but refers to problems solvable in exponential time.
  • Problems in P can be solved by deterministic algorithms in polynomial time. (correct)
  • P includes problems that require exponential time to solve.

A problem is in NP if and only if a solution to that problem can be verified in polynomial time.

True (A)

What does the acronym 'SAT' stand for in the context of theoretical computer science?

Satisfiability

The problem 3SAT is a special case of SAT where each clause in the Boolean formula has exactly ______ literals.

<p>three</p> Signup and view all the answers

Match the following complexity classes with their corresponding definitions:

<p>P = Problems solvable by deterministic algorithms in polynomial time. NP = Problems for which solutions can be verified in polynomial time. 3SAT = A special case of SAT with three literals per clause.</p> Signup and view all the answers

P-reducibility is a special type of reduction that preserves exponential time complexity.

<p>False (B)</p> Signup and view all the answers

What is the significance of P-reducibility in complexity theory?

<p>It helps establish the computational complexity of problems by relating them to other problems with known complexities. (C)</p> Signup and view all the answers

Give an example of a real-world problem that could be modeled as a SAT problem.

<p>Examples: Scheduling, circuit design, logistics, constraint satisfaction problems.</p> Signup and view all the answers

Which of the following statements accurately describes the relationship between P and NP?

<p>P is a subset of NP, meaning that all problems solvable in polynomial time are also verifiable in polynomial time. (B), The relationship between P and NP is unknown, and it is an unsolved problem whether P = NP. (C)</p> Signup and view all the answers

A problem in NP can always be solved in polynomial time.

<p>False (B)</p> Signup and view all the answers

What is the primary question that the P vs. NP problem seeks to answer?

<p>Whether every problem whose solution can be verified in polynomial time can also be solved in polynomial time.</p> Signup and view all the answers

The ______ problem is considered a classic NP-complete problem, where the goal is to determine if there exists a truth assignment that can make a given Boolean formula true.

<p>SAT</p> Signup and view all the answers

Match the following terms with their corresponding definitions:

<p>P = Problems solvable in polynomial time. NP = Problems verifiable in polynomial time. SAT = A decision problem asking if a Boolean formula can be made true. 3SAT = A specific type of SAT problem where each clause has exactly three literals. P-reducibility = A reduction that shows problem X is at least as hard as problem Y, implying that an algorithm for Y would also solve X.</p> Signup and view all the answers

Which of the following is a key property of 3SAT that makes it an NP-complete problem?

<p>Any problem in NP can be reduced to 3SAT in polynomial time. (C)</p> Signup and view all the answers

Problems that are P-reducible to each other have the same computational complexity.

<p>True (A)</p> Signup and view all the answers

Explain the significance of P-reducibility in the context of NP-completeness.

<p>If a problem X can be P-reduced to an NP-complete problem Y, then X is also NP-complete. This means proving an algorithm for X would also imply an algorithm for Y and thus for all problems in NP.</p> Signup and view all the answers

Flashcards

Theory of Computation

Branch of computer science studying limits and possibilities of computation.

P (Polynomial time)

Complexity class for problems solvable by a deterministic algorithm in polynomial time.

NP (Nondeterministic Polynomial time)

Complexity class for problems where solutions can be verified in polynomial time.

SAT (Satisfiability)

Classic NP-complete problem asking if a boolean formula is satisfiable.

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3SAT

Specific SAT problem where each clause has exactly three literals.

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P-reducibility

Concept in complexity theory relating difficulty of problems while preserving polynomial time.

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Deterministic Algorithm

An algorithm that behaves predictably, producing the same output for a given input every time.

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NP-complete

Class of problems that are both in NP and as hard as the hardest problems in NP.

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P vs NP

The core question in computational theory regarding if every problem solvable in polynomial time can also be verified in polynomial time.

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Computational Complexity

Study of the resources needed for algorithms to solve computational problems, typically time or space.

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Turing Machine

A theoretical model of computation that defines an abstract machine capable of performing any computation.

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Polynomial Time

Time complexity where the running time of an algorithm is expressed as a polynomial function of the input size.

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NP-hard

A class of problems that are at least as hard as the hardest problems in NP; solving one in polynomial time means all NP problems can be solved in polynomial time.

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Reduction

A method of converting one problem into another, often used to show relationships between their complexities.

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NP-completeness

A property of decision problems for which a solution can be verified quickly (in polynomial time) and are as hard as any problem in NP.

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3-SAT Reduction

The process of transforming any SAT problem into a 3SAT problem, which is crucial for proving NP-completeness.

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Study Notes

Theory of Computation

  • Theory of computation explores the limits of what computers can solve.
  • It examines algorithms, computational models (like Turing machines), and problem classification by complexity.
  • It's a branch of computer science dealing with the fundamental concepts of calculation.

P

  • P (Polynomial Time) is a complexity class.
  • Problems in P are solvable in polynomial time by a deterministic algorithm.
  • Polynomial time means solution time grows at most polynomially with input size.
  • Problems in P are considered computationally tractable.

NP

  • NP (Nondeterministic Polynomial Time) is a complexity class.
  • NP problems are those where solutions can be verified in polynomial time.
  • Finding a solution might not be possible in polynomial time, but checking a proposed solution is.
  • The core question is whether P = NP. If P = NP, every verifiable problem is also solvable quickly. If P ≠ NP, some problems are inherently more difficult to solve than to verify.

SAT

  • SAT (Satisfiability) is a classic NP-complete problem.
  • SAT determines if a Boolean formula can be made true via variable assignments.
  • Many problems reduce to SAT, making it central to NP-completeness.

3SAT

  • 3SAT is a specific SAT problem type.
  • Each clause in the Boolean formula has exactly three variables.
  • 3SAT retains the computational difficulty of general SAT.
  • 3SAT is NP-complete.

P-reducibility

  • P-reducibility is vital in complexity theory.
  • It establishes relationships between problems' difficulty.
  • P-reducibility is a reduction maintaining polynomial-time complexity.
  • If problem A reduces to problem B in polynomial time, and B is in P, then A is also in P.
  • If X reduces to Y in polynomial time, Y is at least as hard as X. If X is in P, Y is in P.

Implications

  • Theory of computation is applied in cryptography, optimization, and AI.
  • The P vs. NP problem reveals potential limits to efficient computation
  • Understanding P, NP, and NP-complete problems is essential for problem evaluation by computational difficulty. P-reducibility is about comparing algorithms using a mathematical equivalence.

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Description

Explore the fascinating world of computation in this quiz on the Theory of Computation. Understand the key concepts of P and NP complexity classes and delve into algorithm efficiency. Test your knowledge on polynomial time problems and their significance in computer science.

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