Probabilistically checkable proof

Probabilistically Checkable Proofs (PCPs) are a concept in computational complexity theory, a branch of theoretical computer science and mathematics that studies the resources required during computation to solve a given problem. The most notable aspect of PCPs is their role in the PCP theorem, which essentially states that every mathematical proof can be verified by examining a very small portion of the proof. This theorem has profound implications for the field of approximation algorithms and hardness of approximation.

Overview

A Probabilistically Checkable Proof is a type of proof that can be checked by a verifier in a probabilistic manner. Instead of reading the entire proof, the verifier reads only a few randomly selected bits of the proof and decides whether to accept or reject the proof based on this partial information. The remarkable property of PCPs is that they allow for the verification of proofs with high confidence while only looking at a tiny fraction of the proof.

Formal Definition

A PCP is defined by two parameters, r and q, where r denotes the randomness used by the verifier and q denotes the number of queries the verifier makes into the proof. The formal definition of a PCP system involves a verifier that, given a random string (for randomness) and access to a proof (oracle), decides whether to accept or reject the proof by making q queries based on r bits of randomness.

PCP Theorem

The PCP theorem states that for every decision problem in NP, there exists a PCP verifier that uses a logarithmic amount of randomness and queries a constant number of bits in the proof. This theorem implies that NP-complete problems can be verified with very high confidence by checking only a small portion of the proof, leading to significant implications for the field of hardness of approximation.

Applications

The PCP theorem has profound implications in the field of computational complexity theory, especially in the study of approximation algorithms. It provides a framework for proving that certain problems cannot be approximated beyond a certain threshold unless P=NP, which is a major unsolved question in computer science. This has led to the development of hardness of approximation results for many problems, showing that finding near-optimal solutions is computationally hard.

Related Concepts

• Interactive proof system
• Zero-knowledge proof
• NP-completeness
• Hardness of approximation
• Approximation algorithm

See Also

• Computational complexity theory
• P=NP
• Graph theory
• Cryptography

References

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