Explain Advanced Complexity Theory
Probabilistically checkable proof (PCP) is a type of proof that can be checked by a randomized algorithm
using a bounded amount of randomness and reading a bounded number of bits of the proof and the algorithm is then required to accept correct proofs and reject incorrect proofs with very high probability.
A standard proof is used in the formal verification-based definition of the complexity class that satisfies these requirements. The probabilistically checkable proofs can be checked by reading only a few bits of the proof using randomness in an essential way.
Derandomization is the process of removing the randomness and the specific methods that can be employed to derandomize particular randomized algorithms are the Method of conditional probabilities and its generalization, Discrepancy theory which is used to derandomize geometric algorithms.
The exploitation of limited independence in the random variables used by the algorithm such as Pairwise independence is used in universal hashing. The user of the expanded graph is used to amplify a limited amount of initial randomness.
The zero-knowledge protocol is an interactive method for one party to prove to another that a statement is true by revealing only the veracity of the statement and it must satisfy the properties such as Completeness if the statement is true. Soundness if the statement is false and Zero-knowledge if the statement is true and no cheating verifier learns anything other than this fact.
The hardness of approximation is a field that studies the Algorithmic complexity of finding near-optimal solutions to Optimization problems and it complements the study of approximation algorithms by proving, for certain problems.
A limit on the factors with which their solution can be efficiently approximated and the limits show a factor of approximation beyond which a problem becomes NP-Hard implying that finding a polynomial time approximation for the problem is impossible unless NP=P. At the same time, some hardness of approximation results is based on other hypotheses such as the Unique games conjecture.
Harmonic analysis of Boolean function involves the Fourier transform of a Boolean function which is an invertible linear mapping of the values of the function onto a set of coefficients, known as Fourier coefficients and this transformation is such that the Fourier coefficients contain information about the regularities of the function and the computational complexity.
Embedding is a kind of mathematical structure contained within another instance such as a group that is a subgroup and some object X is said to be embedded in another object Y, the embedding is given by some injective and structure-preserving map f: X? Y.
Lattice problems are a kind of optimization problem on the lattice.
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