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The Cook-Levin Theorem establishes that the problem of Circuit Satisfiability is NP-Complete. To understand why this classification is accurate, it is essential to consider the definitions and implications of NP-Completeness.

A problem is classified as NP-Complete if it satisfies two primary criteria: first, it must be in the class NP, meaning that given a proposed solution, it can be verified in polynomial time whether the solution is correct. Circuit Satisfiability fits this description because, for a given assignment of truth values to the inputs of a Boolean circuit, it is possible to evaluate the circuit in polynomial time to check if it outputs true.

Second, an NP-Complete problem must be as hard as the hardest problems in NP, which means that any NP problem can be transformed into this problem in polynomial time. The Cook-Levin Theorem specifically shows that Circuit Satisfiability can be used to represent all other problems in NP, implying that if one could find a polynomial-time solution for Circuit Satisfiability, it would extend to all NP problems (demonstrating that P = NP).

This dual requirement is why Circuit Satisfiability is classified as NP-Complete. It is among the foundational problems in computational complexity theory