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The 3-SAT (3-Satisfiability) problem is a classic problem in computer science, particularly in the field of computational complexity theory. It's a specific type of Boolean satisfiability problem (SAT), which is foundational in the study of algorithmic logic and has significant implications in various areas like cryptography, artificial intelligence, and algorithm design.

  1. Boolean Variables โ€” The problem involves a set of Boolean variables. Each variable can take on one of two values: true or false.

  2. Clauses โ€” The heart of the problem lies in a series of clauses. Each clause is a disjunction (logical OR) of exactly three literals. A literal is either a variable or its negation. For example, a clause might be (xโˆจยฌyโˆจz)(x \lor \neg y \lor z), where x,y,x, y, and zz are Boolean variables, and ยฌy\neg y represents the negation of yy.

  3. Satisfiability โ€” The question posed by the 3-SAT problem is whether there exists an assignment of values to the variables that makes the entire Boolean expression true. In other words, can we assign true/false values to each variable in such a way that every clause has at least one true literal?

  4. NP-Completeness โ€” The 3-SAT problem is famously known for being NP-complete, which means two things:

    • It's in NP (nondeterministic polynomial time), meaning that if a solution exists, it can be verified quickly (in polynomial time).
    • Every problem in NP can be reduced to it in polynomial time. This makes 3-SAT a central problem in complexity theory, as finding a polynomial-time algorithm for it (if one exists) would imply P = NP, solving a major open question in computer science.

The importance of 3-SAT and other SAT problems lies in their applicability to real-world scenarios where complex decision-making is required. They are used in various domains like electronic design automation, model checking, software verification, scheduling, and more. The challenge in solving these problems efficiently has driven much of the research in algorithm development and complexity theory.

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