in computer science called the Boolean formula satisfiability (SAT) problem.
Consider, for example,
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, two variables, x and y,
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, and the logical statement (x OR y) AND ((NOT x) OR (NOT y)). The OR means that the clause (x OR y) is true if either x or y is true, and the AND means that the clause (x AND y) is true only if both x and y are true. Solving the given problem means assigning a value of true or false to each of the two variables so that the entire statement is satisfied. Here,
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, x must be true and y false,
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, or vice versa, for the statement to be true.
"For any given instance of SAT, there is a corresponding TipOver puzzle that can be solved just when the SAT problem can be solved," Hearn says.
SAT is the prototypical NP-complete problem. Roughly speaking, an NP problem is one for which it is relatively easy to check whether a given answer is correct, but may require an impossibly long time to solve by any direct procedure. In general,
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, as the number of elements, n,
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, increases, a computer's solution time grows exponentially in the worst case. In effect, systematically solving such a worst-case problem involving many elements can take an enormous amount of time, even on the fastest available computers.
Hearn's results show that TipOver must be at least as hard as SAT. "It's easy to show that TipOver is also no harder than SAT,
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," Hearn says,
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, "so TipOver must be NP-complete

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