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When Satisfiability Solving Meets Symbolic Computation


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Mathematicians have long been fascinated by objects that exhibit exceptionally nice combinatorial properties. However, it is often difficult to determine whether objects satisfying a given combinatorial property exist. Sometimes, the only feasible method of definitively answering the question of existence is simply to perform a systematic search. A famous example of this is the proof of the four-color theorem—the notion that four colors suffice to color the regions of a planar map with adjacent regions colored differently.3 The theorem has been known to be true since 1977, but every known proof relies on computer calculations in an essential way. Mathematical arguments are used to reduce the search for counterexamples to a finite number of cases, and the cases are then exhaustively checked using a custom-written computer program to rule out any counterexamples.

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Independently, computer scientists have made significant progress over the last 50 years on developing general-purpose programs that can automatically solve many kinds of mathematical problems. Satisfiability solving and symbolic computation are two important branches of computer science that each specialize in solving mathematical problems. Both fields have long histories and have produced impressive tools—satisfiability (SAT) solvers in the former and computer algebra systems (CASs) in the latter. Originally, SAT solvers were designed to solve problems in logic, and CASs were tools to manipulate and simplify algebraic expressions. As we will see, these tools have since found an abundance of new applications outside of these original domains.


 

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