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| SAT (Boolean satisfiability problem) | |
|---|---|
| Name | Boolean satisfiability problem |
| Caption | Boolean formula example |
| Field | Theoretical computer science |
| Introduced | 1960s |
| Notable | Cook–Levin theorem, DPLL algorithm |
SAT (Boolean satisfiability problem) The Boolean satisfiability problem asks whether a Boolean formula can be assigned truth values that make the formula evaluate to true. It is a central decision problem in Alan Turing-era computability studies, pivotal to results by Stephen Cook and Leonid Levin, and foundational for work by researchers at institutions like Princeton University and Massachusetts Institute of Technology.
A Boolean formula is built from variables and logical connectives; the satisfiability decision asks if there exists an assignment to variables that satisfies the formula. Formalizations appear in the context of Alonzo Church's lambda calculus, Kurt Gödel-related decidability inquiries, and model-theoretic questions discussed at venues such as International Congress of Mathematicians meetings. Instances are typically expressed in conjunctive normal form or disjunctive normal form for analysis used by groups at Bell Labs, IBM Research, and AT&T Bell Laboratories.
SAT is the canonical NP-complete problem identified in the Cook–Levin theorem by Stephen Cook and independently characterized by Leonid Levin. Its NP-completeness connects to complexity classes studied by contributors like Richard Karp, Michael Rabin, Juris Hartmanis, and institutions such as University of Toronto and Harvard University. SAT occupies a role in reductions among problems like Hamiltonian path problem, Clique problem, and Subset sum problem, and it informs research in classes including P versus NP problem and conjectures considered at Clay Mathematics Institute workshops. Results about SAT influence algorithmic lower bounds in work by Scott Aaronson, László Babai, and teams at Microsoft Research.
Standard variants include k-SAT (notable for 3-SAT), quantified Boolean formulas (QBF), and restricted forms such as Horn-SAT and XOR-SAT. 3-SAT figures in completeness proofs by Richard Karp and in reductions used by Donald Knuth in algorithm analysis. QBF generalizes to PSPACE-complete problems studied by Christos Papadimitriou and Juraj Hromkovič. Horn clauses appear in logic programming influenced by Alfred Aho and Jeff Ullman. Other normal forms arise in constraint formulations used by groups at Carnegie Mellon University and Stanford University.
Practical SAT solving uses backtracking algorithms such as DPLL originating from Martin Davis and Hilary Putnam, and enhancements like conflict-driven clause learning (CDCL) developed by researchers at Compaq and University of California, Berkeley. Heuristics including VSIDS and restarts come from collaborative work involving Niklas Eén and Niklas Sörensson. Local search methods such as WalkSAT have been advanced by teams at Hebrew University of Jerusalem and University of British Columbia. Industrial-strength solvers like MiniSAT, Glucose, and Lingeling emerged from efforts at Linköping University, Université catholique de Louvain, and Karlsruhe Institute of Technology, and are benchmarked in competitions organized by the SAT Competition community with participation from groups at ETH Zurich and INRIA.
SAT underlies verification tasks in hardware and software performed at companies like Intel and Google and research labs at Bell Labs and NASA. Model checking and bounded model checking employ SAT engines in projects influenced by work at Formal Methods Europe and IBM Research. Cryptanalysis, planning, and synthesis use encodings developed by researchers at University of Cambridge and Massachusetts Institute of Technology. Combinatorial design, automated theorem proving, and bioinformatics pipelines draw on SAT techniques used by teams at Cambridge University Press-affiliated groups and laboratories at Broad Institute.
When a formula is satisfiable, a satisfying assignment acts as a certificate verifiable in polynomial time, a notion central to NP definitions formalized by Stephen Cook and Leonid Levin. For unsatisfiable instances, proofs or unsatisfiable cores serve as compact certificates; resolution proofs and DRAT proofs have been standardized in communities connected to CADE and IJCAR conferences where contributors like Marijn Heule and Matti Järvisalo presented advances. Core extraction techniques are used by verification teams at Microsoft Research and Amazon to explain counterexamples in model checking.
Early decidability and completeness studies trace to work by Emil Post and Alonzo Church. The NP-completeness landmark came from Stephen Cook and was paralleled by Leonid Levin in the Soviet literature, influencing later expositions by Richard Karp and textbooks by Michael Sipser and Christos Papadimitriou. Key algorithmic milestones include DPLL, CDCL, and modern SAT competitions that accelerated solver engineering, with notable contributors like Martin Davis, Hilary Putnam, Martin Fürer, Niklas Eén, Niklas Sörensson, Marijn Heule, and Armin Biere. Major results such as Schaefer's dichotomy theorem and hardness proofs for QBF were proven by researchers including Thomas Schaefer and discussed at institutions like Cornell University and University of California, San Diego.