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Questions tagged [satisfiability]

Satisfiability (SAT) is the problem of determining whether there is a variable assignment that fulfills a given Boolean formula.

Satisfiability (often written in all capitals or abbreviated SAT) is the problem of determining if the variables of a given Boolean formula can be assigned in such a way as to make the formula evaluate to TRUE.

SAT was the first known example of an NP-complete problem. Further, a wide range of other naturally occurring decision and optimization problems can be transformed into instances of SAT. A class of algorithms called SAT solvers can efficiently solve a large enough subset of SAT instances to be useful in various practical areas such as circuit design and automatic theorem proving.

History of Satisfiability
Note: This is the first chapter of "Handbook of Satisfiability" (WorldCat) by Hans van Maaren, Armin Biere, Toby Walsh
Also see the timeline on page 52.

The SAT Association has Tutorials and Survey Articles

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Encoding 1-out-of-n constraint for SAT solvers

I'm using a SAT solver to encode a problem, and as part of the SAT instance, I have boolean variables $x_1,x_2,\dots,x_n$ where it is intended that exactly one of these should be true and the rest should be false. (I've sometimes seen this …
D.W.
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Measuring the difficulty of SAT instances

Given an instance of SAT, I would like to be able to estimate how difficult it will be to solve the instance. One way is to run existing solvers, but that kind of defeats the purpose of estimating difficulty. A second way might be looking a the…
Artem Kaznatcheev
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Why is SAT so important in theoretical computer science?

In my Computability and Complexity class, we are focusing on P, NP, NP-complete, and NP-hard problems and the one thing that keeps coming up is the SAT problem, in the context of reduction from one complexity to another or to explain certain…
Rishav
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Is there a sometimes-efficient algorithm to solve #SAT?

Let $B$ be a boolean formula consisting of the usual AND, OR, and NOT operators and some variables. I would like to count the number of satisfying assignments for $B$. That is, I want to find the number of different assignments of truth values to…
Mark Dominus
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Converting (math) problems to SAT instances

What I want to do is turn a math problem I have into a boolean satisfiability problem (SAT) and then solve it using a SAT Solver. I wonder if someone knows a manual, guide or anything that will help me convert my problem to a SAT instance. Also, I…
Dchris
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Classification of intractable/tractable satisfiability problem variants

Recently I found in a paper [1] a special symmetric version of SAT called the 2/2/4-SAT. But there are many $\text{NP}$-complete variants out there, for example: MONOTONE NAE-3SAT, MONOTONE 1-IN-3-SAT, ... Some other variants are tractable:…
Vor
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What's an example of an unsatisfiable 3-CNF formula?

I'm trying to wrap my head around an NP-completeness proof which seem to revolve around SAT/3CNF-SAT. Maybe it's the late hour but I'm afraid I can't think of a 3CNF formula that cannot be satisfied (I'm probably missing something obvious). Can you…
user11171
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Why do all recent SAT solvers work on CNF instead of circuit SAT?

After the release of the AIGER library to handle and-inverter graphs sometime in 2006 (I think), some circuit SAT solvers were released in 2006-2008, and in a few SAT Races/Competitions there were AIG tracks. However since then it seems the focus…
Sami Liedes
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Proving that the conversion from CNF to DNF is NP-Hard

How can I prove that the conversion from CNF to DNF is NP-Hard? I'm not asking for an answer, just some suggestions about how to go about proving it.
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Why there are no approximation algorithms for SAT and other decision problems?

I have an NP-complete decision problem. Given an instance of the problem, I would like to design an algorithm that outputs YES, if the problem is feasible, and, NO, otherwise. (Of course, if the algorithm is not optimal, it will make errors.) I…
Ribz
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Recipe book for SAT encodings?

SAT solvers are getting more and more efficient in solving large instances and are being used as back-ends in various contexts. Every time one wants to use them to solve a problem in a specific domain, he/she has to come up with an ad-hoc encoding…
Bordaigorl
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Implementing the GSAT algorithm - How to select which literal to flip?

The GSAT algorithm is, for the most part, straight forward: You get a formula in conjunctive normal form and flip the literals of the clauses until you find a solution that satisfies the formula or you reach the max_tries/max_flips limit and find no…
Adeiln
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Supporting data structures for SAT local search

WalkSAT and GSAT are well-known and simple local search algorithms for solving the Boolean satisfiability problem. The pseudocode for the GSAT algorithm is copied from the question Implementing the GSAT algorithm - How to select which literal to…
Juho
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Planarity conditions for Planar 1-in-3 SAT

Planar 3SAT is NP-complete. A planar 3SAT instance is a 3SAT instance for which the graph built using the following rules is planar: add a vertex for every $x_i$ and $\bar{x_i}$ add a vertex for every clause $C_j$ add an edge for every…
Vor
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Is generalized XOR-SAT efficiently solvable?

I've seen how XOR-3-SAT is efficiently solvable (for instance, see the "XOR-satisfiability" section in the Wikipedia entry for Boolean satisfiability problem). I'm wondering a basic question: Is XOR-k-SAT efficiently solvable, for formulas with…
Matt Groff
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