mathematics

XOR-SAT

XOR-SAT is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand XOR-SAT rather than just read about it. In short: In computational complexity, XOR-SAT (also known as XORSAT) is the class of boolean satisfiability problems where each clause contains XOR (i.e. exclusive or, written "⊕") rather than (plain) OR operators. XOR-SAT is in P, since an XOR-SAT formula can also be viewed as a system of linear equations mod 2, and can be solved in cubic time by Gaussian elimination;.

Key takeaways

XOR-SAT belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect XOR-SAT to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of XOR-SAT from memory before moving on to harder problems.

Reference excerpt

In computational complexity, XOR-SAT (also known as XORSAT) is the class of boolean satisfiability problems where each clause contains XOR (i.e. exclusive or, written "⊕") rather than (plain) OR operators. XOR-SAT is in P, since an XOR-SAT formula can also be viewed as a system of linear equations mod 2, and can be solved in cubic time by Gaussian elimination;. This recast is based on the kinship between Boolean algebras and Boolean rings, and the fact that arithmetic modulo two forms the finite field GF(2).

Examples Here is an unsatisfiable XOR-SAT instance of 2 variables and 3 clauses:

(a ⊕ b) ∧ (a) ∧ (b) Here is a satisfiable XOR-SAT instance of 2 variables and 1 clause admitting 2 solutions:

(a ⊕ b) And here is a unique XOR-SAT instance, that is to say a satisfiable XOR-SAT instance of 2 variables and 2 clauses admitting exactly one solution:

(a ⊕ b) ∧ (a)

Comparison with SAT variations

Since a ⊕ b ⊕ c evaluates to TRUE if and only if exactly 1 or 3 members of {a,b,c} are TRUE, each solution of the 1-in-3-SAT problem for a given CNF formula is also a solution of the XOR-3-SAT problem, and in turn each solution of XOR-3-SAT is a solution of 3-SAT; see the picture. As a consequence, for each CNF formula, it is possible to solve the XOR-3-SAT problem defined by the formula, and based on the result infer either that the 3-SAT problem is solvable or that the 1-in-3-SAT problem is unsolvable. Provided that the complexity classes P and NP are not equal, neither 2-, nor Horn-, nor XOR-satisfiability is NP-complete, unlike SAT.

Solving an XOR-SAT example by Gaussian elimination Given formula (the red clause is optional): (x1 ⊕ ¬x2 ⊕ x4) ∧ (x2 ⊕ x4 ⊕ ¬x3) ∧ (x1 ⊕ x2 ⊕ ¬x3) ∧ (x1 ⊕ x2 ⊕ x4)

Equation system "1" means TRUE, "0" means FALSE. Each clause leads to one equation.

Normalized equation system Using properties of Boolean rings (¬x=1⊕x, x⊕x=0)

If the red equation is present, it contradicts the first black one, so the system is unsolvable. Therefore, Gauss' algorithm is used only for the black equations.

Associated coefficient matrix

Transforming to echelon form

Transforming to diagonal form

Variable random assignments For all the variables at the right of the diagonal form (if any), we assign any random value.

Solution If the red clause is present, the instance is unsolvable. Otherwise:

x1 = 1 = TRUE x2 = 0 = FALSE x3 = 1 = TRUE x4 = 1 = TRUE As a consequence, R(x1, ¬x2, x4) ∧ R(x2, x4, ¬x3) ∧ R(x1, x2, ¬x3) ∧ R(¬x1,x2,x4) is not 1-in-3-satisfiable, while (x1 ∨ ¬x2 ∨ x4) ∧ (x2 ∨ x4 ∨ ¬x3) ∧ (x1 ∨ x2 ∨ ¬x3) ∧ (x1 ∨ x2 ∨ x4) is 3-satisfiable with x1=x2=x3=x4=TRUE.

Notes

References

Worked examples

Example 1 — a first encounter with XOR-SAT

Start with the simplest possible case. Write down what XOR-SAT claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to XOR-SAT before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about XOR-SAT ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of XOR-SAT

In research: XOR-SAT appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses XOR-SAT in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: XOR-SAT is common in secondary-school and first-year university syllabi. It links to neighbouring topics Boolean algebra, Electronic design automation, Formal methods, so understanding it makes those chapters shorter.
In everyday life: Look for XOR-SAT outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study XOR-SAT in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what XOR-SAT means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain XOR-SAT out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is XOR-SAT in simple terms?

Why does XOR-SAT matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study XOR-SAT?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on XOR-SAT.