Editing Boolean satisfiability problem (section)

===XOR-satisfiability===

{| align="right" class="wikitable collapsible collapsed" style="text-align:left; margin: 1em; max-width: 95%"
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! Solving an XOR-SAT example<BR>by [[Gaussian elimination]]
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! Given formula
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| ("⊕" means XOR, the {{color|#ff8080|red clause}} is optional)
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| (''a''⊕''c''⊕''d'') ∧ (''b''⊕¬''c''⊕''d'') ∧ (''a''⊕''b''⊕¬''d'') ∧ (''a''⊕¬''b''⊕¬''c'') {{color|#ff8080|∧ (¬''a''⊕''b''⊕''c'')}}
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{| align="left" class="collapsible collapsed" style="text-align:left"
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! colspan="9" | Equation system
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| colspan="9" | ("1" means TRUE, "0" means FALSE)
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| colspan="9" | Each clause leads to one equation.
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|  || ''a'' || ⊕ ||   || ''c'' || ⊕ ||   || ''d'' || = 1
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|  || ''b'' || ⊕ || ¬ || ''c'' || ⊕ ||   || ''d'' || = 1
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|  || ''a'' || ⊕ ||   || ''b'' || ⊕ || ¬ || ''d'' || = 1
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|  || ''a'' || ⊕ || ¬ || ''b'' || ⊕ || ¬ || ''c'' || = 1
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|  {{color|#ff8080|¬}} || {{color|#ff8080|''a''}} || {{color|#ff8080|⊕}} || || {{color|#ff8080|''b''}} || {{color|#ff8080|⊕}} || || {{color|#ff8080|''c''}} || {{color|#ff8080| ≃ 1}}
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{| align="left" class="collapsible collapsed" style="text-align:left"
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! colspan="6" | Normalized equation system
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| colspan="6" | using properties of [[Boolean rings]] (¬''x''=1⊕''x'', ''x''⊕''x''=0)
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| ''a'' || ⊕ || ''c'' || ⊕ || ''d'' || = '''1'''
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| ''b'' || ⊕ || ''c'' || ⊕ || ''d'' || = '''0'''
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| ''a'' || ⊕ || ''b'' || ⊕ || ''d'' || = '''0'''
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| ''a'' || ⊕ || ''b'' || ⊕ || ''c'' || = '''1'''
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| {{color|#ff8080|''a''}} || {{color|#ff8080|⊕}}  || {{color|#ff8080|''b''}} || {{color|#ff8080|⊕}} || {{color|#ff8080|''c''}} || {{color|#ff8080| ≃ '''0'''}}
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| colspan="6" | (If the {{color|#ff8080|red equation}} is present, {{color|#ff8080|it}} contradicts
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| colspan="6" | the last black one, so the system is unsolvable.
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| colspan="6" | Therefore, Gauss' algorithm is
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| colspan="6" | used only for the black equations.)
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{| align="left" class="collapsible collapsed" style="text-align:left"
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! colspan="6" | Associated coefficient matrix
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| &nbsp;
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! ''a'' !! ''b'' !! ''c'' !! ''d'' !!   !! line
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| &nbsp;
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| 1 || 0 || 1 || 1
! 1
| A
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| 0 || 1 || 1 || 1
! 0
| B
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| 1 || 1 || 0 || 1
! 0
| C
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| 1 || 1 || 1 || 0
! 1
| D
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{| align="left" class="collapsible collapsed" style="text-align:left"
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! colspan="6" |Transforming to echelon form
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| &nbsp;
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! ''a'' !! ''b'' !! ''c'' !! ''d'' !!   !! operation
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| &nbsp;
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| 1 || 0 || 1 || 1
! 1
| A
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| 1 || 1 || 0 || 1
! 0
| C
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| 1 || 1 || 1 || 0
! 1
| D
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| 0 || 1 || 1 || 1
! 0
| B (swapped)
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| &nbsp;
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| 1 || 0 || 1 || 1
! 1
| A
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| 0 || 1 || 1 || 0
! 1
| E = C⊕A
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| 0 || 1 || 0 || 1
! 0
| F = D⊕A
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| 0 || 1 || 1 || 1
! 0
| B
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| &nbsp;
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| 1 || 0 || 1 || 1
! 1
| A
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| 0 || 1 || 1 || 0
! 1
| E
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| 0 || 0 || 1 || 1
! 1
| G = F⊕E
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| 0 || 0 || 0 || 1
! 1
| H = B⊕E
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{| align="left" class="collapsible collapsed" style="text-align:left"
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! colspan="6" | Transforming to diagonal form
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| &nbsp;
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! ''a'' !! ''b'' !! ''c'' !! ''d'' !!   !! operation
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| &nbsp;
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| 1 || 0 || 1 || 0
! 0
| I = A⊕H
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| 0 || 1 || 1 || 0
! 1
| E
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| 0 || 0 || 1 || 0
! 0
| J = G⊕H
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| 0 || 0 || 0 || 1
! 1
| H
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| &nbsp;
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| 1 || 0 || 0 || 0
! 0
| K = I⊕J
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| 0 || 1 || 0 || 0
! 1
| L = E⊕J
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| 0 || 0 || 1 || 0
! 0
| J
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| 0 || 0 || 0 || 1
! 1
| H
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{| align="left" class="collapsible collapsed" style="text-align:left"
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! Solution:
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| If the {{color|#ff8080|red clause}} is present: || Unsolvable
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| Else: || ''a'' = 0 = FALSE
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| || ''b'' = 1 = TRUE
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| || ''c'' = 0 = FALSE
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| || ''d'' = 1 = TRUE
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| colspan="2" | '''As a consequence:'''
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| colspan="2" | ''R''(''a'',''c'',''d'') ∧ ''R''(''b'',¬''c'',''d'') ∧ ''R''(''a'',''b'',¬''d'') ∧ ''R''(''a'',¬''b'',¬''c'') {{color|#ff8080|∧ ''R''(¬''a'',''b'',''c'')}}
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| colspan="2" | is not 1-in-3-satisfiable,
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| colspan="2" | while (''a'' ∨ ''c'' ∨ ''d'') ∧ (''b'' ∨ ¬''c'' ∨ ''d'') ∧ (''a'' ∨ ''b'' ∨ ¬''d'') ∧ (''a'' ∨ ¬''b'' ∨ ¬''c'')
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| colspan="2" |  is 3-satisfiable with ''a''=''c''=FALSE and ''b''=''d''=TRUE.
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Another special case is the class of problems where each clause contains XOR (i.e. [[exclusive or]]) rather than (plain) OR operators.{{efn|Formally, generalized conjunctive normal forms with a ternary Boolean function ''R'' are employed, which is TRUE just if 1 or 3 of its arguments is. An input clause with more than 3 literals can be transformed into an equisatisfiable conjunction of clauses á 3 literals similar to [[#3-satisfiability|above]]; i.e. XOR-SAT can be reduced to XOR-3-SAT.}} This is in [[P (complexity class)|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]];<ref>{{citation|title=The Nature of Computation|first1=Cristopher|last1=Moore|author1-link=Cristopher Moore|first2=Stephan|last2=Mertens|publisher=Oxford University Press|year=2011|isbn=9780199233212|page=366|url=https://books.google.com/books?id=z4zMiZyAE1kC&pg=PA366}}.</ref> see the box for an example. This recast is based on the [[Boolean algebra (structure)#Boolean rings|kinship between Boolean algebras and Boolean rings]], and the fact that arithmetic modulo two forms a [[finite field]]. Since ''a'' XOR ''b'' XOR ''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 [[P = NP problem|complexity classes P and NP are not equal]], neither 2-, nor Horn-, nor XOR-satisfiability is NP-complete, unlike SAT.