Constraint programming: Difference between revisions

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Assignment is the association of a variable to a value from its ___domain. A partial assignment is when a subset of the variables of the problem has been assigned. A total assignment is when all the variables of the problem have been assigned.

{{math theorem|Property|Given <math>\mathcal{A} = (\mathcal{X_{\mathcal{A}}}, \mathcal{V_{\mathcal{A}}})</math> an assignment (partial or total) of a CSP <math>P = (\mathcal{X},\mathcal{D},\mathcal{C})</math>, and <math>C_i = (\mathcal{X}_i, \mathcal{R}_i)</math> a constraint of <math>P</math> such as <math>\mathcal{X}_i \subseteq \mathcal{X_{\mathcal{A}}}</math>, the assignment <math>\mathcal{A}</math> satisfies the constraint <math>C_i</math> if and only if all the values <math>\mathcal{V}_{\mathcal{A}_i} = \{ v_i \in \mathcal{V}_{\mathcal{A}} \mbox{ telsuch quethat } x_i \in \mathcal{X}_{i} \}</math> of the variables of the constraint <math>C_i</math> belongs to <math>\mathcal{R}_i</math>.

{{math theorem

|Property

|Given <math>\mathcal{A} = (\mathcal{X_{\mathcal{A}}}, \mathcal{V_{\mathcal{A}}})</math> an assignment (partial or total) of a CSP <math>P = (\mathcal{X},\mathcal{D},\mathcal{C})</math>, and <math>C_i = (\mathcal{X}_i, \mathcal{R}_i)</math> a constraint of <math>P</math> such as <math>\mathcal{X}_i \subseteq \mathcal{X_{\mathcal{A}}}</math>, the assignment <math>\mathcal{A}</math> satisfies the constraint <math>C_i</math> if and only if all the values <math>\mathcal{V}_{\mathcal{A}_i} = \{ v_i \in \mathcal{V}_{\mathcal{A}} \mbox{ tel que } x_i \in \mathcal{X}_{i} \}</math> of the variables of the constraint <math>C_i</math> belongs to <math>\mathcal{R}_i</math>.

}}

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[[Constraint propagation]] in [[Constraint Satisfaction Problems|constraint satisfaction problems]] is a typical example of a refinement model, and formula evaluation in [[spreadsheet]]s are a typical example of a perturbation model.

The refinement model is more general, as it does not restrict variables to have a single value, it can lead to several solutions to the same problem. However, the perturbation model is more intuitive for programmers using mixed imperative constraint object-oriented languages.<ref>Lopez, G., Freeman-Benson, B., & Borning, A. (1994, January). [ftp://trout.cs.washington.edu/tr/1993/09/UW-CSE-93-09-04.pdf Kaleidoscope: A constraint imperative programming language.]{{dead link|date=May 2025|bot=medic}}{{cbignore|bot=medic}} In ''Constraint Programming'' (pp. 313-329). Springer Berlin Heidelberg.</ref>

==Domains==

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* [[Boolean data type|Boolean]] domains, where only true/false constraints apply ([[Boolean satisfiability problem|SAT problem]])

* [[integer]] domains, [[Rational numbers|rational]] domains

* [[Interval_(mathematics)|interval]] domains, in particular for [[Automated planning and scheduling|scheduling]] problems<ref>{{Cite book |last1=Baptiste |first1=Philippe |author-link1=Philippe Baptiste|url=https://books.google.com/books?id=qUzhBwAAQBAJ |title=Constraint-Based Scheduling: Applying Constraint Programming to Scheduling Problems |last2=Pape |first2=Claude Le |last3=Nuijten |first3=Wim |date=2012-12-06 |publisher=Springer Science & Business Media |isbn=978-1-4615-1479-4 |language=en}}</ref>

* [[Linear algebra|linear]] domains, where only [[linear]] functions are described and analyzed (although approaches to [[non-linear]] problems do exist)

* [[wiktionary:finite|finite]] domains, where constraints are defined over [[finite set]]s

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=== Dynamic programming ===

{{Main|Dynamic programming}}

'''Dynamic programming''' is both a [[mathematical optimization]] method and a computer programming method. It refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a [[Recursion|recursive]] manner. While some [[decision problemsproblem]]s cannot be taken apart this way, decisions that span several points in time do often break apart recursively. Likewise, in computer science, if a problem can be solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have [[optimal substructure]].

== Example ==

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*[http://www.a4cp.org/ Association for Constraint Programming]

*[http://www.a4cp.org/events/cp-conference-series CP Conference Series]

*[http://kti.ms.mff.cuni.cz/~bartak/constraints/index.html Guide to Constraint Programming]

*{{webarchive |url=https://archive.today/20121205051244/http://www.mozart-oz.org/ |date=December 5, 2012 |title=The Mozart Programming System}}, an [[Oz (programming language)|Oz]]-based free software ([[X11X Window System]]- style)

*{{webarchive |url=https://archive.today/20130107222548/http://4c.ucc.ie/web/index.jsp |date=January 7, 2013 |title= Cork Constraint Computation Centre}}

{{Programming paradigms navbox}}

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[[Category:Programming paradigms]]

[[Category:Declarative programming]]

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