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| {{cleanup|date=September 2011}}
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| A program in [[Feature Oriented Programming|'''''Feature Oriented Software Development (FOSD)''''']] is
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| a composition of functions (program transformations): a base program (modeled by a nullary function) is composed with
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| increments in program functionality, called '''''features''''' (which are unary functions),
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| to produce a complex program.
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| A '''''software product line (SPL)''''' is a family of related programs.
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| Suppose product line PL has F<sub>0</sub> as a base program, and F<sub>1</sub>..F<sub>n</sub>
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| as features that could be added to F<sub>0</sub>. Different compositions of these functions/transformations yield different programs.
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| For this discussion, let
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| + denote function composition. A program P in PL might have the following expression:
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|
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| <math> P = F_8 + F_4 + F_2 + F_1 + F_0 </math>
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|
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| That is, P extends program F<sub>0</sub> with features F<sub>1</sub>, F<sub>2</sub>, F<sub>4</sub>, and F<sub>8</sub> in this order.
| | Surely the second option would be more beneficial for any website. You may discover this probably the most time-consuming part of building a Word - Press MLM website. The Word - Press Dashboard : an administrative management tool that supports FTP content upload 2. Word - Press also provides protection against spamming, as security is a measure issue. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. <br><br>Right starting from social media support to search engine optimization, such plugins are easily available within the Word - Press open source platform. If you loved this article and you would want to receive more info relating to [http://zpib.com/backup_plugin_663117 backup plugin] generously visit our site. You may either choose to link only to the top-level category pages or the ones that contain information about your products and services. You are able to set them within your theme options and so they aid the search engine to get a suitable title and description for the pages that get indexed by Google. This is identical to doing a research as in depth above, nevertheless you can see various statistical details like the number of downloads and when the template was not long ago updated. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself. <br><br>Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. After age 35, 18% of pregnancies will end in miscarriage. Nonetheless, with stylish Facebook themes obtainable on the Globe Broad Internet, half of your enterprise is done previously. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration. <br><br>Word - Press installation is very easy and hassle free. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. To do this, you should link your posts to other relevant posts that you've created. There are many advantages of hiring Wordpress developers for Wordpress project development:. Word - Press offers constant updated services and products, that too, absolutely free of cost. <br><br>Internet is not only the source for information, it is also one of the source for passive income. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. Offshore Wordpress development services from a legitimate source caters dedicated and professional services assistance with very simplified yet technically effective development and designing techniques from experienced professional Wordpress developer India. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. As with a terminology, there are many methods to understand how to use the terminology. |
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| We can recast P in terms of a projection and contraction of a 1-dimensional array.
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| Let '''F'''<sub>i</sub> = [F<sub>0</sub> .. F<sub>n</sub>] denote the array of transformations that define PL. A projection of '''F'''<sub>i</sub> eliminates
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| unneeded transformations, yielding a shorter array (call it) '''G'''<sub>i</sub>. A contraction of '''G'''<sub>i</sub> composes each
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| transformation of '''G'''<sub>i</sub> in a specific order, to yield a scalar expression. The expression for P becomes:
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| <math> P = \sum_{i=(0,1,2,4,8)} \mathbf{F}_i </math>
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| where the index values accomplish projection and summation is array contraction. This idea
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| generalizes to n-dimensional arrays that model multi-dimensional product lines.
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| == Multi-Dimensional Product Lines ==
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| [[Image:Kube.jpg|thumb|A 2-D Product Line (or Kube)]] | |
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| A '''''multi-dimensional product line''''' is described by multiple interacting sets of features.
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| <ref name="origami">{{cite web| title=Generating Product-Lines of Product-Families | url=ftp://ftp.cs.utexas.edu/pub/predator/Origami.pdf}}</ref>
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| <ref name="sigsoft03">{{cite web| title=Refinements and Multi-Dimensional Separation of Concerns | url=ftp://ftp.cs.utexas.edu/pub/predator/OrigamiMDSC.pdf}}</ref>
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| <ref name="ahead">{{cite web | title=Scaling Step-Wise Refinement | url=ftp://ftp.cs.utexas.edu/pub/predator/TSE-AHEAD.pdf}}</ref>
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| <ref name="ECOOP05">{{cite web| title=Evaluating Support for Features in Advanced Modularization Technologies | url=ftp://ftp.cs.utexas.edu/pub/predator/ECOOP2005.pdf}}</ref>
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| As an elementary 2D example, it is easy to create a product line of calculators, where
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| variants offer different sets of operations. Another variation
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| might offer different presentation front ends to calculators, one with no GUI, another
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| with a Java GUI, a third with a web GUI. These variations interact:
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| each GUI representation references a specific calculator operation, so each GUI
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| feature cannot be designed independently of its
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| calculator feature. Such a design leads to a matrix: columns represent increments in
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| calculator functionality, and rows represent different presentation front-ends. Such a matrix M is shown to the right: columns allow one to pair
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| basic calculator functionality (base) with optional logarithmic/exponentiation (lx)
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| and trigonometric (tg) features. Rows allow one to pair core functionality with no | |
| front-end (core), with optional GUI (gui) and web-based (web) front-ends.
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| An element M<sub>ij</sub> implements the interaction of column feature i and row feature j.
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| For example, the element labeled cb is a base program
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| that implements the core functionality of a calculator. Element gb adds code that
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| displays the core functionality as a GUI; element wb adds code that displays the
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| core functionality via the web. Similarly, element ct adds trigonometric code to the
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| core calculator functionality; elements gt and wt add code to display the trigonometric functionality
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| as a GUI and web front-ends.
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| A calculator is uniquely specified by two sequences
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| of features: one sequence defining the calculator functionality, the other the front-end.
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| For example, calculator C that offers both base and trig functionality in a web format
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| is defined by the expression:
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| <math> C = M_{cb} + M_{ct} + M_{wb} + M_{wt} </math>
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| : Note: Each dimension is a collection of base programs and features. Not all of their compositions are meaningful. A [[Feature Model|feature model]] defines the legal combinations of features. Thus, each dimension would have its own [[Feature Model|feature model]]. It is possible that selected features along one dimension may preclude or require features along other dimensions. In any case, these feature models define the legal combinations of features in a multi-dimensional product line.
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| == Kubes ==
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| In general, a '''''kube''''' is an n-dimensional array. The rank of a kube is its dimensionality.
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| A scalar is a kube of rank 0, a vector is a kube of rank 1, and a matrix is
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| rank 2. Following tensor notation: the number of indices a kube has designates
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| its rank. A scalar S is rank 0 (it has no indices), V<sub>k</sub> is a vector (rank
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| 1), M<sub>ij</sub> is a matrix (rank 2), C<sub>ijk</sub> is a cube (rank 3).
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| '''''Program Cubes''''' or '''''program kubes''''' are n-dimensional arrays of functions
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| (program transformations) that represent n-dimensional product lines.
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| The values along each axis
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| of a kube denote either a base program or a feature that could elaborate a base program.
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| The rank of a product line is the rank of its kube.
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| : Note: program kubes are inspired by [[tensor]]s and [[OLAP cube|data cubes]] in databases. The primary difference is that data cube elements are numerical values that are added during kube contraction; program kube elements are transformations that are composed. Both use [[tensor]] notations and terminology, although kubes satisfy few algebraic properties of tensors.
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| A program in an n-dimensional SPL is uniquely specified by n sequences of features S<sub>1</sub>..S<sub>n</sub>, one per dimension.
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| The '''''design''''' of a program is a scalar (expression) that is formed by (1) projecting the kube
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| of its unneeded elements, and (2) contracting the resultant kube to a scalar:
| |
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| <math> P = \sum_ { i_1 = S_1, i_2 = S_2, .. i_n = S_n} K_{i_1 i_2 .. i_n} </math>
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| Program '''''synthesis''''' is evaluating the scalar expression to produce program P.
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| An interesting property of kube design is that the order in which dimensions are contracted does not
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| matter—any permutation of dimensions during contraction will result in a different
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| scalar expression (i.e. a different program design), but all expressions produce the
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| same value (program). For example, another expression (design) to produce calculator C contracts
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| dimensions in the opposite order from its original specification:
| |
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| <math> C = M_{cb} + M_{wb} + M_{ct} + M_{wt} </math>
| |
| | |
| Or more generally:
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| <math> P = \sum_ { i_n = S_n, .. i_2 = S_2, i_1 = S_1} K_{i_1 i_2 .. i_n} </math>
| |
| | |
| : Note: Underlying kube designs is a [[Commutative diagram|commuting diagram]], such that there are an exponential number of paths from the empty program 0 to program P. Each path denotes a particular contraction of a kube, and corresponds to a unique incremental design of P. Included among these paths are kube aggregations that contract kubes using different dimensional orders.
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| The significance of program kubes is that it provides a structured way in which to
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| express and build multi-dimensional models of SPLs. Further, it provides scalable
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| specifications. If each dimension has k values, an n-kube specification of a program
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| requires O(kn) terms, as opposed to O(k<sup>n</sup>) kube elements that would otherwise
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| have to be identified and then composed. In general, kubes provide a compact way
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| to specify complex programs.
| |
| | |
| == Applications ==
| |
| | |
| The '''''expression problem (EP)''''' (a.k.a. the '''''extensibility problem''''') is a fundamental problem in programming languages aimed at type systems that can add new classes and methods to a program in a type-safe manner.
| |
| <ref name="Reynolds">
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| {{cite web
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| | title=User-defined types and procedural data structures as complementary approaches to data abstraction
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| | url= http://portal.acm.org/citation.cfm?id=186680
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| }}</ref>
| |
| <ref name="Cook">
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| {{cite web
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| | title=Object-Oriented Programming versues Abstract Data Types
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| | url=http://www.cs.utexas.edu/users/wcook/papers/OOPvsADT/CookOOPvsADT90.pdf
| |
| }}</ref>
| |
| <ref name="Wadler">
| |
| {{cite web
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| | title=The Expression Problem
| |
| | url=http://www.daimi.au.dk/~madst/tool/papers/expression.txt
| |
| }}</ref>
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| <ref name="Synth">
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| {{cite web
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| | title=Synthesizing Object-Oriented and Functional Design to Promote Re-Use
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| | url=http://citeseer.ist.psu.edu/krishnamurthi98synthesizing.html
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| }}</ref>
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| It is also a fundamental problem in multi-dimensional SPL design. The expression problem is an example of an SPL of rank 2. The following applications either explain/illustrate the expression problem or show how it scales to product lines of large programs. EP is really a SPL of ~30 line programs; the applications below show how these ideas scale to programs of >30K lines (a 10<sup>3</sup> increase in size).
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| * [ftp://ftp.cs.utexas.edu/pub/predator/ECOOP2005.pdf Expression Problem]
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| * [http://www.cs.utexas.edu/users/schwartz/ATS/EPL/index.html Illustration of Small Expression Problem]
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| * [ftp://ftp.cs.utexas.edu/pub/predator/Origami.pdf Extensible IDEs]
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| * [ftp://ftp.cs.utexas.edu/pub/predator/OrigamiMDSC.pdf Multi-Dimensional Separation of Concerns]
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| * [ftp://ftp.cs.utexas.edu/pub/predator/TSE-AHEAD.pdf Calculator Product Line]
| |
| | |
| Also, [[FOSD metamodels]] can be viewed as special cases of program kubes.
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| == See also ==
| |
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| *[[Feature Oriented Programming]]—basic overview
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| *[[FOSD metamodels|FOSD Metamodels]]—product lines of product lines
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| *[[FOSD Feature Algebras]]—operations from which FOSD features (0-ary and 1-ary) functions are defined
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| *[[FOSD Feature Interactions]]—general concepts for feature interactions
| |
| | |
| == References ==
| |
| <references/>
| |
| | |
| [[Category:Programming paradigms]]
| |
Surely the second option would be more beneficial for any website. You may discover this probably the most time-consuming part of building a Word - Press MLM website. The Word - Press Dashboard : an administrative management tool that supports FTP content upload 2. Word - Press also provides protection against spamming, as security is a measure issue. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed.
Right starting from social media support to search engine optimization, such plugins are easily available within the Word - Press open source platform. If you loved this article and you would want to receive more info relating to backup plugin generously visit our site. You may either choose to link only to the top-level category pages or the ones that contain information about your products and services. You are able to set them within your theme options and so they aid the search engine to get a suitable title and description for the pages that get indexed by Google. This is identical to doing a research as in depth above, nevertheless you can see various statistical details like the number of downloads and when the template was not long ago updated. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself.
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Word - Press installation is very easy and hassle free. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. To do this, you should link your posts to other relevant posts that you've created. There are many advantages of hiring Wordpress developers for Wordpress project development:. Word - Press offers constant updated services and products, that too, absolutely free of cost.
Internet is not only the source for information, it is also one of the source for passive income. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. Offshore Wordpress development services from a legitimate source caters dedicated and professional services assistance with very simplified yet technically effective development and designing techniques from experienced professional Wordpress developer India. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. As with a terminology, there are many methods to understand how to use the terminology.