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[[Feature Oriented Programming]] or '''''Feature Oriented Software Development (FOSD)''''' is a general paradigm for program synthesis in software product lines.  Please read the [[Feature Oriented Programming]] page that explains how an FOSD model of a domain is a tuple of 0-ary functions (called values) and a set of 1-ary (unary) functions called features. This page discusses multidimensional generalizations of FOSD models, which are important for compact specifications of complex programs.


==Origami==
A fundamental generalization of metamodels is ''origami''.
The essential idea is that a program's design
need not be represented by a single expression; multiple expressions
can be used.<ref name="origami">{{cite web| title=Generating Product-Lines of Product-Families | url=ftp://ftp.cs.utexas.edu/pub/predator/Origami.pdf}}</ref><ref name="sigsoft03">{{cite web| title=Refinements and Multi-Dimensional Separation of Concerns | url=ftp://ftp.cs.utexas.edu/pub/predator/OrigamiMDSC.pdf}}</ref><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>  This involves the use of multiple orthogonal GenVoca models.


:: Example: Let T be a tool model, which has features P (parse), H (harvest),D (doclet), and J (translate to Java). P is a value and the rest are unary-functions. A tool T1 that parses a file written in a Java dialect language and translates it to pure Java is modeled by:  T1 = J•P. And a javadoc-like tool T2 parses a file in a Java dialect, harvests comments, and translates harvested comments into an HTML page is: T2 = D•H•P. So tools T1 and T2 are among the products of the product line of T.
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:: A language model L describes a family (product line) of Java dialects. It includes the features: B (Java 1.4), G (generics), S (State machines). B is a value, and the rest are unary functions. So a dialect of Java L1 that has generics (i.e., Java 1.5) is:  L1 = G•B. And a dialect of Java L2 that has language support for state machines is: L2 = S•B. So dialects L1 and L2 are among the products of the product line of L.
 
:: To describe a javadoc like tool (E) for the dialect of Java with state machines requires two expressions: one that defines the tool functionality for E (using the T model) and its Java dialect (using the L model):
 
  E = D•H•P    -- tool equation
  E = S•B      -- language equation
 
:: Models L and T are orthogonal GenVoca models: one expresses the feature-based structure of the E tool, and the other the feature-based structure of its input language.   Note that models T and L really are ''abstract'' in the following sense: the implementation of any feature of T really depends on the tool's dialect (expressed by L), and (symmetrically) the implementation of any feature of L really depends on the tool's functionality (expressed by T). So the only way one could implement E is by knowing both T and L equations.
 
Let U=[U<sub>1</sub>,U<sub>2</sub>,...,U<sub>n</sub>] be a GenVoca model of n features, and  
W=[W<sub>1</sub>,...W<sub>m</sub>] be a GenVoca model of m features. The relationship
between two orthogonal models U and W is a matrix UW, called an
'''''Origami matrix''''', where each
row corresponds to a feature in U and each column corresponds to
a feature in W. Entry UW<sub>ij</sub> is a function that implements the
''combination'' of features U<sub>i</sub> and W<sub>j</sub>.
 
: Note: UW is the [[tensor product]] of U and W (i.e., UW=U&times;W).
 
    <math>  UW = U \times W
= \begin{bmatrix}
UW_{11} & UW_{12} & \cdots & UW_{1n} \\
\vdots & \vdots & \ddots & \vdots \\
UW_{m1} & UW_{m2} & \cdots & UW_{mn}
\end{bmatrix}
</math>
 
:: Example. Recall models T=[P,H,D,J] and L=[B,G,S]. The Origami matrix TL is:
 
    <math> TL = T \times L
  = \begin{bmatrix}
PB & PG & PS \\
HB & HG & HS \\
DB & DG & DS \\
JB & JG & JS
  \end{bmatrix}
</math>
 
:: where PB is a value that implements a parser for Java, PG is a unary-function that extends a Java parser to parse generics, and PS is a unary-function that extends a Java parser to parse state machine specifications. HB is a unary-function that implements a harvester of comments on Java code. HG is a unary-function that implements a harvester of comments on generic code, and HS is a unary-function that implements a harvester of comments on state machine specifications, and so on.
 
To see how multiple equations are used to synthesize a program, again consider
models U and W.  A program F is described by two equations, one per model. We can
write an equation for F in two different ways: referencing features by name or
by their index position, such as:
 
    <math> F = U_1 \cdot U_2 \cdot U_4 = \sum_{i=1,2,4} U_i </math>  -- U expression of F
   
    <math> F = W_1 \cdot W_3 = \sum_{j=1,3} W_i</math>       -- W expression of F
 
The UW model defines how models U and W are implemented. Synthesizing program F involves projecting UW of unneeded columns and rows, and aggregating (a.k.a. [[tensor contraction]]):
 
    <math> F = UW_{11} \cdot UW_{21} \cdot ... \cdot UW_{33} = \sum_{i=1,2,3} \sum_{j=1,3} UW_{i,j} = \sum_{j=1,3} \sum_{i=1,2,3} UW_{i,j}</math>
 
A fundamental property of origami matrices, called '''''orthogonality''''', is that the order in which dimensions are contracted does not matter. In the above equation, summing across the U dimension (index i) first or the W dimension (index j) first does not matter. Of course, orthogonality is a property that must be verified. Efficient (linear) algorithms have been developed to verify that origami matrices (or tensors/n-dimensional arrays) are orthogonal.<ref name="sahil">{{cite web| title=Design and Analysis of Multidimensional Program Structures | url=ftp://ftp.cs.utexas.edu/pub/predator/SahilThesis.pdf}}</ref> The significance of orthogonality is one of view consistency.  Aggregating (contracting) along a particular dimension offers a 'view' of a program. Different views should be consistent: if one repairs the program's code in one view (or proves properties about a program in one view), the correctness of those repairs or properties should hold in all views.
 
In general, a product of a product line may be represented by n expressions, from  n ''orthogonal'' and ''abstract'' GenVoca models G<sub>1</sub> ... G<sub>n</sub>. The Origami
matrix (or cube or tensor) is an n-dimensional array A:
 
      <math> A =  G_1 \times ... \times G_n =  \prod_{k=1}^n G_k </math>
 
A product H of this product line is formed by eliminating unnecessary rows, columns, etc.  
from A, and aggregating (contracting) the n-cube into a scalar:
 
    <math> H =  \sum_{i_1} \sum_{i_2} ... \sum_{i_n} G_{i_1,i_2...i_n} </math>
 
:: Example.  Recall program E and model T=[P,H,D,J]. E=D•H•P=T<sub>2</sub>•T<sub>1</sub>•T<sub>0</sub>. Similarly, E's representation in model L=[B,G,S] is E=S•B=L<sub>2</sub>•L<sub>0</sub>. Synthesizing E given Origami model TL is evaluating the following expression: <math>E = \sum_{i=2,0} \sum_{j=2,0} TL_{i,j}  = \sum_{j=2,0} \sum_{i=2,0} TL_{i,j}</math>.
 
==Applications==
There are several of product line applications developed using Origami.  Among them include:
 
* [ftp://ftp.cs.utexas.edu/pub/predator/Origami.pdf AHEAD Tool Suite and Extensible Java Preprocessors]
* [ftp://ftp.cs.utexas.edu/pub/predator/ECOOP2005.pdf Expression Problem or the Extensibility Problem]
* [ftp://ftp.cs.utexas.edu/pub/predator/OrigamiMDSC.pdf Refinements and Multi-Dimensional Separation of Concerns]
 
More applications to be supplied.
 
==See also==
*[[Feature Oriented Programming]]
*[[FOSD metamodels]]
*[[FOSD Feature Interactions]]
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Fosd Origami}}
[[Category:Programming paradigms]]

Latest revision as of 17:27, 2 July 2014


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