FOSD origami

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 designneed not be represented by a single expression; multiple expressionscan be used cite web| title=Generating Product-Lines of Product-Families | url=ftp://ftp.cs.utexas.edu/pub/predator/Origami.pdf] cite web| title=Refinements and Multi-Dimensional Separation of Concerns | url=ftp://ftp.cs.utexas.edu/pub/predator/OrigamiMDSC.pdf] cite web| title=Evaluating Support for Features in Advanced Modularization Technologies | url=ftp://ftp.cs.utexas.edu/pub/predator/ECOOP2005.pdf] . 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.

:: 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= [U₁,U₂,...,U_n] be a GenVoca model of n features, and W= [W₁,...W_m] be a GenVoca model of m features. The relationshipbetween two orthogonal models U and W is a matrix UW, called an"Origami matrix", where eachrow corresponds to a feature in U and each column corresponds toa feature in W. Entry UW_ij is a function that implements the"combination" of features U_i and W_j.

: Note: UW is the tensor product of U and W (i.e., UW=U×W).

:: Example. Recall models T= [P,H,D,J] and L= [B,G,S] . The Origami matrix TL is:

:: 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 canwrite an equation for F in two different ways: referencing features by name orby their index position, such as:

$F\; =\; U\_1\; cdot\; U\_2\; cdot\; U\_4\; =\; sum\_\{i=1,2,4\}\; U\_i$ -- U expression of F $$ $F\; =\; W\_1\; cdot\; W\_3\; =\; sum\_\{j=1,3\}\; W\_i$ -- 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):

$$$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\}$

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. cite web| title=Design and Analysis of Multidimensional Program Structures | url=ftp://ftp.cs.utexas.edu/pub/predator/SahilThesis.pdf] 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 G₁ ... G_n. The Origamimatrix (or cube or tensor) is an n-dimensional array A:

$A\; =\; G\_1\; imes\; ...\; imes\; G\_n\; =\; prod\_\{k=1\}^n\; G\_k$

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:

$H\; =\; sum\_\{i\_1\}\; sum\_\{i\_2\}\; ...\; sum\_\{i\_n\}\; G\_\{i\_1,i\_2...i\_n\}$

:: Example. Recall program E and model T= [P,H,D,J] . E=D•H•P=T₂•T₁•T₀. Similarly, E's representation in model L= [B,G,S] is E=S•B=L₂•L₀. Synthesizing E given Origami model TL is evaluating the following expression: $E\; =\; sum\_\{i=2,0\}\; sum\_\{j=2,0\}\; TL\_\{i,j\}\; =\; sum\_\{j=2,0\}\; sum\_\{i=2,0\}\; TL\_\{i,j\}$.

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