Coarea formula

Template:Expert-subject Template:Orphan 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. 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 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_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).

    $UW=U\times W={\begin{bmatrix}UW_{11}&UW_{12}&\cdots &UW_{1n}\\\vdots &\vdots &\ddots &\vdots \\UW_{m1}&UW_{m2}&\cdots &UW_{mn}\end{bmatrix}}$

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

    $TL=T\times L={\begin{bmatrix}PB&PG&PS\\HB&HG&HS\\DB&DG&DS\\JB&JG&JS\end{bmatrix}}$

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:

    $F=U_{1}\cdot U_{2}\cdot U_{4}=\sum _{i=1,2,4}U_{i}$   -- 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. 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 Origami matrix (or cube or tensor) is an n-dimensional array A:

     $A=G_{1}\times ...\times G_{n}=\prod _{k=1}^{n}G_{k}$

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:

• AHEAD Tool Suite and Extensible Java Preprocessors
• Expression Problem or the Extensibility Problem
• Refinements and Multi-Dimensional Separation of Concerns

More applications to be supplied.

See also

• Feature Oriented Programming
• FOSD metamodels
• FOSD Feature Interactions

References

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