Feature-oriented programming or feature-oriented software development (FOSD) is a general paradigm for program synthesis in software product lines. The feature-oriented programming page is recommended, it 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.
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.[1] [2] [3] 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=[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 relationshipbetween two orthogonal models U and W is a matrix UW, called anOrigami matrix, where eachrow corresponds to a feature in U and each column corresponds toa feature in W. Entry UWij is a function that implements thecombination of features Ui and Wj.
Note: UW is the tensor product of U and W (i.e., UW=U×W).
UW=U x W =\begin{bmatrix} UW11&UW12& … &UW1n\\ \vdots&\vdots&\ddots&\vdots\\ UWm1&UWm2& … &UWmn\end{bmatrix}
Example. Recall models T=[P,H,D,J] and L=[B,G,S]. The Origami matrix TL is:
TL=T x 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 canwrite an equation for F in two different ways: referencing features by name orby their index position, such as:
F=U1 ⋅ U2 ⋅ U4=\sumi=1,2,4Ui
F=W1 ⋅ W3=\sumj=1,3Wi
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=UW11 ⋅ UW21 ⋅ ... ⋅ UW33=\sumi=1,2,3\sumj=1,3UWi,j=\sumj=1,3\sumi=1,2,3UWi,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.[4] 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 ... Gn. The Origami matrix (or cube or tensor) is an n-dimensional array A:
A=G1 x ... x Gn=
| n | |
| \prod | |
| k=1 |
Gk
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 | |
| i1 |
| \sum | |
| i2 |
...
| \sum | |
| in |
| G | |
| i1,i2...in |
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=\sumi=2,0\sumj=2,0TLi,j=\sumj=2,0\sumi=2,0TLi,j
There are several of product line applications developed using Origami. Among them include:
More applications to be supplied.