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 design
need not be represented by a single expression; multiple expressions
can be used. This involves the use of multiple orthogonal GenVoca models.
 E = D•H•P -- tool equation
 E = S•B -- language equation
Let U=[U1,U2...,Un] be a GenVoca model of n features, and 
W=[W1...Wm] 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 UWij is a function that implements the
"combination" of features Ui and Wj.
 formula_1
 formula_2
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:
 formula_3—U expression of F
 formula_4—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):
 formula_5
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 ... Gn. The Origami
matrix (or cube or tensor) is an n-dimensional array A:
 formula_6
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:
 formula_7
Applications.
There are several of product line applications developed using Origami. Among them include:
More applications to be supplied.