Material failure theory

Feature Oriented Programming (FOP) or Feature Oriented Software Development (FOSD) is a general paradigm for program synthesis in software product lines.

Connection Between Layer Stacks and Transformation Compositions

FOSD arose out of layer-based designs and levels of abstraction in network protocols and extensible database systems in the late-1980s. A program was a stack of layers. Each layer added functionality to previously composed layers and different compositions of layers produced different programs. Not surprisingly, there was a need for a compact language to express such designs. Elementary algebra fit the bill: each layer was function (program transformation) that added new code to an existing program to produce a new program, and a program's design was modeled by an expression, i.e., a composition of transformations (layers). The figure below illustrates the stacking of layers h, j, and i (where h is on the bottom and i is on the top). The algebraic notations i(j(h))and i•j•h express these designs.

Over time, the idea of layers was generalized to features, where a feature is an increment in program development or functionality. The paradigm for program design and synthesis was recognized to be a generalization of relational query optimization, where query evaluation programs were defined as relational algebra expressions, and query optimization was expression evaluation. A software product line (SPL) is a family of programs where each program is defined by a unique composition of features, and no two programs have the same combination. FOSD has since evolved into the study of feature modularity, tools, analyses, and design techniques to support feature-based program synthesis.

Further advances in FOSD arose from recognizing the following facts: Every program has multiple representations (e.g., source, makefiles, documentation, etc.) and adding a feature to a program should elaborate each of its representations so that all representations are consistent. Additionally, some of these representations could be generated (or derived) from other representations. In this article, the mathematics of the three most recent generations of FOSD, namely GenVoca, AHEAD, and FOMDD are described, and links to product lines that have been developed using FOSD tools are provided. Also, four additional results that apply to all generations of FOSD are presented elsewhere: MetaModels, Program Cubes, Feature Algebras, and Feature Interactions.

GenVoca

GenVoca (a meld of the names Genesis and Avoca) is a compositional paradigm for defining programs of a product lines. Base programs are 0-ary functions or transformations called values:

  f      -- base program with feature f
  h      -- base program with feature h

and features are unary functions/transformations that elaborate (modify, extend, refine) a program:

  i • x  -- adds feature i to program x
  j • x  -- adds feature j to program x

where • denotes function composition. The design of a program is a named expression, e.g.:

  p1 = j • f       -- program p1 has features j and f
  p2 = j • h       -- program p2 has features j and h
  p3 = i • j • h   -- program p3 has features i, j, and h

A GenVoca model of a domain or software product line is a collection of base programs and features (see MetaModels and Program Cubes). The programs (expressions) that can be created defines a product line. Expression optimization is program design optimization, and expression evaluation is program synthesis.

Note: GenVoca is based on the stepwise development of programs: a process that emphasizes design simplicity and understandability, which are key to program comprehension and automated program construction. Consider program p₃ above: it begins with base program h, then feature j is added (read: the functionality of feature j is added to the codebase of h), and finally feature i is added (read: the functionality of feature i is added to the codebase of j•h).

Note: not all combinations of features are meaningful. Feature models (which can be translated into propositional formulas) are graphical representations that define legal combinations of features.

Note: A more recent formulation of GenVoca is symmetric: there is only one base program, 0 (the empty program), and all features are unary functions. This suggests the interpretation that GenVoca composes program structures by superposition, the idea that complex structures are composed by superimposing simpler structures. Yet another reformulation of GenVoca is as a monoid: a GenVoca model is a set of features with a composition operation (•); composition is associative and there is an identity element (namely 1, the identity function). Although all compositions are possible, not all are meaningful as mentioned above.

GenVoca features were originally implemented using C preprocessor (#ifdef feature ... #endif) techniques. A more advanced technique, called mixin layers, showed the connection of features to object-oriented collaboration-based designs.

AHEAD

Algebraic Hierarchical Equations for Application Design (AHEAD) generalized GenVoca in two ways. First it revealed the internal structure of GenVoca values as tuples. Every program has multiple representations, such as source, documentation, bytecode, and makefiles. A GenVoca value is a tuple of program representations. In a product line of parsers, for example, a base parser f is defined by its grammar g_f, Java source s_f, and documentation d_f. Program f is modeled by the tuple f=[g_f, s_f, d_f]. Each program representation may have subrepresentations, and they too may have subrepresentations, recursively. In general, a GenVoca value is a tuple of nested tuples that define a hierarchy of representations for a particular program.

Hierarchical Relationships among Program Artifacts

Example. Suppose terminal representations are files. In AHEAD, grammar g_f corresponds to a single BNF file, source s_f corresponds to a tuple of Java files [c₁…c_n], and documentation d_f is a tuple of HTML files [h₁…h_k]. A GenVoca value (nested tuples) can be depicted as a directed graph: the graph for program f is shown in the figure to the right. Arrows denote projections, i.e., mappings from a tuple to one of its components. AHEAD implements tuples as file directories, so f is a directory containing file g_f and subdirectories s_f and d_f. Similarly, directory s_f contains files c₁…c_n, and directory df contains files h₁…h_k.

Note: Files can be hierarchically decomposed further. Each Java class can be decomposed into a tuple of members and other class declarations (e.g., initialization blocks, etc.).

Second, AHEAD expresses features as nested tuples of unary functions called deltas. Deltas can be program refinements (semantics-preserving transformations), extensions (semantics-extending transformations), or interactions (semantics-altering transformations). We use the neutral term “delta” to represent all of these possibilities, as each occurs in FOSD.

To illustrate, suppose feature j extends a grammar by $\mathrm{\Delta }$g_j (new rules and tokens are added), extends source code by $\mathrm{\Delta }$s_j (new classes and members are added and existing methods are modified), and extends documentation by $\mathrm{\Delta }$d_j. The tuple of deltas for feature j is modeled by j=[$\mathrm{\Delta }$g_j,$\mathrm{\Delta }$s_j,$\mathrm{\Delta }$d_j], which we call a delta tuple. Elements of delta tuples can themselves be delta tuples. As an example, $\mathrm{\Delta }$s_j represents the changes that are made to each class in s_f by feature j, i.e., $\mathrm{\Delta }$s_j=[$\mathrm{\Delta }$c₁…$\mathrm{\Delta }$c_n]. The representations of a program are computed recursively by composing tuples element-wise. The representations for parser p (whose GenVoca expression is j•f) are:

  p2 = j • f                            -- GenVoca expression
     = [$\mathrm{\Delta }$gj, $\mathrm{\Delta }$sj, $\mathrm{\Delta }$dj] • [gf, sf, df]   -- substitution
     = [$\mathrm{\Delta }$gj•gf, $\mathrm{\Delta }$sj•sf, $\mathrm{\Delta }$dj•df]         -- compose tuples element-wise

That is, the grammar of p is the base grammar composed with its extension ($\mathrm{\Delta }$g_j•g_f), the source of p is the base source composed with its extension ($\mathrm{\Delta }$s_j•s_f), and so on. As elements of delta tuples can themselves be delta tuples, composition recurses, e.g., $\mathrm{\Delta }$s_j•s_f= [$\mathrm{\Delta }$c₁…$\mathrm{\Delta }$c_n]•[c₁…c_n]=[$\mathrm{\Delta }$c₁•c₁…$\mathrm{\Delta }$c_n•c_n]. Summarizing, GenVoca values are nested tuples of program artifacts, and features are nested delta tuples, where • recursively composes them. This is the essence of AHEAD.

The ideas presented above concretely expose two FOSD principles. The Principle of Uniformity states that all program artifacts are treated and refined in the same way. (This is evidenced by deltas for different artifact types above). The Principle of Scalability states all levels of abstractions are treated uniformly. (This gives rise to the hierarchical nesting of tuples above).

The original implementation of AHEAD is the AHEAD Tool Suite and Jak language, which exhibits both the Principles of Uniformity and Scalability. Next-generation tools include CIDE and FeatureHouse.

FOMDD

Derivational and Refinement Relationships among Program Artifacts

Feature Oriented Model Driven Design (FOMDD) combines the ideas of AHEAD with Model Driven Design (MDD) (a.k.a. Model-Driven Architecture (MDA)). AHEAD functions capture the lockstep update of program artifacts when a feature is added to a program. But there are other functional relationships among program artifacts that express derivations. For example, the relationship between a grammar g_f and its parser source s_f is defined by a compiler-compiler tool, e.g., javacc. Similarly, the relationship between Java source s_f and its bytecode b_f is defined by the javac compiler. A commuting diagram expresses these relationships. Objects are program representations, downward arrows are derivations, and horizontal arrows are deltas. The figure to the right shows the commuting diagram for program p₃ = i•j•h = [g₃,s₃,b₃].

A fundamental property of a commuting diagram is that all paths between two objects are equivalent. For example, one way to derive the bytecode b₃ of parser p₃ (lower right object in the above figure) from grammar g_f of parser f (upper left object) is to derive the bytecode b_f and refine to b₃, while another way refines g_f to g₃, and then derive b₃:

  $\mathrm{\Delta }$bi • $\mathrm{\Delta }$bj • javac • javacc = javac • javacc • $\mathrm{\Delta }$gi • $\mathrm{\Delta }$gj

There are $(\genfrac{}{}{0ex}{}{4}{2})$ possible paths to derive the bytecode b₃ of parser p₃ from the grammar g_f of parser f. Each path represents a metaprogram whose execution synthesizes the target object (b₃) from the starting object (g_f). There is a potential optimization: traversing each arrow of a commuting diagram has a cost. The cheapest (i.e., shortest) path between two objects in a commuting diagram is a geodesic, which represents the most efficient metaprogram that produces the target object from a given object.

Note: A “cost metric” need not be a monetary value; cost may be measured in production time, peak or total memory requirements, some informal metric like “ease of explanation”, or a combination of the above (e.g., multi-objective optimization). The idea of a geodesic is quite general, and should be understood and appreciated from this more general context.

Note: It is possible for there to be m starting objects and n ending objects in a geodesic; when m=1 and n>1, this is the Directed Steiner Tree Problem, which is NP-hard.

Commuting diagrams are important for at least two reasons: (1) there is the possibility of optimizing the synthesis of artifacts (e.g., geodesics) and (2) they specify different ways of constructing a target object from a starting object. A path through a diagram corresponds to a tool chain: for an FOMDD model to be consistent, it should be proven (or demonstrated through testing) that all tool chains that map one object to another in fact yield equivalent results. (If different paths/tool chains yield different results, then either there is a bug in one or more of the tools or the FOMDD model is wrong).

Note: the above ideas were inspired by category theory.

Applications

• Network Protocols
• Extensible Database Systems
• Data Structures
• Distributed Army Fire Support Simulator
• Production System Compiler
• Graph Product Line
• Extensible Java Preprocessors
• Web Portlets
• SVG Applications

See also

• FOSD MetaModels—product lines of product lines
• FOSD Program Cubes—multi-dimensional product lines
• FOSD Feature Algebras—basic operations from which FOSD features (0-ary and 1-ary) functions are defined
• FOSD Feature Interactions—general concepts for feature interactions