What is "orthogonality"

n. (context mathematics statistics English) the property of being orthogonal

1. n. the relation of opposition between things at right angles [syn: perpendicularity, orthogonal opposition]

2. the quality of lying or intersecting at right angles

In mathematics, orthogonality is the relation of two lines at right angles to one another ( perpendicularity), and the generalization of this relation into n dimensions; and to a variety of mathematical relations thought of as describing non-overlapping, uncorrelated, or independent objects of some kind.

The concept of orthogonality has been broadly generalized in mathematics (including in the areas of mathematical functions, calculus and linear algebra), as well as in areas such as chemistry, and engineering.

Orthogonality as a property of term rewriting systems describes where the reduction rules of the system are all left-linear, that is each variable occurs only once on the left hand side of each reduction rule, and there is no overlap between them.

Orthogonal term rewriting systems have the consequent property that all reducible expressions (redexes) within a term are completely disjoint -- that is, the redexes share no common function symbol.

For example, the term rewriting system with reduction rules

is orthogonal -- it is easy to observe that each reduction rule is left-linear, and the left hand side of each reduction rule shares no function symbol in common, so there is no overlap.

Orthogonal term rewriting systems are confluent.

In computer programming, orthogonality in a programming language means that a relatively small set of primitive constructs can be combined in a relatively small number of ways to build the control and data structures of the language. The term is most-frequently used regarding assembly instruction sets, as orthogonal instruction set.

Orthogonality is an important concept, addressing how a relatively small number of components can be combined in a relatively small number of ways to get the desired results. It is associated with simplicity; the more orthogonal the design, the fewer exceptions. This makes it easier to learn, read and write programs in a programming language.[1] The meaning of an orthogonal feature is independent of context; the key parameters are symmetry and consistency (for example, a pointer is an orthogonal concept).

An example from IBM Mainframe and VAX highlights this concept. An IBM mainframe has two different instructions for adding the contents of a register to a memory cell (or another register). These statements are shown below:

A Reg1, memory_cell AR Reg1, Reg2 In the first case, the contents of Reg1 are added to the contents of a memory cell; the result is stored in Reg1. In the second case, the contents of Reg1 are added to the contents of another register (Reg2) and the result is stored in Reg1.

In contrast to the above set of statements, VAX has only one statement for addition:

ADDL operand1, operand2

In this case the two operands (operand1 and operand2) can be registers, memory cells, or a combination of both; the instruction adds the contents of operand1 to the contents of operand2, storing the result in operand1.

VAX’s instruction for addition is more orthogonal than the instructions provided by IBM; hence, it is easier for the programmer to remember (and use) the one provided by VAX.

The design of C language may be examined from the perspective of orthogonality. The C language is somewhat inconsistent in its treatment of concepts and language structure, making it difficult for the user to learn (and use) the language. Examples of exceptions follow:

Structures (but not arrays) may be returned from a function. An array can be returned if it is inside a structure. A member of a structure can be any data type (except void), or the structure of the same type. An array element can be any data type (except void). Everything is passed by value (except arrays). Void can be used as a type in a structure, but a variable of this type cannot be declared in a function.