What is “Orthogonality”?

Question

What does \"orthogonality\" mean when talking about programming languages?

What are some examples of Orthogonality?

Answer 1

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. Furthermore, every pos- sible combination of primitives is legal and meaningful. For example, consider data types. Suppose a language has four primitive data types (integer, float, double, and character) and two type operators (array and pointer). If the two type operators can be applied to themselves and the four primitive data types, a large number of data structures can be defined. The meaning of an orthogonal language feature is independent of the context of its appearance in a program. (the word orthogonal comes from the mathematical concept of orthogonal vectors, which are independent of each other.) Orthogonality follows from a symmetry of relationships among primi- tives. A lack of orthogonality leads to exceptions to the rules of the language. For example, in a programming language that supports pointers, it should be possible to define a pointer to point to any specific type defined in the language. However, if pointers are not allowed to point to arrays, many potentially useful user-defined data structures cannot be defined. We can illustrate the use of orthogonality as a design concept by compar- ing one aspect of the assembly languages of the IBM mainframe computers and the VAX series of minicomputers. We consider a single simple situation: adding two 32-bit integer values that reside in either memory or registers and replacing one of the two values with the sum. The IBM mainframes have two instructions for this purpose, which have the forms

A Reg1, memory_cell
AR Reg1, Reg2

where Reg1 and Reg2 represent registers. The semantics of these are

Reg1 ← contents(Reg1) + contents(memory_cell)
Reg1 ← contents(Reg1) + contents(Reg2)

The VAX addition instruction for 32-bit integer values is

ADDL operand_1, operand_2

whose semantics is

operand_2 ← contents(operand_1) + contents(operand_2)

In this case, either operand can be a register or a memory cell. The VAX instruction design is orthogonal in that a single instruction can use either registers or memory cells as the operands. There are two ways to specify operands, which can be combined in all possible ways. The IBM design is not orthogonal. Only two out of four operand combinations possibilities are legal, and the two require different instructions, A and AR . The IBM design is more restricted and therefore less writable. For example, you cannot add two values and store the sum in a memory location. Furthermore, the IBM design is more difficult to learn because of the restrictions and the additional instruction. Orthogonality is closely related to simplicity: The more orthogonal the design of a language, the fewer exceptions the language rules require. Fewer exceptions mean a higher degree of regularity in the design, which makes the language easier to learn, read, and understand. Anyone who has learned a sig- nificant part of the English language can testify to the difficulty of learning its many rule exceptions (for example, i before e except after c).