Definition 1   Given a set , a straight-line program (SLP) is a family of functions

for some fixed . An SLP is evaluated on a tuple by recursion in the following way: is evaluated on as a function. The remaining evaluations are recursive. So denotes

and in general

The final output is denoted . In this way we treat as function from .

SLPs arrise from the multiple meansings of expressions of the sort in a some algebraic structure . First of all, one can formally treat as the word in . Secondly this can be interpreted as the actual result of this mulitplication.

In the former meaning, actually storing a word of the form as is difficult hence it is abreviated. Other examples include words such as where the values of are continually changing or even unknown. Here an SLP can encode this word in such a way that if we replace by , then the resulting new word would result simply by evaluation the SLP at the input instead of .

In the second treatment where we whish to actually evaluate and the like, we find the problem of understanding what means as a program. Certainly we may have etc. However this equivalence neglects the problem of selecting a method of computing the result. Usually an efficient method is desired. An SLP developed from simple functions such as and formally address this problem.

The term straight-line reflects the fact that evaluating an SLP can be achieved by a program which does not branch or loop so its execution is a straight-line. It is common for SLPs to be built entirely from simple functions such as or .

Because each element of an SLP is evaluated externally only on and the remaining inputs come internally from previous , , it is convient to write definitions for as taking inputs only from and implicitly allowing for the use of the outputs of previous 's.

SLPs can be defined in contexts other than semigroups including rings, modules, and polynomials. Although they arise naturally to compress computations, they are also useful in describing smaller bounds for many combinatorial theorems related to algebraic objects.

It is possible for a function to be defined equivalently by multiple SLPs and so a notion of equality of SLPs is stronger than equivalence of final outputs.

Definition 2   Two SLPs and are equivalent if and can be evaluated on the same inputs and for every input , . When and SLP is equivalent to a trivial SLP we say that is an SLP representation of .

Every function can be expressed as an SLP trivially by . However, this SLP is typically the least optimal for the actual evaluation of the output for a given input. This leads to a hierarchy imposed on equivalent SLPs based on their associated computational length.

Definition 3   If an SLP represents an algebraic expression (alternatively a word in the generators) in the semigroup then the computational length or simply length of the SLP is the maximum number of multiplications in the semigroup performed to evaluate the expression using the SLP.

It is evident that the trivial SLP of an algebraic expression has length equal to the length of the word.