straight-line program

Definition 1.

Given a set S, a straight-line program (SLP) is a family of functions F={fi:1im}


for some fixed nN. An SLP is evaluated on a tuple (s1,,sn) by recursion in the following way: f1 is evaluated on (s1,,sn) as a function. The remaining evaluations are recursive. So f2(s1,,sn) denotes


and in general


The final output fm(s1,,sn) is denoted F(s1,,sn). In this way we treat F as function from SnS.

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

In the former meaning, actually storing a word of the form an as aa is difficult hence it is abreviated. Other examples include words such as a10(bc)6 where the values of a,b,c are continually changing or even unknown. Here an SLP can encode this word in such a way that if we replace a by bc, then the resulting new word would result simply by evaluation the SLP at the input (bc,b,c) instead of (a,b,c).

In the second treatment where we whish to actually evaluate an and the like, we find the problem of understanding what an means as a program. Certainly we may have a4=(aa)(aa)=a(a(aa)) 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 functionsMathworldPlanetmath such as f(x)=x2 and f(x,y)=x+y 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 f(x,y)=x+y or f(x)=x×x.

Because each element fi of an SLP is evaluated externally only on s1,,sn and the remaining inputs come internally from previous fj, 1ji, it is convient to write definitions for fi as taking inputs only from (s1,,sn) and implicitly allowing for the use of the outputs of previous fj’s.

SLPs can be defined in contexts other than semigroupsPlanetmathPlanetmath including rings, modules, and polynomialsMathworldPlanetmathPlanetmath. 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 f:SnS 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 F={fi:1im} and G={gi:1ik} are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath if F and G can be evaluated on the same inputs and for every input (s1,,sn), F(s1,,sn)=G(s1,,sn). When and SLP F is equivalent to a trivial SLP {f:SnS} we say that F is an SLP representation of f.

Every function f:SnS can be expressed as an SLP trivially by ={f}. 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 generatorsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath) in the semigroup S 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.

Title straight-line program
Canonical name StraightlineProgram
Date of creation 2013-03-22 16:16:02
Last modified on 2013-03-22 16:16:02
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 10
Author Algeboy (12884)
Entry type Definition
Classification msc 08A99
Classification msc 20A05
Classification msc 20-00
Related topic word
Related topic Word
Defines straight-line program
Defines SLP
Defines SLP representation