# What is the relation between arithmetic circuits and straight line programs?

One definition of arithmetic circuits is as follows:

An arithmetic circuit $\Phi$ over the field $\mathbb F$ and the set of variables $X$ usually, $X = \{x_1, \dots , x_n\}$) is a directed acyclic graph as follows. The vertices of $\Phi$ are called gates. Every gate in $\Phi$ of in-degree $0$ is labeled by either a variable from $X$ or a field element from $\mathbb F$. Every other gate in $\Phi$ is labeled by either $\times$ or $+$ and has in-degree $2$.

A straight-line program seems to be a succession of basic operations without any if or while statement. So it seems to be a term from the associated free algebra expressed suitably "compressed" as an acyclic graph. But a (reference to a) nice definition of straight-line programs would already help me a bit with respect to this question.

What I really want to know is whether an arithmetic circuit is a special case of a straight-line program? If yes, then why is arithmetic circuit complexity considered so important, while even finding a reference for a proper definition of straight-line programs seems to be quite challenging? (I found a definition in The Extraordinary Power of Division in Straight Line Programs, but this is a paper from 2012, and the notion itself seems to be much older.)

• +1 for reference to the Borwein Hobart paper. Not sure if it helps but Lipton posted a write up on his blog regarding that paper. – J C Jan 4 '15 at 20:15

Straight-line programs and arithmetic circuits are two equivalent ways of describing the same computational model. A straight-line program typically has the following instructions:

1. Reading the input: $t_i \gets x_j$.
2. Initialization by constants: $t_i \gets r$ for all $r$ in the ambient field.
3. Addition, subtraction, multiplication, division: $t_t \gets t_j \pm t_k$ and so on.
4. Output: $o_i \gets t_j$.

The input and output instructions aren't really necessary. The inputs are $x_1,\ldots,x_n$ and the outputs are $o_1,\ldots,o_m$.

Exercise: show that (multiple-output) arithmetic circuits and straight-line programs are equivalent: they compute the same functions with the same size complexity (if you define size in the correct way, and allow the same sets of gates and lines).

• Is a straight line program natural way to 'computationally categorize' rings? – 1.. Jan 5 '15 at 10:39
• Perhaps, though I'm not sure what computationally categorize means. – Yuval Filmus Jan 5 '15 at 11:42