# Linear Arithmetic - why calling it linear?

This might sound like a stupid question, but I could not figure out, why Linear Arithmetic is called linear.

Linear Arithmetic often comes in play when talking about SMT Solving. For example, in this PDF linear arithmetic is defined as formulas using =, <=, < within the atoms. Somethink like 3*x_1 + 2*x_2 ≤ 5*x_3 is a linear arithmetic formula.

Because of which property is this arithmetic called linear?

-
This arithmetic preserves the linearity of the atoms. en.wikipedia.org/wiki/Linearity –  vadim123 Jul 3 '13 at 14:03

## 2 Answers

It seems that linear in that document is related to linear programming, which is optimization with linear constraints.

-
Yes, a point made explicit in Sec. 5.1.1 of the PDF, Solvers for Linear Arithmetic. Apparently the term linear arithmetic is being used to pose (first order polynomial) constraints, whose satisfiability then constitute decision problems. –  hardmath Jul 3 '13 at 14:10

A function $F(x)$ is linear when $F(ax+by)=aF(x)+bF(y)$. This means that solutions to $F(x)=0$ are linear spaces and solutions to $F(x)\le 0$ are half spaces. The point, here, is that all your formulas are constructed to be linear.

-
It seems more apt to construe "linear" as referring to first-order polynomials, rather than to linearity of a function on a vector space. Affine functions, in particular, are allowed in this linear arithmetic. –  hardmath Jul 3 '13 at 14:12
Well, but why first order polynomials are called linear, after all? –  Emanuele Paolini Jul 3 '13 at 14:13