Arithmetic Indicators
Idea
Arithmetic indicators are
elementary functions that simulate
indicator functions.
[1][1] This article uses Iverson brackets, $[\ \cdot\ ]$, to notate indicator functions. I.e; for any proposition, $\phi$, we have $$[\phi]=\begin{cases}1 &: \phi,\\0 &: \neg\phi.\end{cases}$$
List of arithmetic indicators
In combination with eachother, arithmetic indicators can perform a wide variety of operations. They are usually computationally inefficient, however their utility is in their novelty, ability to have their existence stand as explicit examples / counter-examples, and for the allowance of
turing-completeness of a system which is limited to elementary operations, like most calculators.
Logic
- $[\top]=1$.
- $[\bot]=0$.
- $[p \wedge q]=[p][q]$.
- $[\neg p]=1-[p]$.
- $[p\vee q]=[p]+[q]-[p][q]$.
Equalities and inequalities
- $[a=b]=\left\lfloor\frac{1}{1+|a-b|}\right\rfloor$ .
- Alternatively, $[a=b]=0^{|a-b|}$ if $0^0:=1$,
- or $[a=b]=\left\lfloor(1+\epsilon)^{-|a-b|}\right\rfloor$ (for any $\epsilon>0$)..
- $[a\leq b]=\left\lfloor\frac{2}{\pi}\cot^{-1}(a-b)\right\rfloor$.
- Alternatively, $[a\leq b]=\left\lfloor\frac{2}{1+(1+\epsilon)^{a-b}}\right\rfloor$ (for any $\epsilon>0$).
- $[a > b]=\left\lceil\frac{2}{\pi}\tan^{-1}(a-b)\right\rceil$.
Number theory
- $[p\text{ is prime}]=[\sigma_0(p)=2]$, where $\sigma_0$ is the divisor function.
Articles referencing here
