Difficulties in creating strict and robust equivalence between two symbols using the Notation package

Question

I am investigating the use of the Notation package facilities to create strict the equivalence between pretty-printed symbols such as OverBar[SuperStar[af]] that formats as $\overline{af^*}$, and easy to type (but hard to read) symbols such as afstarbar.

My issue is that I fail at making two such symbols strictly equivalent.

I would like that, after typing Notation[ParsedBoxWrapper[\[Beta]]\[DoubleLongLeftRightArrow]ParsedBoxWrapper[beta]], I get the following behavior:

 In[] beta = 3 
Out[] 3

 In[] \[Beta] 
Out[] 3

 In[] \[Beta] = 0 
Out[] 0

 In[] beta     
Out[] 0

 In[] beta = 4 
Out[] 4

In[] Information[beta] 
Out[] \[Beta]
      \[Beta] = 4

It is to be noted that with Mathematica 8.0.0.0, the Notation behaves differently on Windows and Linux and on Mac, as I figured out by starting this reddit discussion, but on all platforms, I could not manage to get assignments to one symbol propagate to the other symetrically.

The behavior that I get instead of the above is:

 In[] beta = 3 
Out[] 3

 In[] \[Beta] 
Out[] 3

 In[] \[Beta] = 0 
Out[] 0

 In[] beta     
Out[] 3

 In[] beta = 4
Out[] 4

In[] Information[beta] 
Out[] \[Beta]
      \[Beta] = 4

Is there a way to get the scrict and robust equivalence between symbols instead of the assymetrical behavior shown above?

Answer 1

You appear to be looking for the functionality of $PreRead:

$PreRead = # /. "beta" -> "\[Beta]" &;

Answer 2

What you are really looking for is InputAutoReplacements.

If I understand your question correctly you are looking for a simple/quick way to input mildly complex strings of characters. In your example you want to find an easy way to input $\beta$. But there is no point in creating another symbol beta, since the 2 symbols are meant to always be "equivalent" as you say. So you need to create an InputAutoReplacements option for the notebook.

This creates a new notebook. As soon as you type beta and then something else like space or "[" or an operation symbol you will get $\beta$.

nb = CreateDocument[{}, 
       InputAutoReplacements -> {"beta" -> "\[Beta]"}];

In case you just want to add this feature to your current notebook, you can use the Option Inspector, change the InputAutoReplacements option and save the notebook, to fix the changes.

A somewhat alternative route would be to use the AddInputAlias template from the Notation package. At the end of the process you would have to type: Esc beta Esc to get $\beta$. Obviously in this simple case it does not make much sense.

To recap: if you do not want to use Esc, use the InputAutoReplacements system, if do you want to use Esc, use the AddInputAlias system.

Why doesn't the Notation template work in this case? Well, first the LHS and the RHS of the Notation template have different meanings (external form versus internal form), so even if they are linked by a DoubleLongLeftRightArrow they are NOT equivalent in a mathematical sense. In your case you want to parse beta and get $\beta$ which displays as $\beta$, so perhaps this should work better:

Notation[ParsedBoxWrapper["beta"] \[DoubleLongRightArrow] 
  ParsedBoxWrapper["\[Beta]"]]

And indeed it works as long as you leave beta without own- or down- values. But this route is not really what you want to do I guess.