zlacker

Is the unary encoding of the indices in some way optimal or could one compress the representations further using some [variable length] binary encoding, maybe at the loss of some other desirable properties?

>>danbru+(OP)
A unary encoding is certainly optimal in simplicity.

For encoding size, it would be optimal if index i occurs roughly with frequency 2^-i. In many lambda terms of practical interest, one does see higher indices occurring much less frequently, so it's not terribly far from optimal. Some compression is certainly possible; within n binding lambdas, index n could be encoded as 1^n instead of 1^n 0, but again that severely complicates the interpreter itself.

>>tromp+3r2
My first comment was worded stupidly, or at least I forgot to actually ask what I was really wondering.

I noticed that some of the program lengths ended up in expressions of lower and upper bounds. Also lambda terms represented with De Bruijn indices are essentially lists of numbers and a binary encoding could give exponentially shorter representation as compared to an unary encoding at the price of some overhead when dealing with the binary numbers which I thought might be a constant.

But I did admittedly not read the page too carefully and would probably need to refresh my knowledge to properly understand the details. The programs there are also mostly short, so a binary encoding would probably make them longer. And if it really mattered, than this is of course such an obvious thing to do, that it would certainly not have been overlooked.

>>danbru+bu3
> I noticed that some of the program lengths ended up in expressions of lower and upper bounds.

Several of the most celebrated theorems in AIT, such as Symmetry of Information, employ programs that need to interpret other programs. So constants in these theorems significantly suffer from complicated index encodings.

A binary encoding is not as simple as you might think, as the code needs to be self-delimiting. So you first need to encode the number of bits used in the binary code. And that must also be self-delimiting. As you can see, you need to fall back to a unary encoding at some point, as that is naturally self-delimiting.

An example of an efficient binary self-delimiting code can be found at [1]. Note that the program for decoding it into a Church numeral (the very best case) is already about 60% the size of the entire self-interpreter.

Many programs used in AIT proofs use only single digit indices (the self-interpreter e.g. only uses 1-5) and these would be negatively impacted by a binary encoding.

[1] https://gist.github.com/tromp/86b3184f852f65bfb814e3ab0987d8...