> I wonder what should be the most likely computational complexity of:
> 
>      $sequence[$N]

Highly dependent on the circumstances.

Let's assume $N is known statically to be an integer.

Let's assume Saxon 8.2.

If there is only one reference to $sequence, and it isn't in a loop, then
the expression from which $sequence is calculated is effectively inlined;
this expression is then evaluated iteratively, and execution is terminated
when the N'th item is reached.

If there is more than one reference to $sequence, or if the reference is in
a loop, then the value of $sequence is calculated progressively: any
reference to $sequence[$N] ensures that at least $N items of $sequence have
been evaluated, and then returns the $N'th item by calling Java's
ArrayList.get($N-1), which I believe executes in constant time.

> Another question is whether the functions on sequences are faster that
> manipulating them "by hand".

I think the answer is likely to be: sometimes yes, sometimes no. Sometimes
Saxon may be able to exploit knowledge that's not available to the user,
sometimes the user may be able to exploit knowledge that's not available to
Saxon.

> 
> One translation in more practical terms: would it be realistic to try
> to perform binary search in a sorted sequence?

Yes, I think that's a reasonable thing to attempt.

Michael Kay
http://www.saxonica.com/