[DannyBee]

Since this may be confusing at first (why does squaring buy you anything here) - the reason squaring makes it expspace complete is, basically, squaring allows you to express an exponentially large regex in less than exponential input size.

This in turn means polynomial space in the size of the input is no longer enough to deal with the regex.

If you only allow repetition, than an exponentially large regex requires exponential input size, and thus polynomial space in the size of the input still suffices to do equivalence.

This is generally true - operators that allow you to reduce the size of the input necessary to express a regex by a complexity class will usually increase the size complexity class necessary to determine equivalence by a corresponding amount.

[less_less]

But the site does allow squaring, and in fact also general exponentiation? Like you can write "fo{2}" to match "foo", where the {2} is squaring.

[telotortium]

I don't think that's squaring, since "o" is a literal. Squaring is more like /f../ where the first and second /./ must match equal strings. Squaring is actually supported by the Perl regex /f(.)\g1/, where /\g1/ is a backreference[1] to the first match group (here just /(.)/, but could be arbitrarily large). It's easy to see how this feature requires backtracking, and therefore how it can lead to an exponential running time.

/f(.)\g1/ is equivalent to the non-backtracking regex /f(?:\x00\x00|…|aa|bb|cc|…|\U10FFFF\U10FFFF)/ - you've already made a regex over 2 million Unicode codepoints long from an input 6 bytes long. /f(..)\g1/ would create a regex 2 billion codepoints long. If you restrict the regex to Latin-1 or another 1-byte encoding, the exponent base is smaller, but the growth is still exponential.

Supporting features like backreferences is convenient, so that's why Perl has them. You can at least use backtracking to reduce the space blowup (only need linear memory to do a DFS), but it's easy to make the regex match take exponential time with a suitable match text. That's why backtracking regexes are not safe against hostile clients unless you sandbox their time and memory usage carefully.

[1] https://perldoc.perl.org/perlretut#Backreferences

[less_less]

Squaring isn't backreferences, though you can square more than a literal. An example squaring is /(foo|bar*){2}/ which is equivalent to /(foo|bar*)(foo|bar*)/, not /(foo|bar*)\g1/.

Backreferences aren't technically regular, and as you say they can take exponential time to match. But the theorem that regular expression equivalence is EXPSPACE-complete applies to real regular expressions, not just perl "regexes".

IIRC the proof is something like, given a Turing machine that runs in space S, to consider traces of the the computation, which are concatenations of S-long strings (plus a head indicator holding the machine's internal state etc). You can make a regex that matches invalid executions of the machine, which are basically of the form

/.* (foo1.{S}bar1 | foo2.{S}bar2 | ...) .*/

where foo1->bar1, foo2->bar2 etc are all the types of invalid transitions (basically, either something on the tape not next to the head changed, or the head changed in a way not indicated in the state table).

You also set rules to enforce the initial state. Then you ask whether:

/wrong initial state | invalid execution | execution that doesn't end in "accept"/

is the same regex as

/.*/

If it's the same, then there are no strings which don't match the first regex; such a string must have the right initial state, a valid execution, and end in "accept". So if the two are the same, then all valid executions end in "reject". Since you aren't constrained to allow only a single state transition rule, this actually shows NEXPSPACE hardness, but that's equal to EXPSPACE by Savitch's theorem. Anyway I could be misremembering this, it's been a while, but it doesn't require backreferences.

The squaring is required to make the {S} part without exponential blowup. Otherwise, I think the problem is only PSPACE-hard (PSPACE-complete? I don't remember).