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> But if you require your parens to be balanced, that's no longer a regular language From a certain pedantic perspective, it still is a regular language, since

> But if you require your parens to be balanced, that's no longer a regular language From a certain pedantic perspective, it still is a regular language, since there is a finite (implementation dependent) upper bound on nesting depth, and balanced parens with a finite upper bound of nesting is regular. OTOH, one can appeal to the linguistic distinction between competence and performance, and say that regexes are irregular in pure competence, even though all (practically usable) languages turn out to be regular once we constrain them by performance as well.

eru 1y ago
> From a certain pedantic perspective, it still is a regular language, since there is a finite (implementation dependent) upper bound on nesting depth, and balanced parens with a finite upper bound of nesting is regular. That's a silly point, because it makes all languages trivially finite, even monsters like C++. (Finity is even more of a restriction than being regular.)
- skissane 1y ago
  Is it silly? Well, if a person feels drawn to an ultrafinist philosophy of mathematics, they may feel it is important to emphasise the inherent finitude of all creations of inherently finite human minds, formalisms such as programming languages included.
  - eru 1y ago
    Oh, finitism is perfectly fine by me. I have a lot of sympathy for it! What is silly is not finitism (or ultrafinitism), but trying to apply concepts like 'regular language' or 'context free language' without modification in such a setting. I suspect you should be able to suitably alter most of these concepts to make sense in the strictly finite setting. But they are going to become a lot more complicated. For a similar flavour, you can also check out the differences and similarities between differential equations and difference equations.
    - skissane 1y ago
      Right, but the “finite setting” is the real world. Yes, if we talk about C++ as a mathematical abstraction, maybe a C++ program containing a googolplex tokens could be in a theoretical sense syntactically and semantically valid. But such a program can’t exist in the real world Let’s say a compiler has a (very plausible) implementation restriction that the source code of a program can be no longer than 2^32 bytes. Then, even if the abstract grammar is (e.g.) an unbounded context-free language, the actually accepted language has a finite upper bound on length-and it is well-known that applying a length constraint turns a context-free language into a regular language. This doesn’t require us to “to suitably alter most of these concepts to make sense in the strictly finite setting”, because formal language theory in an infinite setting permits constraining languages to be finite, and we have a very good understanding of what that the results of that are - this doesn’t require any new or different math, just applying the same math. Now, it is true that, we will say that certain algorithms only work for regular languages, not context-free - but once we impose a finite bound on length, those algorithms do actually work in principle. In practice, of course, they are likely impractically slow - but the formalism we are talking about (the Chomsky hierarchy) is based on computability (can it answer the question in a finite but unbounded amount of time), not asymptotic computational complexity (nor real-world performance, which isn’t the same thing, as e.g. galactic algorithms demonstrate)
      - eru 1y ago
        And those formalisms all become silly. Computability also becomes trivial, if you only have a finite amount of memory: either your program halts after a finite amount of time, or it enters a loop after at most 2^(number of bits in your memory) steps.
        skissane 1y ago
        > Computability also becomes trivial, if you only have a finite amount of memory I disagree. Computability isn't trivial if you have a finite amount of memory. In the real world, we have machines with finite space and time. It is far from trivial to work out what problems they can and can't solve. In very many cases, all we can say is "we don't know how to solve this problem, but for all we know there is a way of solving it which nobody has discovered yet". There are many problems for which, a century will pass from today, and we'll likely be no closer to knowing whether they are solvable than we are now. Yes, if you actually had a machine with infinite storage, and you had an eternity to wait for its computations to complete, then you could always eventually get the decisive answer to the question of whether a machine could solve a problem given some finite bound on storage and time. But that's a very unusual sense of "trivial", essentially "this problem is trivial if we assume humans are immortal beings with infinite resources"

eru 1y ago

> From a certain pedantic perspective, it still is a regular language, since there is a finite (implementation dependent) upper bound on nesting depth, and balanced parens with a finite upper bound of nesting is regular. That's a silly point, because it makes all languages trivially finite, even monsters like C++. (Finity is even more of a restriction than being regular.)

skissane 1y ago
Is it silly? Well, if a person feels drawn to an ultrafinist philosophy of mathematics, they may feel it is important to emphasise the inherent finitude of all creations of inherently finite human minds, formalisms such as programming languages included.
- eru 1y ago
  Oh, finitism is perfectly fine by me. I have a lot of sympathy for it! What is silly is not finitism (or ultrafinitism), but trying to apply concepts like 'regular language' or 'context free language' without modification in such a setting. I suspect you should be able to suitably alter most of these concepts to make sense in the strictly finite setting. But they are going to become a lot more complicated. For a similar flavour, you can also check out the differences and similarities between differential equations and difference equations.
  - skissane 1y ago
    Right, but the “finite setting” is the real world. Yes, if we talk about C++ as a mathematical abstraction, maybe a C++ program containing a googolplex tokens could be in a theoretical sense syntactically and semantically valid. But such a program can’t exist in the real world Let’s say a compiler has a (very plausible) implementation restriction that the source code of a program can be no longer than 2^32 bytes. Then, even if the abstract grammar is (e.g.) an unbounded context-free language, the actually accepted language has a finite upper bound on length-and it is well-known that applying a length constraint turns a context-free language into a regular language. This doesn’t require us to “to suitably alter most of these concepts to make sense in the strictly finite setting”, because formal language theory in an infinite setting permits constraining languages to be finite, and we have a very good understanding of what that the results of that are - this doesn’t require any new or different math, just applying the same math. Now, it is true that, we will say that certain algorithms only work for regular languages, not context-free - but once we impose a finite bound on length, those algorithms do actually work in principle. In practice, of course, they are likely impractically slow - but the formalism we are talking about (the Chomsky hierarchy) is based on computability (can it answer the question in a finite but unbounded amount of time), not asymptotic computational complexity (nor real-world performance, which isn’t the same thing, as e.g. galactic algorithms demonstrate)
    - eru 1y ago
      And those formalisms all become silly. Computability also becomes trivial, if you only have a finite amount of memory: either your program halts after a finite amount of time, or it enters a loop after at most 2^(number of bits in your memory) steps.
      - skissane 1y ago
        > Computability also becomes trivial, if you only have a finite amount of memory I disagree. Computability isn't trivial if you have a finite amount of memory. In the real world, we have machines with finite space and time. It is far from trivial to work out what problems they can and can't solve. In very many cases, all we can say is "we don't know how to solve this problem, but for all we know there is a way of solving it which nobody has discovered yet". There are many problems for which, a century will pass from today, and we'll likely be no closer to knowing whether they are solvable than we are now. Yes, if you actually had a machine with infinite storage, and you had an eternity to wait for its computations to complete, then you could always eventually get the decisive answer to the question of whether a machine could solve a problem given some finite bound on storage and time. But that's a very unusual sense of "trivial", essentially "this problem is trivial if we assume humans are immortal beings with infinite resources"