Hacker News new | comments | show | ask | jobs | submit login

> No real-world thing can actually be Turing complete

But when you say Turing-complete, I assume something along the lines of "a programming language can be built on top of it and you can create interactive applications using such a programming language instead of javascript".

It seems like that's not the case at all.




Everyone's being a little cagey about what Turing Completeness really means, in layman's terms, because I think the assumption is that if you're here, you should already have a deep understanding of it.

So no, as others have said, you are completely off base as to what the significance of being Turing complete is. What Turing complete means is that given an infinite amount of memory and time, a language can simulate anything that is computable. (I won't get into what computable means, for that you will need to read and reread the relevant Wikipedia articles, along with a good book on Theory of Computation and probably a good one on Abstract Algebra.)

But the real significance of this is that you cannot write a program that can tell you meaningful facts about the behavior of an arbitrary Turing machine. For any program you write to determine the behavior of HTML + CSS, I could write a piece of HTML + CSS that your program will be unable to say anything about.

However, this particular proof actually only shows HTML + CSS is equivalent to a Linear Bounded Automaton, which unlike a Turing Machine can be predicted. And strictly speaking, even if you have a Turing-machine-equivalent language like C++, you can theoretically make some predictions about programs provided you can bound its memory, i.e., on a machine with less than 4 GB of memory, it is equivalent to a Linear Bounded Automaton. Now, of course actually writing a second program to do this may be impossible to do before the heat death of the universe, because Linear Bounded Automatons are still very powerful computing objects.


"But the real significance of this is that you cannot write a program that can tell you meaningful facts about the behavior of an arbitrary Turing machine. For any program you write to determine the behavior of HTML + CSS, I can write a piece of HTML + CSS that your program will be unable to say anything about."

Sorry, but this is false and for similar reasons is not reasonably considered Turing complete. The encoding presented here produces, for any particular HTML + CSS, a finite number of cells of an automata (and not only that, the number of cells is linear in code size). That means that deciding any property about a given piece of code amounts to search over a finite space.

This is very different from true Turing completeness where a small set of rules might require unbounded space and computation to decide, and in general will not even be decidable. Now, the author of this code says in a parent post that we can't build a real Turing machine. That's true, but doesn't mean that we can't specify a real Turing complete process; it just means that for some executions we have to give up on running them.

In some cases, it makes sense to consider languages that are not actually Turing complete as approximately Turing complete. For example, if you take the C programming language, with a 64 bit memory space, then again there's technically only a finite space of possible program states. But in that case, you can have a 100 line C program that can generate 2^64 possible program states. Turing completeness is a reasonable approximation. If you had to allocate every bit of memory explicitly with a unique line of C code, it would no longer be very reasonable to consider C approximately Turing complete.


Yes, you're right, I've edited my post a bit to make it clearer that HTML+CSS is not Turing complete.


What Turing complete means is that given an infinite amount of memory and time, a language can simulate anything that is computable.

Change "simulate" to "compute" and I agree fully. I don't think it's necessary to introduce the concept of simulation. If a language (or process, mechanism or whatever) is Turing Complete, then you can use it to express all things computable. Given sufficient resources, that expression could actually do the computation.

edit: I recalled in a sibling post why simulation comes up, and I think it's a source of confusion. The easiest way to prove a formalized process is Turing complete is to simulate a Turing machine. Since a Turing machine can compute all that is computable, and your process can simulate a Turing machine, your process can compute all that is computable. Hence, it is Turing complete.


Looking through the Wikipedia page for LBAs, they're a Turing machine where the tape is limited to a linear function of the size of the input. This is still beyond what realistic computers are since the tape would still need to be infinite to hold as big of an input as you want to feed into it.

Aren't practical computers just simply Finite State Machines?


Aren't you confusing computability with Godel's incompleteness theorem?


When I say "Turing complete", I mean Turing equivalent to a Turing machine. Since there are infinitely many Turing machines but I'm pretty sure humanity has access to a finite amount of state. I mean it strictly in the theoretical sense.


>I'm pretty sure humanity has access to a finite amount of state

There is at least one man out there who isn't:

http://en.wikipedia.org/wiki/Omega_point_(Tipler)#The_Omega_...

I doubt I'll be drinking that koolaid to be honest.


Then you assumed a falsehood. Turing completeness is a mathematical concept that has nothing to do with interactive applications.


No, mathematically nothing is "Turing Complete". The only usefulness of saying that something is "Turing Complete" is in knowing that you can treat it as if it was a normal computer for all intents and purposes.

So mathematically a laptop is not Turing Complete, but for all practical purposes, it is.


Nope. The common usefulness is to determine whether a given computation-describing system reaches the threshold of universal computability. Modulo finite size is assumed, else there is no usefulness of the concept given finite resources.

For example, SKI combinators are Turing-equivalent. Regular expressions aren't. The lambda-calculus is. Context-independent grammars probably are. Most data description languages are not.

This is worthy of an HN post because a style-description language would not normally be expected to be Turing-equivalent.

When it became widely known that the C++ template system was Turing-equivalent, it was a shock to many. These meant that the C++ compiler code itself could be exploited to compute arbitrary things such as factorials, or shortest-paths in graphs using A*.

Actually, as the SKI system demonstrates, the threshold is quite low. The core needed components are just "constant" or "if" (they are often equivalent), and "repeat" (equivalent to both "iterate" and "recurse").


I could be mistaken, but I think there's a fundamental distinction between the computing power required to recognize a language and the kinds of functions a language can compute. AFAIK (which isn't much), usually the distinction between regular, context-free, context-sensitive and recursively-enumerable languages turns on the power of the recognizer rather than the power of the language to express functions. In other words, the languages are limited in their syntactic expressivity but not their semantic expressivity. I've always understood the latter to be the key to determining the computational power. Take lambda calculus as an example. It is Turing complete, but it is amenable to a context-free syntactic implementation (see http://www.soe.ucsc.edu/classes/cmps112/Spring03/readings/la...). For C++, I always thought it was that not only is it capable of expressing semantically any computable function, but that a [EDIT: Turing Complete] parser was required to parse C++ strings.


You are right that the power necessary to parse a language is completely different from the power of the language itself.

There is however a different correspondence between recognizing things and computing functions. If we encode a function as taking a bit string as input and generating a bit string as output, we can turn it into a series of recognition problems as follows: "recognize given S as input, whether bit i of f(S) is 1".

Given all these recognizers, we can compute the function, and given a method to compute the function, we can implement the recognizers.

That is, you can reformulate the question "is f computable" to a series of language recognition problems.


>Context-independent grammars probably are.

No they are not. Recursively Enumerable grammars are however. A context free grammar can't be turning complete because it can be decided with only a PDA.


> because a style-description language would not normally be > expected to be Turing-equivalent.

Wouldn't it? XSLT, the only other style-description language in anything like common use on the web, has been Turing-complete for a while now (possibly since it started existing; I'm not familiar with the history)...


XSLT is the transformation part of XSL (hence the T), meant for transforming XML into other formats, including things like XSL-FO which is the formatting part of XSL. As it is a general purpose language (with iteration, recursion and conditionals) it's not that surprising that it is Turing-equivalent.

XSL-FO is the formatting part of XSL (i.e. the bit most like CSS) - I'd be pretty surprised if this was shown to be Turing-equivalent.


XSLT performs transformations (i.e., takes input and produces output). This sounds to me like a "program" so I wouldn't be surprised that a language like XSLT might be Turing complete.

HTML and CSS are much more "static" in flavor (hence the OP's use of the term "descriptive") and so it might be more unexpected that they can perform universal computation.


The concept of Turing Complete does not apply to a laptop because the laptop itself is not a means to express computation. It is a means to execute computations. There is a difference. Turing Complete is a statement about the computational generality of languages. (Or, even more general, formalized processes.)

What you mean to say is that a laptop is not an actual Turing machine, but for all intents and purposes, it is. On that point, I agree.


Also incorrect. Counterexample: Turing machines are Turing complete. You are correct that a laptop is not Turing complete, though.


Seriously? It's obvious I meant "nothing real is mathematically Turing complete".


I know, I answered (incorrect) nitpicking with (correct) nitpicking.


There's plenty of things that are Turing Complete but that shouldn't be used for interactive applications.

http://en.wikipedia.org/wiki/Brainfuck

http://en.wikipedia.org/wiki/Template_metaprogramming

http://en.wikipedia.org/wiki/Turing_machine

and now apparently HTML + CSS



Is it not possible to write a C compiler that outputs Brainfuck code that can be executed inside a Brainfuck interpreter?


What people usually mean by "Turing complete" is "would be Turing complete if it were infinite." That does imply that a programming language can be built on top of it, and that programming language is fully capable of performing any given algorithm.

It says nothing about interactivity, which is a function of ability to (directly or indirectly) access I/O hardware.




Guidelines | FAQ | Support | API | Security | Lists | Bookmarklet | DMCA | Apply to YC | Contact

Search: