5 ms·

I am not talking about limitations of find or mkdir like other commenters are. I can write a Python program that simulates rule 110 with unbounded state width

I am not talking about limitations of find or mkdir like other commenters are. I can write a Python program that simulates rule 110 with unbounded state width and unbounded iteration depth. I might not be able to execute it on any computer (it's going to run out of memory at some point), but I can still reason about it and its behavior with a (theoretical) infinite memory. After reading the blog post, I am not convinced I can write such a program using `find` and `mkdir`, since the provided example uses explicit limits for WIDTH and ITER in the program itself.

camel-cdr 2y ago
The same argument would make C non turing complete. Because the size of pointers is a compile time constant and because everything needs to have an address that puts a large, but hard limit on tape length. There are ways to argue arround that, e.g. C might be able to interface with a infinite tape file via the stantard library, and maybe strict aliasing and pointer provenance let's you create a system where bit identical pointers can be different. But the mental model most people have of C wouldn't be turing complete.
- tomsmeding 2y ago
  On the other hand, a C running on a machine with a significantly larger address space would have appropriately larger pointers. The C standard does not specify any particular pointer bitwidth. With these things together, C as a language has a decent claim to Turing-completeness.
  - camel-cdr 2y ago
    Yes, but you still configure (choose a compiler) it to a fixed size before running, that is in my mind no different than specifying a fixed tape size, like in the find + mkdir example.
  - oecumena 2y ago
    For any C program there is a number N, that depends on the program, compiler, architecture, etc., but does not depend on the program input, such that the program won't be able to access more than N bits of state at any moment in any of its possible executions. Hence, the program is equivalent to a finite state automaton.
- deredede 2y ago
  While strictly speaking true I don't think it is the same argument at all. You are talking about a restriction of the runtime (much like mkdir argument length or maximum filesystem depth), even though it leaks into the standard because standard people care about physical hardware, not theoretical ones. The WIDTH and ITER limit being actual constants that are part of the program makes all the difference compared to C pointer limitations that are part of the execution environment.
  - camel-cdr 2y ago
    The difference is very small though, you could say that WIDTH amd ITER must be defined in the execution enviroment (shell)before execution and that the rest of the code is the program, and we are at the same situation as in C.
    - deredede 2y ago
      If you define WIDTH and ITER prior to execution you are just giving arguments to your program. Maybe using the term "environment" was not the best choice; what I mean is that WIDTH and ITER are program variables that impact program behavior and output (appear in regexes etc.) whereas (most) C programs don't actually reference or depend on the pointer width (other than crashing if it's too small); it is an internal detail of the C compiler and underlying hardware that only happens to be visible to the programmer due to leaky abstractions. I don't think those are comparable.
      - horsawlarway 2y ago
        I mean... isn't this just `sizeof` in c? Honestly, I'm struggling to think of any real world code base I've worked with in C that didn't care about pointer size.
  - Joker_vD 2y ago
    But C is not Turing-complete, because its numbers (and pointers) are required to have an arbitrary limit which itself must be representable and useable in C. Python, on the other hand, has no such requirement and you could imagine an implementation that could grow indefinitely (well, until it ran out of the physical universe). C is prohibited from doing that, there is always a theoretical limit present. You can set that limit hella high, but it's still there.
- oecumena 2y ago
  C is definitely not Turing complete. The standard library provides no escape, because file sizes are also limited (due to ftell (3)), and there is no chdir in the C standard library, so the total number of files is also limited. I have a recollection of an attempt to construct a possible Turing-complete interpretation of the C standard involving recursion and va_arg, but I don't think it went anywhere.
- osmarks 2y ago
  C is indeed not Turing-complete for more or less this reason.
  - ori_b 2y ago
    Neither is the universe: we have (as far as we know) a limited number of matter and energy that can be converted to computation, which limits the size of an implementable system. Real Turing completeness is necessarily theoretical.
    - osmarks 2y ago
      Yes. C is not Turing-complete even in theory. Other languages are. It doesn't especially matter.
    - layer8 2y ago
      We don’t know that at all. It’s a sensible assumption that the universe is infinite in size, and we have no indication to the contrary. The biggest impediments are the accelerating expansion of the universe (which however isn’t fully explained and thus may not be inevitable) and the heat death, which limits time for meaningful causal interaction.
thih9 2y ago
Looks like you're treating Python as a spec; in this case I'd say we should treat mkdir+find as a spec too. Programming language implementations often have some hard limits built in. E.g.: https://peps.python.org/pep-0611/ https://peps.python.org/pep-0611/
- deredede 2y ago
  I'm treating mkdir+find as a spec. The values for WIDTH and ITER are in the program itself and will impact any implementation of mkdir and find you use, including theoretical ones.
  - thih9 2y ago
    Thanks for explaining. I missed that WIDTH and ITER are in the program.
cryptonector 2y ago
I interpreted `-maxdepth` as a safety device.
ogiekako2 2y ago
Thank you for your comment on my article. I think I've managed to fix the proof by implementing a tag system. Would you (anyone) mind reviewing my code [1][2]? The key point is using back references, which I think gave us capabilities beyond regular expressions to achieve Turing completeness. However, I'm not very familiar with tag systems, and I'm worried I might have missed something. [1] https://onecompiler.com/bash/42mux2442 https://onecompiler.com/bash/42mux2442 [2] https://onecompiler.com/bash/42mux3nf8 https://onecompiler.com/bash/42mux3nf8
- uzerfcwn 2y ago
  The definition of tag systems says "add P(x) to the end" but doesn't define x. I can't spot errors in the logic or code, though I'm not an expert either.
  - ogiekako2 2y ago
    Thank you! I have fixed the sentence and some edge case handling.
- deredede 2y ago
  I am not an expert in either tag systems (or `find`), but this seems right. Using backreferences to handle copies sounds right (it does add expressive power to regular expressions but does not give Turing completeness, afaik). I think your first example is missing the "any word of length < 2 is a halting word" condition but it is present in your second example.
  - ogiekako2 2y ago
    Thank you! I've updated the article with a link to my comment.
golol 2y ago
I think it's not that simple. It's always confusing to talk about Turing machines and the requirement of infinite memory vs. the reality of finite memory. I think "Turing completeness" is not so obvious to define rigorously and the way people use it does maybe not exactly capture the idea of "arbitrary computation" being possible. I'll try to clarify some things for myself and maybe others. First of all, recall that a dynamical system is a set X with a map f: X -> X. The evolution of the system is given by the iterated application of f. A dynamical system is finite if the set X is finite. I think it is useful to broaden this concept and define an IO-system as three sets X and I and O with a map f: I × X -> O × X. This means at every evolution step an "input" value i ∈ I has to be provided and an "output" value o ∈ O is obtained. A Turing machine m consists of a finite alphabet A of symbols and a finite IO-system h: A × S -> O × S, where O = {move left, move right, print symbol a ∈ A}. This represents how the "head" of the Turing machine updates its internal state s ∈ S when reading a symbol from the alphabet I. We call this IO-system h the head of the Turing machine. You could specify the Turing machine with the data T = (A, S, O, h). You now couple this Turing machine with another IO-system, which we call the "tape". It is either an infinite (N = ∞) tape or a finite, circular tape of length N. It has states X = {1, ..., N} × I × ... × I where the product I × ... × I has length N. It's set of inputs is the set O and its set of outputs is A. It's operation is given by a function t: O × X -> A × X, which describes the intended reaction of the type to the instructions from the head, i.e. depending on the instruction in O it either moves the "position counter" of the tape to the left, to the right, or it prints a symbol onto the tape. After it has performed this it reads the symbol at the current position and gives this output back to the head. We can now combine the head h and the tape t into a "machine" dynamical system m: X × S × O -> X × S × O where h(x, s, o) = (t(o, x)_X, h(t(o, x)_A, s)_S, h(t(o, x)_A, s)_O). This represents the evolution of the Turing machine together with the tape. We call this the [machine dynamicals system with memory N of the Turing machine T]. Definition 1. Let's say that [the dynamical system f: X -> X simulates another dynamical system g: Y -> Y] if there exists an injective map u: Y -> X such that g(y) = f(u(y)). In order to compute the evolution g(g(...(g(y))...)) we can instead compute f(u(f(u(...(f(u(y))...)) and use injectivity of u to get back a result in Y. Lemma 2. Any finite dynamical system is simulated by the machine dynamical system of some Turing machine with tape length N = 1. proof: Just set the head of the Turing machine to be the desired dynamical system and trivialize all the other objects. This is a triviality result and tells us that this is not a good attempt to investigate universality of Turing machines in a "finite memory" setting. False Hypothesis 3. There exists a universal Turing machine U in the sense that this Turing machine has the property that its machine dynamical system with infinite memory simulates the machine dynamical system with infinite memory of any other Turing machine T. As far as I know this hypothesis is false because the sense of simulation mentioned above is far too strong. At this point I think there are many definitions one can make so let's stick with the one of Alan Turing. Definition 4. We say that [the dynamical system f: X -> X simulates another dynamical system g: Y -> Y with respect to the "result" functions R: X -> {null, 0, 1} and Q: Y -> {null, 0, 1}] if there exists an injective map u: Y -> X such that the sequences Q(g^n(y)) and R((f ∘ u)^n(y)) are "result equivalent", meaning they are equal if you delete all instances of "null". We now extend the concept of a Turing machine T by adding to it a result function r: O -> {null, 0, 1}. Definition 5 (A. Turing, 1936). We say that [the Turing machine T with result function r: O -> {null, 0, 1} (N,M)-simulates another Turing machine T' with result function r': O' -> {null, 0, 1}] if the machine dynamical system of T with memory N simulates the machine dynamical system of T' with memory M, with respect to the result functions R: X × S × O -> {null, 0, 1} given by R(x, s, o) = r(o) and R': X' × S' × O' -> {null, 0, 1} given by R'(x, s, o) = r'(o). Definition 6. We say that [a Turing machine U with result function r is (N,M)-universal] if it (N,M)-simulates any other Turing machine with result function. Theorem 7 (A. Turing, 1936). There exists a (∞,∞)-universal Turing machine. Definition 8. We say that [a Turing machine U with result function r is finite-weakly universal] if for any finite M there exists some finite N such that it (N,M) simulates any other Turing machine with result function. Now it gets difficult becasue I don't actually know the answers anymore. I am pretty sure that any (∞,∞)-universal Turing machine is also finite-weakly universal. Even more so, it might be the case that finite-weak universality is equivalent to (∞,∞)-universality. Most certainly finite-weak universality is not a trivial concept and captures an interesting aspect of the concept of computation. I want to make the point that in my opinion infinite memory should not be seen as requirement in order to talk about these concepts of computation like Turing machines and universality. It is also unclear how exactly to define the "Turing completeness" of a system, as I don't think there exists a definition of Turing completeness for dynamical systems. You have to specify how you are allowed to put an input into the dynamical system at least. I think that in some sense one could use what OP found and prove a rigorous result that with `find` + `mkdir` one can somehow construct a finite-weakly universal Turing machine.

camel-cdr 2y ago

The same argument would make C non turing complete. Because the size of pointers is a compile time constant and because everything needs to have an address that puts a large, but hard limit on tape length. There are ways to argue arround that, e.g. C might be able to interface with a infinite tape file via the stantard library, and maybe strict aliasing and pointer provenance let's you create a system where bit identical pointers can be different. But the mental model most people have of C wouldn't be turing complete.

tomsmeding 2y ago
On the other hand, a C running on a machine with a significantly larger address space would have appropriately larger pointers. The C standard does not specify any particular pointer bitwidth. With these things together, C as a language has a decent claim to Turing-completeness.
- camel-cdr 2y ago
  Yes, but you still configure (choose a compiler) it to a fixed size before running, that is in my mind no different than specifying a fixed tape size, like in the find + mkdir example.
- oecumena 2y ago
  For any C program there is a number N, that depends on the program, compiler, architecture, etc., but does not depend on the program input, such that the program won't be able to access more than N bits of state at any moment in any of its possible executions. Hence, the program is equivalent to a finite state automaton.
deredede 2y ago
While strictly speaking true I don't think it is the same argument at all. You are talking about a restriction of the runtime (much like mkdir argument length or maximum filesystem depth), even though it leaks into the standard because standard people care about physical hardware, not theoretical ones. The WIDTH and ITER limit being actual constants that are part of the program makes all the difference compared to C pointer limitations that are part of the execution environment.
- camel-cdr 2y ago
  The difference is very small though, you could say that WIDTH amd ITER must be defined in the execution enviroment (shell)before execution and that the rest of the code is the program, and we are at the same situation as in C.
  - deredede 2y ago
    If you define WIDTH and ITER prior to execution you are just giving arguments to your program. Maybe using the term "environment" was not the best choice; what I mean is that WIDTH and ITER are program variables that impact program behavior and output (appear in regexes etc.) whereas (most) C programs don't actually reference or depend on the pointer width (other than crashing if it's too small); it is an internal detail of the C compiler and underlying hardware that only happens to be visible to the programmer due to leaky abstractions. I don't think those are comparable.
    - horsawlarway 2y ago
      I mean... isn't this just `sizeof` in c? Honestly, I'm struggling to think of any real world code base I've worked with in C that didn't care about pointer size.
- Joker_vD 2y ago
  But C is not Turing-complete, because its numbers (and pointers) are required to have an arbitrary limit which itself must be representable and useable in C. Python, on the other hand, has no such requirement and you could imagine an implementation that could grow indefinitely (well, until it ran out of the physical universe). C is prohibited from doing that, there is always a theoretical limit present. You can set that limit hella high, but it's still there.
oecumena 2y ago
C is definitely not Turing complete. The standard library provides no escape, because file sizes are also limited (due to ftell (3)), and there is no chdir in the C standard library, so the total number of files is also limited. I have a recollection of an attempt to construct a possible Turing-complete interpretation of the C standard involving recursion and va_arg, but I don't think it went anywhere.
osmarks 2y ago
C is indeed not Turing-complete for more or less this reason.
- ori_b 2y ago
  Neither is the universe: we have (as far as we know) a limited number of matter and energy that can be converted to computation, which limits the size of an implementable system. Real Turing completeness is necessarily theoretical.
  - osmarks 2y ago
    Yes. C is not Turing-complete even in theory. Other languages are. It doesn't especially matter.
  - layer8 2y ago
    We don’t know that at all. It’s a sensible assumption that the universe is infinite in size, and we have no indication to the contrary. The biggest impediments are the accelerating expansion of the universe (which however isn’t fully explained and thus may not be inevitable) and the heat death, which limits time for meaningful causal interaction.

thih9 2y ago

Looks like you're treating Python as a spec; in this case I'd say we should treat mkdir+find as a spec too. Programming language implementations often have some hard limits built in. E.g.: https://peps.python.org/pep-0611/ https://peps.python.org/pep-0611/

deredede 2y ago
I'm treating mkdir+find as a spec. The values for WIDTH and ITER are in the program itself and will impact any implementation of mkdir and find you use, including theoretical ones.
- thih9 2y ago
  Thanks for explaining. I missed that WIDTH and ITER are in the program.

cryptonector 2y ago

I interpreted `-maxdepth` as a safety device.

ogiekako2 2y ago

Thank you for your comment on my article. I think I've managed to fix the proof by implementing a tag system. Would you (anyone) mind reviewing my code [1][2]? The key point is using back references, which I think gave us capabilities beyond regular expressions to achieve Turing completeness. However, I'm not very familiar with tag systems, and I'm worried I might have missed something. [1] https://onecompiler.com/bash/42mux2442 https://onecompiler.com/bash/42mux2442 [2] https://onecompiler.com/bash/42mux3nf8 https://onecompiler.com/bash/42mux3nf8

uzerfcwn 2y ago
The definition of tag systems says "add P(x) to the end" but doesn't define x. I can't spot errors in the logic or code, though I'm not an expert either.
- ogiekako2 2y ago
  Thank you! I have fixed the sentence and some edge case handling.
deredede 2y ago
I am not an expert in either tag systems (or `find`), but this seems right. Using backreferences to handle copies sounds right (it does add expressive power to regular expressions but does not give Turing completeness, afaik). I think your first example is missing the "any word of length < 2 is a halting word" condition but it is present in your second example.
- ogiekako2 2y ago
  Thank you! I've updated the article with a link to my comment.

golol 2y ago

I think it's not that simple. It's always confusing to talk about Turing machines and the requirement of infinite memory vs. the reality of finite memory. I think "Turing completeness" is not so obvious to define rigorously and the way people use it does maybe not exactly capture the idea of "arbitrary computation" being possible. I'll try to clarify some things for myself and maybe others. First of all, recall that a dynamical system is a set X with a map f: X -> X. The evolution of the system is given by the iterated application of f. A dynamical system is finite if the set X is finite. I think it is useful to broaden this concept and define an IO-system as three sets X and I and O with a map f: I × X -> O × X. This means at every evolution step an "input" value i ∈ I has to be provided and an "output" value o ∈ O is obtained. A Turing machine m consists of a finite alphabet A of symbols and a finite IO-system h: A × S -> O × S, where O = {move left, move right, print symbol a ∈ A}. This represents how the "head" of the Turing machine updates its internal state s ∈ S when reading a symbol from the alphabet I. We call this IO-system h the head of the Turing machine. You could specify the Turing machine with the data T = (A, S, O, h). You now couple this Turing machine with another IO-system, which we call the "tape". It is either an infinite (N = ∞) tape or a finite, circular tape of length N. It has states X = {1, ..., N} × I × ... × I where the product I × ... × I has length N. It's set of inputs is the set O and its set of outputs is A. It's operation is given by a function t: O × X -> A × X, which describes the intended reaction of the type to the instructions from the head, i.e. depending on the instruction in O it either moves the "position counter" of the tape to the left, to the right, or it prints a symbol onto the tape. After it has performed this it reads the symbol at the current position and gives this output back to the head. We can now combine the head h and the tape t into a "machine" dynamical system m: X × S × O -> X × S × O where h(x, s, o) = (t(o, x)_X, h(t(o, x)_A, s)_S, h(t(o, x)_A, s)_O). This represents the evolution of the Turing machine together with the tape. We call this the [machine dynamicals system with memory N of the Turing machine T]. Definition 1. Let's say that [the dynamical system f: X -> X simulates another dynamical system g: Y -> Y] if there exists an injective map u: Y -> X such that g(y) = f(u(y)). In order to compute the evolution g(g(...(g(y))...)) we can instead compute f(u(f(u(...(f(u(y))...)) and use injectivity of u to get back a result in Y. Lemma 2. Any finite dynamical system is simulated by the machine dynamical system of some Turing machine with tape length N = 1. proof: Just set the head of the Turing machine to be the desired dynamical system and trivialize all the other objects. This is a triviality result and tells us that this is not a good attempt to investigate universality of Turing machines in a "finite memory" setting. False Hypothesis 3. There exists a universal Turing machine U in the sense that this Turing machine has the property that its machine dynamical system with infinite memory simulates the machine dynamical system with infinite memory of any other Turing machine T. As far as I know this hypothesis is false because the sense of simulation mentioned above is far too strong. At this point I think there are many definitions one can make so let's stick with the one of Alan Turing. Definition 4. We say that [the dynamical system f: X -> X simulates another dynamical system g: Y -> Y with respect to the "result" functions R: X -> {null, 0, 1} and Q: Y -> {null, 0, 1}] if there exists an injective map u: Y -> X such that the sequences Q(g^n(y)) and R((f ∘ u)^n(y)) are "result equivalent", meaning they are equal if you delete all instances of "null". We now extend the concept of a Turing machine T by adding to it a result function r: O -> {null, 0, 1}. Definition 5 (A. Turing, 1936). We say that [the Turing machine T with result function r: O -> {null, 0, 1} (N,M)-simulates another Turing machine T' with result function r': O' -> {null, 0, 1}] if the machine dynamical system of T with memory N simulates the machine dynamical system of T' with memory M, with respect to the result functions R: X × S × O -> {null, 0, 1} given by R(x, s, o) = r(o) and R': X' × S' × O' -> {null, 0, 1} given by R'(x, s, o) = r'(o). Definition 6. We say that [a Turing machine U with result function r is (N,M)-universal] if it (N,M)-simulates any other Turing machine with result function. Theorem 7 (A. Turing, 1936). There exists a (∞,∞)-universal Turing machine. Definition 8. We say that [a Turing machine U with result function r is finite-weakly universal] if for any finite M there exists some finite N such that it (N,M) simulates any other Turing machine with result function. Now it gets difficult becasue I don't actually know the answers anymore. I am pretty sure that any (∞,∞)-universal Turing machine is also finite-weakly universal. Even more so, it might be the case that finite-weak universality is equivalent to (∞,∞)-universality. Most certainly finite-weak universality is not a trivial concept and captures an interesting aspect of the concept of computation. I want to make the point that in my opinion infinite memory should not be seen as requirement in order to talk about these concepts of computation like Turing machines and universality. It is also unclear how exactly to define the "Turing completeness" of a system, as I don't think there exists a definition of Turing completeness for dynamical systems. You have to specify how you are allowed to put an input into the dynamical system at least. I think that in some sense one could use what OP found and prove a rigorous result that with `find` + `mkdir` one can somehow construct a finite-weakly universal Turing machine.