Are Turing Machines Programmable?


Turing machines will also be represented by easy strings:


Much less compactly nonetheless more transparently, they’ll even be represented as tables:

  A B C D E
0 1RB 1RC 1RD 1LA 1RH
1 1LC 1RB 0LE 1LD 0LA

What’s the character of these representations? Are they more like schematics of sure discrete devices, or are they more like applications for a computer? This amounts to asking: what precisely is a Turing “machine”? Is it presupposed to be like for sure this form of mature-timey online sport consoles that only plays one sport, or is it presupposed to be like a single online sport console with a library of swappable video games? And what about a program for executing these representations? Is one of these program a simulator, or is it itself a Turing machine?

In a contrivance, it doesn’t subject. The variation between “program” and “machine” is “correct” a subject of standpoint1, and finally it’s “correct” a subject of terminology. But I’m a stickler for terminology, and pondering via this ask led me to a dramatically faster Turing machine implementation. The leisure of this put up will jog via this line of pondering.

At this level, readers who comprise no longer already completed so also can honest aloof halt and put in pressure a program for operating Turing machines. It goes to also honest aloof gather Turing machine strings or tables and then lift out regardless of is prescribed. There are a diversity of a minute diversified definitions for Turing machines2, nonetheless the critical points aren’t terribly critical.

Whatever language is inclined, nonetheless it’s structured, regardless of bells and whistles are included, one of these program will ultimately boil down to common sense like this:

while explain != HALT: 
    color, shift, explain = table[state][tape[position]]
    tape[position] = color
    plot += shift

This form of program is incessantly described as a “Turing machine simulator”. This makes it sound just like the program is simulating a portion of hardware, like an elevator simulator or a pinball machine simulator. On this be taught about, a string or table just like the one above is a schematic description, and it describes a single Turing machine that does precisely one thing, namely whatever the schematic says it does. Working a search over some field of strings amounts to simulating a field of sure machines.

Perchance it’s because I’m no longer a hardware particular person, nonetheless I fetch this level of be taught about and its connected terminology unfamiliar. Once I prance a search over a list of Turing machine strings, I feel about a single instrument successively loading and operating a list of applications. On this be taught about, the program with while explain != HALT is the Turing machine, and the strings are applications telling that machine what to serve out. That’s how I take into memoir Turing machines, and the terminology in my applications (like variable and purpose names) mostly reflects that.

So a “Turing machine” is either a instrument to be simulated or a program to be completed. In either case, our access to the “machine” is mediated via utilizing one other program. What would it suggest to comprise inform access to a Turing machine?

This “inform access” is qualified for 2 reasons. The first is velocity. Working a program to prance one other program is clearly slower than correct operating the program straight. When you happen to love to comprise to prance via an extended list of Turing machine strings looking out for something explicit (like a Busy Beaver), you can’t give you the money for to prance a program that prance applications. You’ve got got to prance the applications straight.3

The 2nd reason is that simulation stifles the spirit of the program. A Turing machine representation describes some computational course of. That course of will also be simulated with the while loop mentioned above, nonetheless that’s no longer what the outline says to serve out. What precisely it says to serve out will also be out of the ordinary to train, since Turing machines are able to arbitrary administration drift. The simulation loop obscures the ravishing nature of the Turing machine being prance by imposing on it a structure that will also be understood by programmers. If we desire to listen to what the computational processs has to train, the Turing machine comprise to be free to talk in its devour verbalize.4

So, the Turing machine will be applied straight. What language comprise to be inclined? A serious requirement is that the language comprise so as to enhance arbitrary administration drift, and that narrows the selections down seriously. I feel about it’d be possible to explicit a Turing machine’s compuational course of straight in Plot or some diversified language with tail call elimination, nonetheless for maximum velocity I went with C. C points the worthy-maligned goto operator, and that makes it trivial to transcribe a Turing machine into executable C code.

For occasion, preserve mumble of the Turing machine mentioned above: 1RB 1LC 1RC 1RB 1RD 0LE 1LA 1LD 1RH 0LA. Starting with a easy tape, it runs for 47,176,870 steps sooner than halting. That’s a ponderous computation, and on my outdated college pc it takes my Python simulator 45 seconds to prance it.5

Here is the design in which it appears to be like to be like in C, with just a few well-liked metrics captured:


#account for PROG "1RB 1LC 1RC 1RB 1RD 0LE 1LA 1LD 1RH 0LA"

#account for TAPE_LEN ((100000 2) + 10)

int POS = TAPE_LEN / 2;

#account for ZERO_TAPE for (int i = 0; i < TAPE_LEN; i++) {TAPE[i] = 0;}

#define L POS--
#define R POS++

#define INSTRUCTION(c0, s0, t0, c1, s1, t1)     
  if (TAPE[POS])                                
    {TAPE[POS] = c1; s1; goto t1;}              
    {TAPE[POS] = c0; s0; goto t0;}

int XX, AA, BB, CC, DD, EE;

#define ZERO_COUNTS XX = AA = BB = CC = DD = EE = 0;

int main (void) {

  INSTRUCTION(1, R, B, 1, L, C);

  INSTRUCTION(1, R, C, 1, R, B);

  INSTRUCTION(1, R, D, 0, L, E);

  INSTRUCTION(1, L, A, 1, L, D);

  INSTRUCTION(1, R, H, 0, L, A);

  printf("%s | %d | %d %d %d %d %dn", PROG, XX, AA, BB, CC, DD, EE);

On my feeble laptop, this takes half a second to compile and run, which is quite an improvement.

Fast though it may be, that code is not general-puprose. It runs exactly one program, or equivalently, it models exactly one machine. Either way, it isn’t all that useful. How can a whole list of Turing machines be run in this manner?

My first idea was to mass-produce single-purpose machines as follows. Write a Python script containing a template of the C program above with holes for the instructions that reads in and parses a Turing machine string, fills in the holes, and spits out a complete C program. Dump that output into a temp file, pass the file to the C compiler, then run the resulting executable. Do that over a list of Turing machine strings. Coordinate the process with a simple shell script:

cat tm-list.txt | while read tm_string; do
    python "$tm_string" > temp.c
    cc temp.c -o temp

This notion sounded factual in my head, nonetheless grew to turn into out to be a spectacular failure. Inside an hour, my pc bought scorching, like I-would-comprise-sustained-injuries-if-it-had-been-on-my-lap scorching, and for some reason the fan wasn’t turning on. It used to be also leisurely, so an expedient quantity of effort used to be being expended with nothing to mumble for it.

Pretty than a single-reason machine with some mounted directions burned into it or regardless of, what if the directions had been configurable? There shall be correct one machine, and it’ll be programmed no longer to simulate one other machine, nonetheless to serve out correct precisely what that machine would lift out. The mental describe I comprise is like for sure this form of mountainous early computers with a bunch of toggles and switches, the form of machine that used to be customarily operated by females.6

It isn’t out of the ordinary to glance how the program above shall be generalized, nonetheless there could be a technical snag when enforcing it in C. In customary C, goto targets (or “labels”) comprise to be identified at bring collectively time. But the Turing machine to be completed gained’t be identified till runtime, and so neither will its administration drift. One technique to procure round this constraint is to use GCC’s labels-as-values extension.7 This permits for passing round label addresses as values, striking them into variables, and quite a lot of others. With that and a minute input handling, a programmable Turing machine is understated:


#account for TAPE_LEN ((100000 2) + 10)

unsigned int POS;
unsigned int TAPE[TAPE_LEN];

#account for ZERO_TAPE                               
  POS = TAPE_LEN / 2;                           
  for (int i = 0; i < TAPE_LEN; i++) {          
    TAPE[i] = 0;                                

#account for L POS--;
#account for R POS++;

#account for ACTION(c, s, t) {                       
    TAPE[POS] = c - 48;                         
    if (s - 76) { R } else { L };               
    goto *dispatch[t - 65];                     

#account for INSTRUCTION(c0, s0, t0, c1, s1, t1)     
  if (TAPE[POS])                                
    ACTION(c1, s1, t1)                          
      ACTION(c0, s0, t0)

unsigned int XX, AA, BB, CC, DD, EE;
unsigned int PP = 0;

#account for ZERO_COUNTS XX = AA = BB = CC = DD = EE = 0; PP++;
#account for INCREMENT(COUNT) XX++; COUNT++;

int c0, c1, c2, c3, c4, c5,
  c6, c7, c8, c9, c10, c11,
  c12, c13, c14, c15, c16, c17,
  c18, c19, c20, c21, c22, c23,
  c24, c25, c26, c27, c28, c29;

#account for READ(VAR) if ((VAR = getc(stdin)) == EOF) goto EXIT;

#account for LOAD_PROGRAM                                                
  READ(c0); READ(c1); READ(c2); READ(c3); READ(c4); READ(c5);       
  READ(c6); READ(c7); READ(c8); READ(c9); READ(c10); READ(c11);     
  READ(c12); READ(c13); READ(c14); READ(c15); READ(c16); READ(c17); 
  READ(c18); READ(c19); READ(c20); READ(c21); READ(c22); READ(c23); 
  READ(c24); READ(c25); READ(c26); READ(c27); READ(c28); READ(c29); 

int major (void) {
  static void* dispatch[] = { &&A, &&B, &&C, &&D, &&E, &&F, &&G, &&H };


  INSTRUCTION(c0, c1, c2, c3, c4, c5);

  INSTRUCTION(c6, c7, c8, c9, c10, c11);

  INSTRUCTION(c12, c13, c14, c15, c16, c17);

  INSTRUCTION(c18, c19, c20, c21, c22, c23);

  INSTRUCTION(c24, c25, c26, c27, c28, c29);

  printf("%d | %d | %c%c%c %c%c%c %c%c%c %c%c%c %c%c%c %c%c%c %c%c%c %c%c%c %c%c%c %c%c%c | %d %d %d %d %dn",
         PP, XX,
         c0, c1, c2, c3, c4, c5,
         c6, c7, c8, c9, c10, c11,
         c12, c13, c14, c15, c16, c17,
         c18, c19, c20, c21, c22, c23,
         c24, c25, c26, c27, c28, c29,
         AA, BB, CC, DD, EE);



  goto EXIT;

On my outdated college pc, this thing burns via Pascal Michel’s list of the 23 longest-operating 5-explain Turing machines in below 4 seconds.

The programmable Turing machine mannequin is both faster than the simulator mannequin and affords a more legit expression of computational processes. I due to this reality lift out that Turing machines are programmable, and the acceptable name for Turing machine representations is “program”.

  1. What format does the programmable Turing machine demand for its applications?
  2. The programmable Turing machine only runs 5-explain 2-color applications. Can or no longer it is extended to handle diversified cases?
  3. What precisely went inappropriate with the “mass production” contrivance? May maybe maybe or no longer it is salvaged into something workable?
  4. Why did the “mass production” contrivance develop the writer’s pc so scorching?
  5. Is it moral to put in writing nonstandard C?
  6. Why doesn’t customary C allow striking labels into variables?
  1. Implement 1RB 1LC 1RC 1RB 1RD 0LE 1LA 1LD 1RH 0LA straight (that’s, with out any mediating structure) in Plot.
  2. Implement the programmable Turing machine in Plot.
  3. Implement a Turing machine simulator in Python that’s faster than the writer’s.

1Martin Davis build it this contrivance:

Earlier than Turing the approved supposition used to be that in facing such machines the three classes, machine, program, and recordsdata, had been fully separate entities. The machine used to be a bodily object; at the present time we would call it hardware. The program used to be the notion for doing a computation, perchance embodied in punched cards or connections of cables in a plugboard. At final, the suggestions used to be the numerical input. Turing’s universal machine showed that the distinctness of these three classes is an phantasm. A Turing machine is first and foremost envisioned as a machine with mechanical parts, hardware. But its code on the tape of the universal machine functions as a program, detailing the directions to the universal machine wished for the acceptable computation to be applied. At final, the universal machine in its step-by-step actions sees the digits of a machine code as correct more recordsdata to be worked on. This fluidity amongst these three ideas is key to up-to-the-minute computer be aware.

2 In particular, there are the so-called “4-tuple” and “5-tuple” definitions. The 5-tuple definition is the customary for the Busy Beaver opponents, and it’s what’s inclined in this put up.

3 For sure, a Python program is prance by an interpreter, so a simulator written in Python is a program that runs a program that runs a program. The same goes for any “interpreted language”.

4 This reason will only be compelling to you whenever you are some form of hippie weirdo, like me.

5 If anyone has a approved-reason Turing machine simulator written in Python that may maybe well lift out better, I’d love to glance it.

6 Leer High Secret Rosies: The Female “Computers” of WWII.


7 In customary C, this same common sense shall be expressed with a switch assertion, nonetheless that’s a minute too stop to the mediative simulation contrivance for my style. It’s also slower.

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