Turing Machines – what they are, what they aren't.

It’s the anniversary of the birth of Alan Turing. So there’s a ton of people writing commemorative stories. And naturally, a ton of those stories get it wrong. And they get it wrong in a very sad way.

Of course, when you talk about Turing, you talk about Turing machines. Despite the fact that Turing did lots of stuff besides just that machine, it’s always the machine that people focus on. Partly that’s laziness – the machine has his name after all, so it’s one of the first things you find if you Google Turing. It’s also the easiest thing to write about.

What do they say about the Turing machine? It’ “the simplest computing device”. It’s the “basis for modern computers”. It’s “the theoretical model for the microchips in your laptop”. It’s the “mathematical description of your computer”. None of those things are true. And they all both over and under-state what Turing really did. In terms of modern computers, the Turing machine’s contribution to the design of real computers is negligible if not non-existent. But his contrubition to our understanding of what computers can do, and our understanding of how mathematics really works – they’re far, far more significant than the architecture of any single machine.

The Turing machine isn’t a model of real computers. The computer that you’re using to read this has absolutely nothing to do with the Turing machine. As a real device, the turing machine is absolutely terrible.

The turing machine is a mathematical model not of computers, but of computation. That’s a really important distinction. The Turing machine is an easy to understand model of a computing device. It’s definitely not the simplest model. There are simpler computing devices (for example, I think that the rule 111 machine is simpler) – but their simplicitly makes them harder to understand.

The reason that the Turing machine is so important comes down to two important facts. First, which machine you use to talk about computation doesn’t matter. There’s a limit to what a mechanical device can do. There are lots of machines out there – but ultimately, no machine can go past the limit. Any machine that can reach that limit is, for the purposes of the theory of computation, pretty much the same. When we talk about studying computation, what we’re talking about is the set of things that can be done by a machine – not by a specific machine, but by any machine. The specific choice of machine isn’t important. And that’s the point: computation is computation. That’s what Turing figured out.

The Turing machine is a brilliant creation. It’s a simple machine. It’s really easy to understand. And it’s easy to tweak – that is, it’s easy to do experiments where you can modify the machine, and see what effect it has.

So let’s take a step back, and see: what is a Turing machine?

The Turing machine is a very simple kind of theoretical computing device. In fact, it’s almost downright trivial. But according to everything that we know and understand about computation, this trivial device is capable of any computation that can be performed by any other computing device.

The basic idea of the Turing machine is very simple. It’s a machine that runs on top of a tape, which is made up of a long series of little cells, each of which has a single character written on it. The machine is a read/write head that moves over the tape, and which can store a little bit of information. Each step, the machine looks at the symbol on the cell under the tape head, and based on what it sees there, and whatever little bit of information it has stored, it decides what to do. The things that it can do are change the information it has store, write a new symbol onto the current tape cell, and move one cell left or right.

That’s really it. People who like to make computing sound impressive often have very complicated explanations of it – but really, that’s all there is to it. The point of it was to be simple – and simple it certainly is. And yet, it can do anything that’s computable.

To really understand how that trivial machine can do computations, it helps to be a bit formal. In formal terms, we talk about a turing machine as a tuple: (S, s0, A, T), where:

  • S is a finite list of possible states that the machine can be in. The state is the information that the machine can store in the head to make decisions. It’s a very limited amount of information; you can think of it as either a specific list of states, or a small set of small numbers. For most Turing machines, we’ll use it as a specific list of states. (You’ll see what I mean in a minute.)
  • s0S is the initial state – the state that the machine will be in when it starts a computation.
  • A is the machine’s alphabet, which is the set of symbols which can appear on the machine’s tape.
  • T is the machines transition function. This is the real heart of the machine, where the computation is defined. It’s a function from the machine state and the alphabet character on the current tape cell to the action that the machine should take. The action is a triple consisting of a new value for the machine’s state, a character to write onto the current tape cell, and a direction to move the tape head – either left or right.

So, for example, let’s look at a simple machine. This is one of the classic examples: a Turing machine that does subtraction using unary numbers. A unary number “N” is written as a series of N 1s. For the program, we’ll give the machine an input in the format “N-M=” written onto the tape; after running the machine, the tape will contain the value of M subtracted from N. So, for example, we could use “1111-11=” as an input; the output would be “11”.

The alphabet is the characters “1”, ” ” (blank space), “-” (minus sign), and “=” (equal sign). The machine has four states: {“scanright”, “eraseone”, “subone”, “skip”}. It starts in the state “scanright”. It’s transition function is given by the following table:

FromState Symbol ToState WriteChar Dir
scanright space scanright space right
scanright 1 scanright 1 right
scanright minus scanright minus right
scanright equal eraseone space left
eraseone 1 subone equal left
eraseone minus HALT space n/a
subone 1 subone 1 left
subone minus skip minus left
skip space skip space left
skip 1 scanright space right

What this machine does is move to the right until it sees the equal sign; it erases the equal sign and moves to the left, erases one digit off of the second number, and replaces it with the equal sign (so the second number is reduced by one, and the equal sign is moved over one position). Then it scans back to the minus sign (which separates the two numbers), and erases one digit of the first number, and then switches back to scanning to the right for the equal. So one at a time, it erases one digit from each of the two numbers. When it reaches the equal sign, and starts going back to erase a digit from the second number, and hits the “-” before it finds a digit, that means its done, so it stops. Let’s look at a simple execution trace. In the trace, I’ll write the machine state, followed by a colon, followed by the tape contents surrounded by “[]”, with the current tape cell surrounded by “{}”.

	scanright:  [ {1}1111111-111= ]"
	scanright:  [ 1{1}111111-111= ]"
	scanright:  [ 11{1}11111-111= ]"
	scanright:  [ 111{1}1111-111= ]"
	scanright:  [ 1111{1}111-111= ]"
	scanright:  [ 11111{1}11-111= ]"
	scanright:  [ 111111{1}1-111= ]"
	scanright:  [ 1111111{1}-111= ]"
	scanright:  [ 11111111{-}111= ]"
	scanright:  [ 11111111-{1}11= ]"
	scanright:  [ 11111111-1{1}1= ]"
	scanright:  [ 11111111-11{1}= ]"
	scanright:  [ 11111111-111{=} ]"
	eraseone :  [ 11111111-11{1}  ]"
	subone   :  [ 11111111-1{1}=  ]"
	subone   :  [ 11111111-{1}1=  ]"
	subone   :  [ 11111111{-}11=  ]"
	skip     :  [ 1111111{1}-11=  ]"
	scanright:  [ 1111111 {-}11=  ]"
	scanright:  [ 1111111 -{1}1=  ]"
	scanright:  [ 1111111 -1{1}=  ]"
	scanright:  [ 1111111 -11{=}  ]"
	eraseone :  [ 1111111 -1{1}   ]"
	subone   :  [ 1111111 -{1}=   ]"
	subone   :  [ 1111111 {-}1=   ]"
	skip     :  [ 1111111{ }-1=   ]"
	skip     :  [ 111111{1} -1=   ]"
	scanright:  [ 111111 { }-1=   ]"
	scanright:  [ 111111  {-}1=   ]"
	scanright:  [ 111111  -{1}=   ]"
	scanright:  [ 111111  -1{=}   ]"
	eraseone :  [ 111111  -{1}    ]"
	subone   :  [ 111111  {-}=    ]"
	skip     :  [ 111111 { }-=    ]"
	skip     :  [ 111111{ } -=    ]"
	skip     :  [ 11111{1}  -=    ]"
	scanright:  [ 11111 { } -=    ]"
	scanright:  [ 11111  { }-=    ]"
	scanright:  [ 11111   {-}=    ]"
	scanright:  [ 11111   -{=}    ]"
	eraseone :  [ 11111   {-}     ]"
	Halt:       [ 11111  { }-     ]"

See, it works!

One really important thing to understand here is that we do not have a program. What we just did was define a Turing machine that does subtraction. The machine does not take any instructions: the states and the transition function are an intrinsic part of the machine. So the only thing this machine can do is to subtract.

The real genius of Turing wasn’t just defining this simple computing machine. It was realizing the concept of the program: you could design a Turing machine whose input tape contained a description of a Turing machine – that is a program – followed by an input to the program. This single machine – the Universal Turing machine – could simulate any other Turing machine; one machine which could perform any computation.

Turing machines are a lot of fun to play with. Figuring out how to do things can be fascinating. Figuring out how to define a Universal turing machine’s transition function is an amazing thing to do; it’s astonishing how simple the universal machine can be!

As I said earlier – you can’t make a mechanical computing device that does anything that a Turing machine can’t do. One of the beauties of the Turing machine is that it lets you explore that. You can try adding and removing features to the basic machine, and see what happens.

For example: if you can do lots of great stuff with a Turing machine with one tape, what if you had a two-tape turing machine? That is, take the basic turing machine, and say that it’s got two tapes, each with a read/write head. Each state transition rule on this machine depends on the pair of values found on the two tapes. For now, we’ll say that the tapes move together – that is, the transition rule says “move the heads right” or “move the heads left”.

Seems like this should represent a real increase in power, right? No. Take a single-tape turing machine. Take the alphabet for tape one, and call it A1; take the alphabet for tape 2, and call it A2. We can create a single-tape turing machine whose alphabet is the cross-product of A1 and A2. Now each symbol on the tape is equivalent of a symbol on tape 1 and a symbol on tape 2. So we’ve got a single-tape machine which is equivalent to the two-tape machine. Bingo.

We can lift the restriction on the heads moving together, but it’s a lot more work. A two-tape machine can do things a lot faster than a one-tape, and the simulation will necessarily adapt to that. But it’s entirely doable. How about a two-dimensional tape? We can simulate that pretty easily with a two-tape machine, which means we can do it with a one-tape machine. For a two tape machine, what we do is map the two-D tape onto the one-D-tape, as seen in the diagram below – so that cell 0 on the one-D tape corresponds to cell (0,0) on the two tape; cell (0,1) on the two-D corresponds to cell 1 on the one-D; cell (1,1) on the 2-D is cell 2 on the 1-D; etc. Then we use the second tape for the book-keeping necessary to do the equivalent of T-D tape moves. And we’ve got a two-D turing machine simulated with a two-tape one-D; and we know that we can simulate a two-tape one-D with a one-tape one-D.

This is, to me, the most beautiful thing about the Turing machine. It’s not just a fundamental theoretical construction of a computing device; it’s a simple construction of a computing device that’s really easy to experiment with. Consider, for a moment, lambda calculus. It’s more useful that a Turing machine for lots of purposes – we write real programs in lambda calculus, when no one would build a real application using a Turing machine program. But imagine how you’d try things like the alternate constructions of the Turing machine? It’s a whole lot harder to build experiments like those in lambda calculus. Likewise for Minsky machines, Markov machines, etc.

For your enjoyment, I’ve implemented a Turing machine programming language. You feed it a Turing machine description, and an input string, and it will give you a trace of the machines execution like the one above. Here’s the specification of the subtraction machine written in my little Turing language:

states { "scanright" "eraseone" "subtractOneFromResult" "skipblanks" } initial "scanright"
alphabet { '1' ' ' '=' '-' } blank ' '
trans from "scanright" to "scanright" on (' ') write ' ' move right
trans from "scanright" to "scanright" on ('1') write '1' move right
trans from "scanright" to "scanright" on ('-') write '-' move right
trans from "scanright" to "eraseone" on ('=') write ' ' move left
trans from "eraseone" to "subtractOneFromResult" on ('1') write '=' move left
trans from "eraseone" to "Halt" on ('-') write ' ' move left
trans from "subtractOneFromResult" to "subtractOneFromResult" on ('1') write '1' move left
trans from "subtractOneFromResult" to "skipblanks" on ('-') write '-' move left
trans from "skipblanks" to "skipblanks" on (' ') write ' ' move left
trans from "skipblanks" to "scanright" on ('1') write ' ' move right

I think it’s pretty clear as a syntax, but it still needs explanation.

  • The first line declares the possible states of the machine, and what state it starts in. This machine has four possible states – “scanright”, “eraseone”, “subtractOneFromResult”, and “skipblanks”. When the machine starts, it will be in the state “skipright”.
  • The second line declares the set of symbols that can appear on the tape, including what symbol will initially appear on any tape cell whose value wasn’t specified by the input. For this machine the symbols are the digit 1, a blank space, the equal sign, and the subtraction symbol; the blank symbol is on any tape cell whose initial value wasn’t specified.
  • After that is a series of transition declarations. Each declaration specifies what the machine will do for a given pair of initial state and tape symbol. So, for example, if the machine is in state “scanright”, and the current tape cell contains the equal-to sign, then the machine will change to state “eraseone”, write a blank onto the tape cell (erasing the “=” sign), and move the tape cell one position to the left.

Finally, the code. You’ll need GHCI to run it; at the moment, it won’t work in Hugs (I don’t have the parsing library in my version of Hugs), and I haven’t worked out the linkage stuff to make it work under the GHC compiled mode.

-- A Simple Turing machine interpreter
-- Copyright 2007 by Mark C. Chu-Carroll
--    markcc@gmail.com
--   http://scienceblogs.com/goodmath
-- You can do anything you want with this code so long as you
-- leave this copyright notice intact.
module Turing where
import Text.ParserCombinators.Parsec
import qualified Text.ParserCombinators.Parsec.Token as P
import Text.ParserCombinators.Parsec.Language
import System.Environment

data Motion = MoveLeft  | MoveRight deriving (Show)

-- Transition from to on move write
data Transition = Transition String String [Char] Motion  Char deriving (Show)

-- TuringMachine states initial alphabet blank transitions
data TuringMachine = Machine [String] String   [Char]    Char    [Transition] deriving (Show)

data MachineState = TMState [Char] Char [Char]  String  deriving (Show)
--                           tape-left curcell  curstate

getBlankSym :: TuringMachine -> Char
getBlankSym (Machine _ _ _ bl _) = bl

findMatchingTransition :: [Transition] -> String -> Char -> Maybe Transition
findMatchingTransition [] _ _ =  Nothing
findMatchingTransition translist state c =
     let matches = filter (transitionMatches state c) translist
     in case matches of
          [] -> Nothing
          t:[] -> Just t
          _ -> error "Ambiguous transition"
       where transitionMatches state c (Transition from to on move write) =
                        (from == state) && (elem c on)

runTransition :: TuringMachine -> [Char] -> Char -> [Char] -> String -> Transition -> MachineState
runTransition m (l:left) c right state (Transition from tostate on MoveLeft write) =
   TMState left l (write:right) tostate
runTransition m left c [] state (Transition from tostate on MoveRight write) =
   TMState (write:left) (getBlankSym m) [] tostate
runTransition m left c (r:right) state (Transition from tostate on MoveRight write) =
   TMState (write:left) r right tostate

stepMachine :: TuringMachine -> MachineState -> MachineState
stepMachine machine@(Machine _ _ _ _ translist) st@(TMState tapeleft c taperight state) =
       let trans = findMatchingTransition translist state c
       in case trans of
          (Just t) -> runTransition machine tapeleft c taperight state t
          Nothing -> error "No applicable transition from state"

getMachineStateString (TMState left c right state) =
	(state ++ ":[" ++ (reverse left) ++ "{" ++ [c] ++ "}" ++ right ++ "]")

runTracedMachine :: TuringMachine -> [Char] -> [String]
runTracedMachine tm@(Machine states initial alphabet blank transitions) (t:tape) =
    runTracedMachineSteps tm (TMState [] t tape initial) where
        runTracedMachineSteps machine state =
           let newstate@(TMState left c right st) = stepMachine machine state
           in if st == "Halt"
               then [getMachineStateString newstate]
               else let trace=runTracedMachineSteps machine newstate
                    in ((getMachineStateString newstate):trace)

runMachine :: TuringMachine -> [Char] -> [Char]
runMachine tm@(Machine states initial alphabet blank transitions) (t:tape) =
    runMachineSteps tm (TMState [] t tape initial) where
        runMachineSteps machine state =
           let newstate@(TMState left c right st) = stepMachine machine state
           in if st == "Halt"
               then (concat [(reverse left), [c], right])
               else runMachineSteps machine newstate

-- Parsing code - implemented using the Parsec monadic parser library.

-- Basic setup stuff - use a standard haskell style lexer; set up the reserved words
-- and symbols in the lexer.
lexer :: P.TokenParser ()
lexer = P.makeTokenParser (haskellDef
                        { P.reservedNames = ["states","alphabet","trans","from","to","on","write","move","left","right","initial","blank"] })

reserved = P.reserved lexer
symbol = P.symbol lexer
braces = P.braces lexer
parens = P.parens lexer
charlit = P.charLiteral lexer
strlit = P.stringLiteral lexer
ident = P.identifier lexer
whitespace = P.whiteSpace lexer

states = reserved "states"
alphabet = reserved "alphabet"
trans = reserved "trans"
from = reserved "from"
to = reserved "to"
on = reserved "on"
write = reserved "write"
move = reserved "move"
initial = reserved "initial"
blank = reserved "blank"

left = do { reserved "left"
          ; return MoveLeft

right = do { reserved "right"
           ; return MoveRight

-- statesDecl ::= "states" "{"  strlit+  "}" "initial" strlit
statesDecl = do { states
                ; strlst <- braces (many1 strlit)
                ; initial
                ; i <- strlit
                ; return (i,strlst)

-- alphaDecl ::= "alphabet" "{" charlit+  "}" "blank" charlit
alphaDecl = do { alphabet
               ; charlst <- braces (many1 charlit)
               ; blank
               ; bl <- charlit
               ; return (bl, charlst)

-- transDecl ::= "trans" "from" strlit "to" strlit "on" "(" charlit+ ")" "write" charlit "move" ("left" | "right")
transDecl = do { trans
               ; from
               ; fromState <- strlit
               ; to
               ; targetState <- strlit
               ; on
               ; chrs <- parens (many1 charlit)
               ; write
               ; wrchar <- charlit
               ; move
               ; dir <- ( left <|> right)
      	   ; return (Transition fromState targetState chrs dir wrchar)

-- machine ::= statesDecl alphaDecl transDecl+
machine = do { (i,sts) <- statesDecl
             ; (bl,alpha) <- alphaDecl
             ; trans <- many1 transDecl
             ; return (Machine sts i alpha bl trans)

run :: (Show a) => Parser a -> String -> IO a
run p input
	= case (parse p "" input) of
		Left err -> do{ putStr "parse error at "
		; print err
		; error "Parse error"
		Right x -> return x

runTParser ::  String -> IO TuringMachine
runTParser input =
	run (do { whitespace
	        ; x <- machine
	        ; eof
	        ; return x })  input

-- A sample Turing program - first in the comment, and then coded into
-- a string literal, to make it easy to run tests. This sample program
-- is a basic Turing subtract - it takes a unary equation "111111-111=",
-- and leaves the result of subtracting the second from the first.
-- states { "one" "two" "three" "four" } initial "one"
-- alphabet { '1' ' ' '=' '-' } blank ' '
-- trans from "one" to "one" on (' ') write ' ' move right
-- trans from "one" to "one" on ('1') write '1' move right
-- trans from "one" to "one" on ('-') write '-' move right
-- trans from "one" to "two" on ('=') write ' ' move left
-- trans from "two" to "three" on ('1') write '=' move left
- trans from "two" to "Halt" on ('-') write '-' move left
-- trans from "three" to "three" on ('1') write '1' move left
-- trans from "three" to "four" on ('-') write '-' move left
-- trans from "four" to "four" on (' ') write ' ' move left
-- trans from "four" to "one" on ('1') write ' ' move right

sampleMachine = concat ["states { "one" "two" "three" "four" } initial "one"n ",
                        " alphabet { '1' ' ' '=' '-' } blank ' 'n ",
                        "trans from "one" to "one" on (' ') write ' ' move rightn ",
                        "trans from "one" to "one" on ('1') write '1' move rightn ",
                        "trans from "one" to "one" on ('-') write '-' move rightn ",
                        "trans from "one" to "two" on ('=') write ' ' move leftn ",
                        "trans from "two" to "three" on ('1') write '=' move leftn ",
                        "trans from "two" to "Halt" on ('-') write '-' move leftn ",
                        "trans from "three" to "three" on ('1') write '1' move leftn ",
                        "trans from "three" to "four" on ('-') write '-' move leftn ",
                        "trans from "four" to "four" on (' ') write ' ' move leftn ",
                        "trans from "four" to "one" on ('1') write ' ' move right"  ]

runTracedMachineOnString :: String -> String -> IO ([String])
runTracedMachineOnString m str =
		tm <- runTParser m
		return (runTracedMachine tm str)

runMachineOnString :: String -> String -> IO String
runMachineOnString m str =
	    tm <- runTParser m
	    return (runMachine tm str)

sampleInput = " 11111111-111= "

-- Main program execution scaffolding
-- main still needs a bit of work so that ghci will link correctly;
-- runs fine in GHCI, but linkage errors in GHC. For now, just load
-- this file, and then execute "runFromFile" from the command line.
main =
       [file] <- getArgs
       m <- parseFromFile (do { whitespace
                              ; x <- machine
                              ; eof
                              ; return x }) file
       case m of
          Right machine -> do
             print "Enter input for parser:"
             s <- getLine
             result <- return (runMachine machine s)
             print (concat ["Result:[", result, "]"])
          Left x -> do
	        print (concat ["Parse error"])

runFromFile :: String -> IO ()
runFromFile filename =
       m <- parseFromFile (do { whitespace
                              ; x <- machine
                              ; eof
                              ; return x }) filename
       case m of
          Right machine -> do
             print "Enter input for parser:"
             s <- getLine
             result <- return (runMachine machine s)
             print (concat ["Result:[", result, "]"])
          Left x -> do
	        print (concat ["Parse error"])

7 thoughts on “Turing Machines – what they are, what they aren't.

  1. Ørjan Johansen

    Hi, the reason you couldn’t get it to compile is probably because of the “module Turing where” line. By default, the main function is in the module named Main. (Although the filename can be anything, since it doesn’t need to imported from any other.) Alternatively, compile with the option “-main-is Turing.main”.

    Also, the line
    – trans from “two” to “Halt” on (‘-‘) write ‘-‘ move left
    needs another – at the beginning.

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