# Turing machine simulator (OCaml)

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We describe a simple OCaml program for simulating an abstract Turing machine. This demonstrates that OCaml is Turing-complete (with the caveat that limitations in word size limit the effective addressable memory).

## Contents |

## Formal problem description and assumptions

We define a single-tape Turing machine as a 6-tuple , where

*Q*is a finite set of states;- Γ is the finite tape alphabet (the symbols that can occur on the tape);
- is the initial state;
- is the blank symbol;
- is the set of accepting (final) states;
- is the transition function which determines the action performed at each step.

Initially the tape has the input string on it, followed by an infinite number of blank symbols (), and the head is at the left end of the tape. At each step we use the transition function to determine the next state, the symbol written on the tape just prior to moving, and the direction to move the head, left (L) or right (R). If we ever reach a final state, the machine halts.

## Main simulator

### State representation

We will represent the tape by a list of OCaml characters. States are represented by an OCaml `option`

type -- normal states are associated with `Some`

value, while `None`

marks the invalid state. Directions will be represented with an algebraic data type that is either `Left`

or `Right`

<<types>>=typedir = Left | Right

### Simulation

The main function of our simulator is `simulate`

which performs a single execution step.
The first three parameters represent the current execution state, the remainder describes the Turing machine
and remains constant throughout the calculation. `simulate`

shall return `true`

if the machine halts on an accepting state and `false`

if it reaches an invalid state.

<<simulator>>=letrecsimulate tape state head_position transition_func accepting_states blank_symbol = trace current state; act according to current state

In a single simulation step, we have three possible cases depending on the current state value:

<<act according to current state>>=matchstatewith| machine in invalid state | machine in accepting state | machine running

In invalid state, the machine stops and returns `false`

.

<<machine in invalid state>>=None -> false

Likewise, if the current state is in the set of accepting states, we stop and return `true`

.

<<machine in accepting state>>=Some swhenList.mem s accepting_states -> true

As long as we're neither in invalid or accepting state, we read the symbol at the current tape position and call the transition function to determine new state, the symbol to write to the tape, and the head move direction:

<<machine running>>=Some s ->letsymbol = read symbol at current head positioninletnewstate, newsymbol, movedir = transition_func s symbolinnext simulation step

To read a symbol from the tape, we define a small utility function that takes a few special cases into account.

<<utility functions>>=letsymbol_at tape position blank_symbol =ifposition < 0thenfailwith "Invalid tape position"elseifposition >= List.length tapethenblank_symbolelseList.nth tape position

With its help, reading the current symbol from the tape is straightforward:

<<read symbol at current head position>>=symbol_at tape head_position blank_symbol

For the next simulation step, we call `simulate`

recursively with an updated tape and head position.

<<next simulation step>>=simulate(update tape)newstate(determine new head position)transition_func accepting_states blank_symbol

The head position is updated according to the symbol in `movedir`

.

<<determine new head position>>=matchmovedirwith| Left -> pred head_position | Right -> succ head_position

Before the next step, we write the symbol returned by the transition function to the old head position.

<<update tape>>=write_to_tape tape head_position newsymbol

This is done by another utility function `write_to_tape`

we define as follows:

<<utility functions2>>=letwrite_to_tape tape position symbol =letl = List.length tapeinifposition < 0 || position > lthenfailwith "Invalid tape position"elseifposition = lthentape @[symbol]elsereplace symbol on tape

If the symbol is in the middle of the tape, we copy the tape up to the current position and append the new symbol and the rest of the tape.

<<replace symbol on tape>>=replace_symbol tape position symbol<<utility functions>>=letreplace_symbol tape position symbol =letrecreplace_helper(x::xs)n acc =ifn = 0thenList.rev acc @ symbol :: xselsereplace_helper xs(pred n)(x :: acc)inreplace_helper tape position[]

### Tracing State

For diagnostic purposes, it's also useful to print out some of the details of a state. We only show the first `trace_tape_chars`

characters of the tape:

<<constants>>=lettrace_tape_chars = 78<<trace current state>>=trace_state tape head_position blank_symbol<<utility functions2>>=lettrace_state tape head_position blank_symbol =letprint_n_times s n =fori = 1tondoprint_char sdoneinletrecprint_tape tape n =matchtape, nwith_, 0 |[], _ ->()| x :: xs, _ -> print_char x; print_tape xs(pred n)inifhead_position < trace_tape_charsthenbeginprint_n_times ' ' head_position; print_endline "v"end; print_tape tape trace_tape_chars; print_n_times blank_symbol(max 0(trace_tape_chars - List.length tape)); print_newline()

### Files

Finally, we put it all together into a source file:

<<simulate_turing_machine.ml>>=types constants utility functions utility functions2 simulator

This completes the simulator implementation.

## Test driver

To test the simulation, we'll implement the simple Turing machine shown to the right, which is based roughly on this Turing machine example. This diagram is a *state graph*, meaning that each vertex represents a state in the machine's finite control, and each edge indicates the input character that must be read to follow that edge, the character to write over it, and the direction to move the head. The initial state is 0 and the only accepting state is 5. It recognizes the following context-free but nonregular language:

The pumping lemma says that this is nonregular, but a Turing machine to recognize it is fairly straightforward. The main task is to transform the state graph into a transition function. We could code such a function for each program we run on the simulator, but a much better idea is to write a generic function which reads the appropriate return values from a data table.

To keep it simple, we use a Hashtbl that maps an input to its corresponding output values. For inputs that do not match any entry in the table, the invalid state is returned.

<<utility functions>>=letfind_trans_state states state symbol blank_symbol =tryletstate, symbol, dir = Hashtbl.find states(state, symbol)inSome state, symbol, dirwithNot_found -> None, blank_symbol, Left

Now we're ready to define our a^{n}b^{n} test. We pick `#`

for the blank symbol.

<<test_driver.ml>>=openSimulate_turing_machine;;lettest_states = Hashtbl.create 10;; Hashtbl.add test_states(0,'#')(4,'#',Right);; Hashtbl.add test_states(0,'a')(1,'#',Right);; Hashtbl.add test_states(4,'#')(5,'#',Right);; Hashtbl.add test_states(1,'a')(1,'a',Right);; Hashtbl.add test_states(1,'b')(1,'b',Right);; Hashtbl.add test_states(1,'#')(2,'#',Left);; Hashtbl.add test_states(2,'b')(3,'#',Left);; Hashtbl.add test_states(3,'a')(3,'a',Left);; Hashtbl.add test_states(3,'b')(3,'b',Left);; Hashtbl.add test_states(3,'#')(0,'#',Right);;lettest_anbn_trans_func state symbol = find_trans_state test_states state symbol '#'letanbn_test initial_tape = simulate initial_tape(Some 0)0 test_anbn_trans_func[5]'#';; anbn_test['a'; 'b'];;

Now we can see, for example:

v ab############################################################################ v #b############################################################################ v #b############################################################################ v #b############################################################################ v ############################################################################## v ############################################################################## v ############################################################################## v ##############################################################################

See Turing machine simulator (Scheme)/Example output for a longer example output.

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