529 | Run the machine to see it in action. 530 | At any time, you can step or pause to get a closer look, 531 | or reset to start over. 532 |
533 |There are over a dozen different example machines to explore.
534 |535 | Most of the examples take input. Experiment with different inputs to see what happens! 536 | Edit the code and click Load machine to sync your changes. 537 |
538 |The colored circles are states. The squares underneath are tape cells.
541 |The current state and tape cell are highlighted.
542 |At each step, a Turing machine reads its current state and tape symbol, 543 | and looks them up in its transition table for an instruction. 544 | Each instruction does 3 things:
545 |That’s it!
551 |This repeats step after step, until the machine reaches a combination of state and symbol 552 | that don’t have an instruction defined. At that point it halts. 553 |
554 |A Turing machine is an abstract device to model computation as rote symbol manipulation.
571 |Each machine has a finite number of states, and a finite number of possible symbols. 572 | These are fixed before the machine starts, and do not change as the machine runs.
573 |There are an infinite number of tape cells, however, extending endlessly to the left and right. 574 | Each tape cell bears a symbol. Any cell not part of the input or not yet written to bears the blank symbol by default. 575 | Notice that at any step, only finitely many cells bear a non-blank symbol.
576 |(As a mathematical model, a Turing machine has infinite memory (an infinite tape) so as to not artificially restrict its power. 577 | In practice, many machines of interest take finite memory, and can be fully simulated for manageable input sizes. 578 | Even machines that use infinite memory—and hence never halt—use at most one new tape cell per step, and so can be simulated to an extent.)
579 |The formal definition of a Turing machine has slight variations but essentially is a tuple (ordered list) comprising
593 |where Q, Σ, and Γ are finite nonempty sets. 602 | Some definitions also require that the blank symbol not be part of the input (b ∉ Σ).
603 | 604 |As an example, a formal description for the “binary increment” machine is as follows:
605 |
606 | Q = { right, carry, done }
607 |
q₀ = right
608 |
Σ = { 1, 0 }
609 |
Γ = { 1, 0, ' ' }
610 |
b = ' '
611 |
613 | δ(right, 1) = (1, R, right)
614 |
δ(right, 0) = (0, R, right)
615 |
δ(right, ' ') = (' ', L, carry)
616 |
δ(carry, 1) = (0, L, carry)
617 |
δ(carry, 0) = (1, L, done)
618 |
δ(carry, ' ') = (1, L, done)
619 |
Note that for simplicity, the simulator limits each symbol to one character. 621 | Furthermore, input is not checked for conformance with an input alphabet, in exchange for not having to define input and tape alphabets.
622 |(The behavior of a Turing machine can also be described in mathematical terms if desired. 623 | Without going into too much detail, this involves defining how a configuration 624 | of a machine—its state, tape contents, and tape head position (current cell)—leads to the next configuration, based on the transition function. 625 | To start with, the tape’s contents may be defined as a function from integers to symbols, 626 | with the head position as an integer, and each move {L, R} adding -1 or +1 respectively to the head position.)
627 |Some aspects of the definition vary from author to author, but the differences come down to preference and do not affect computational power. 641 | That is, machines on one model can be simulated or converted to run on another model.
642 |Some of the variations you may come across:
643 |The simulator was designed with these and other considerations in mind.
653 |670 | For a very readable introduction to Turing machines, including their significance and interesting implications, 671 | see the excellent entry in the Stanford Encyclopedia of Philosophy. 672 |
673 |Make spinoffs from the examples or your own creations by using 680 | Edit > Duplicate document. 681 | You can also start from scratch with a new blank document.
682 |All it takes to describe a Turing machine is a start state, blank symbol, and transition table.
683 |
685 | # Adds 1 to a binary number.
686 | input: '1011'
687 | blank: ' '
688 | start state: right
689 | table:
690 | # scan to the rightmost digit
691 | right:
692 | 1: R
693 | 0: R
694 | ' ': {L: carry}
695 | # then carry the 1
696 | carry:
697 | 1: {write: 0, L}
698 | 0: {write: 1, L: done}
699 | ' ': {write: 1, L: done}
700 | done:
701 |
702 |
703 | Here the states are right, carry, and done.
704 | The symbols are '1', '0', and ' '.
705 |
707 | We designate one state the start state, 708 | and one symbol the blank symbol (found on unmarked tape cells). 709 |
710 |
711 | A state and a symbol together map to an instruction.
712 | In the carry state, for instance, the symbol 1 maps to the instruction {write: 0, L}.
713 | When no instruction is defined, as in the done state, the machine halts.
714 |
{write: symbol, move: state}
718 | {write: 1, L: done} writes the symbol 1,
720 | moves the tape head left (L), and goes to the done state.
721 | For brevity, you can omit the symbol and state if they stay the same.
724 |{L: carry} writes the same symbol that was read,
726 | moves the tape head left, and goes to the carry state.R (shorthand for {R}) simply moves the tape head right.
728 | It writes back the same symbol and stays in the same state.More keyboard shortcuts are available on the full list.
745 |The code is written in YAML 1.2, a general-purpose data format.
761 |
762 | If you’re familiar with JSON, YAML is similar, but designed to be more readable.
763 | Mappings can use indent instead of brackets {}, and strings can often be included directly without quotes.
764 |
766 | Some strings need to be quoted: for example, those that start/end with a space, or include certain characters that have special meaning in YAML. 767 | If a string is causing problems, try putting quotes around it. 768 |
769 |