Turing Machine Simulator
Define a Turing machine with states and transitions, run it step by step on an infinite tape, and see busy beaver machines and the halting problem made concrete.
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The model is deliberately impoverished
A Turing machine has a tape, a head that reads and writes one cell, and a table saying what to do for each combination of state and symbol. There is no memory beyond the tape, no arithmetic, no addressing. Turing chose this because the point was never efficiency but a proof: anything computable at all can be computed by this, so a limit on what this can do is a limit on computation itself. Every argument about what computers cannot do rests on the model being as weak as it plausibly could be.
Busy beavers make non-computability concrete
The busy beaver function asks: among all halting machines with n states, what is the greatest number of steps any of them takes? For two states the answer is 6, for three 21, for four 107, and for five it is 47,176,870, which took until 2024 to establish. For six states the value is known to exceed ten to the ten to the ten to the fifteen. The function grows faster than any computable function, which is not a statement about difficulty but a proof: if you could compute it you could solve the halting problem.
You cannot tell whether a machine will stop
Watching a machine run for a million steps tells you nothing about whether it halts, because there is no bound to compare against. That is the halting problem, and it is undecidable in general rather than merely hard. A step limit is therefore not a workaround but an admission: a simulator can report that a machine did not halt within the budget, and that is a different statement from saying it never will.
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Frequently Asked Questions
What is a Turing machine?
A tape, a head that reads and writes one cell at a time, and a table of transitions. It has no other memory and no arithmetic, and that weakness is the point of the model.
What is a busy beaver?
The halting machine with n states that runs the longest. The values are 6, 21, 107 and 47,176,870 for two to five states, and the five-state value was only proved in 2024.
Why does the busy beaver function matter?
It grows faster than any computable function, which is a proof rather than an observation: computing it would let you solve the halting problem.
Can the simulator tell me if a machine halts?
No, and neither can anything else in general. It can report that a machine did not halt within a step budget, which is a different statement from saying it never will.
Why does the tape look infinite?
Because it is, in the model. The simulator extends it as the head moves, which is the practical equivalent for any machine that halts.
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How to Use
Choose a machine and run it to watch the tape evolve.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.