Probabilistic computation

In computability theory, a probabilistic Turing machine is a non-deterministic Turing machine which chooses between the available transitions at each point according to some probability distribution.

In the case of equal probabilities for the transitions, it can be defined as a deterministic Turing machine having an additional "write" instruction where the value of the write is uniformly distributed in the Turing Machine's alphabet (generally, an equal likelihood of writing a '1' or a '0' on to the tape.) Another common reformulation is simply a deterministic Turing machine with an added tape full of random bits called the random tape.

As a consequence, a probabilistic Turing machine can (unlike a deterministic Turing Machine) have results; on a given input and instruction state machine, it may have different run times, or it may not halt at all; further, it may accept an input in one execution and reject the same input in another execution.

Therefore, the notion of acceptance of a string by a probabilistic Turing machine can be defined in different ways. Various polynomial-time randomized complexity classes that result from different definitions of acceptance include RP, co-RP, BPP and ZPP. If the machine is restricted to logarithmic space instead of polynomial time, the analogous RL, co-RL, BPL, and ZPL complexity classes are obtained. By enforcing both restrictions, RLP, co-RLP, BPLP, and ZPLP are yielded.

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