In computer science, a computation history is a sequence of steps taken by an abstract machine in the process of computing its result. Computation histories are frequently used in proofs about the capabilities of certain machines, and particularly about the undecidability of various formal languages.

Formally, a computation history is a (normally finite) sequence of configurations of a formal automaton. Each configuration fully describes the status of the machine at a particular point. To be valid, certain conditions must hold:

• the first configuration must be a valid initial configuration of the automaton and
• each transition between adjacent configurations must be valid according to the transition rules of the automaton.

In addition, to be complete, a computation history must be finite and

• the final configuration must be a valid terminal configuration of the automaton.

The definitions of "valid initial configuration", "valid transition", and "valid terminal configuration" vary for different kinds of formal machines.

A deterministic automaton has exactly one computation history for a given initial configuration, though the history may be infinite and therefore incomplete.

Finite State Machines

For a finite state machine ${\displaystyle M}$, a configuration is simply the current state of the machine, together with the remaining input. The first configuration must be the initial state of ${\displaystyle M}$ and the complete input. A transition from a configuration ${\displaystyle (S,I)}$ to a configuration ${\displaystyle (T,J)}$ is allowed if ${\displaystyle I=aJ}$ for some input symbol ${\displaystyle a}$ and if ${\displaystyle M}$ has a transition from ${\displaystyle S}$ to ${\displaystyle T}$ on input ${\displaystyle a}$. The final configuration must have the empty string ${\displaystyle \epsilon }$ as its remaining input; whether ${\displaystyle M}$ has accepted or rejected the input depends on whether the final state is an accepting state. ^[1]

Turing Machines

Computation histories are more commonly used in reference to Turing machines. The configuration of a single-tape Turing machine consists of the contents of the tape, the position of the read/write head on the tape, and the current state of the associated state machine; this is usually written

${\displaystyle ...0011010101q00110101010...}$

where ${\displaystyle q}$ is the current state of the machine, represented in some way that's distinguishable from the tape language, and where ${\displaystyle q}$ is positioned immediately before the position of the read/write head.

Consider a Turing machine ${\displaystyle M}$ on input ${\displaystyle w}$. The first configuration must be ${\displaystyle q_{0}w_{0}w_{1}...}$, where ${\displaystyle q_{0}}$ is the initial state of the Turing machine. The machine's state in the final configuration must be either ${\displaystyle q_{a}}$ (the accept state) or ${\displaystyle q_{r}}$ (the reject state). A configuration ${\displaystyle c_{i+1}}$ is a valid successor to configuration ${\displaystyle c_{i}}$ if there's a transition from the state in ${\displaystyle c_{i}}$ to the state in ${\displaystyle c_{i+1}}$ which manipulates the tape and moves the read/write head in a way that produces the result in ${\displaystyle c_{i+1}}$.^[2]

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