Computation history

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:

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

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.

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.

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