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A path ''u''<sub>1</sub>...''u''<sub>''n''</sub> ∈ ''U''<sup>[[Kleene star|*]]</sup> is a string of path components ''u''<sub>''i''</sub> ∈ ''U''; ''n'' may be 0, with the empty path denoted by ε.
A thread has the form ''u''<sub>1</sub>...''u''<sub>''n''</sub>:''A'', where ''u''<sub>1</sub>...''u''<sub>''n''</sub> ∈ ''U''<sup>*</sup> is a path, and ''A'' ∈ ''N'' is a state.
A thread store ''S'' is a finite set of threads, viewed as a partial function from ''U''<sup>*</sup> to ''N'', such that ''dom''(''S'') is [[closure (mathematics)|closed]] by [[prefix (computer science)|prefix]].
A thread automaton configuration is a triple ‹''l'',''p'',''S''›, where ''l'' denotes the current position in the input string, ''p'' is the active thread, and ''S'' is a thread store containing ''p''.
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