The computation of an SL₂ automaton A = ( Σ, T) on a string w is the maximal sequence of IDs in which each sequential pair of IDs is related by |-_A and which starts with the initial con?guration of A on w: (p₁,w₁), where p₁ .
w₁ = ?w?.

Since w is ?nite, the computation of A on w will be ?nite. Since it is required to be maximal, the last ID will be one that does not directly compute any other ID. This will either be of the form (σ_iσ_j) , wii where σiσj ∈ T, or of the form (σ_n?, ε), in which σ_n? ∈ T but all the input has been consumed. In the ?rst case we will say that the computation is rejecting (or that it crashes). In the second we will say that it is accepting. Note that we have adopted the convention that the automaton halts with FALSE as soon as it encounters a pair of symbols that are not in T.