rule = {{1, 0} -> 2, {1, 1} -> 1, {2, 0} -> 1, {2, 1} -> 2};
automatonStep[rule_, state_, s_] := {rule, {state, ToExpression[StringPart[s, 1]]} /. rule, StringDrop[s, 1]}
automatonStep[rule, 1, "010"]
{{{1, 0} -> 2, {1, 1} -> 1, {2, 0} -> 1, {2, 1} -> 2}, 2, "10"}
automatonCompute[rule_, initState_, s_, finalStates_] := acceptWordAutomaton[ NestWhileList[ automatonStep[#[[1]], #[[2]], #[[3]]] &, {rule, initState, s}, ToString[#[[3]]] != "" &], finalStates]
automatonCompute[rule, 1, "010", {1}]
acceptWordAutomaton[result_, finalStates_List] := MemberQ[finalStates, result[[Length[result], 2]]]