The Use of Formal Language Theory in Studies of Artificial - CiteSeerX

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which asserts that there is some position x in which a B occurs and that allpositions y in which B occurs are that same position. As One-B witnesses,FO(+1) definitions are more expressive than k-expressions. 10It turns out that a stringset is FO(+1) definable iff it is Locally ThresholdTestable (LTT) (Thomas 1982). Such stringsets distinguish strings onlyon the multiplicity of the k-factors which occur in them and only relativeto a threshold t above which multiplicities of the k-factors cannot be distinguished.Thischaracterization is the key to identifying stringsets that are notFO(+1) definable.An example of a non-LTT, hence non-FO(+1)-definable, stringset is theset of strings over {A, B, C} in which some B occurs before any C:(7)B-before-C def = {w ∈ {A, B, C} ∗ | at least one B precedes any C} ∉ LTT.To see this, note that, for any k, A k BA k CA k and A k CA k BA k have exactlythe same number of occurrences of every k-factor and are therefore indistinguishablein the LTT sense for any threshold t.4.4 FO(
which is satisfied either by strings which consist of a substring in which noC occurs followed by one which starts with B or by those in which, simply,no C occurs at all.If the strings are presented sequentially, this corresponds to being able toapply a fixed set of sequences of LTT-style threshold counting recognitionstrategies. Effectively, it corresponds to being able to count occurrences ofevents up to some threshold, coupled with the capacity to reset the counterssome fixed number of times.The most useful abstract characterization of the structure of the FO(
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