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

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which there is at least one B, a stringset we call Some-B:(3) Some-B def = {w ∈ {A, B} ∗ | |w| B≥ 1}.To see that this is not SL k for any k, note that, for any k, strings of the formA...A·A...A } {{ } ·BA...A and those of the form A...AB ·A...A } {{ } ·A...Ak−1are all in Some-B, but the result of substituting the suffix of a string of thesecond form, starting at the marked sequence of k − 1 As, for the suffix ofa string of the first form is A...A · } A...A {{ } ·A...A which is not in Some-B. So, Some-B does not exhibit suffix substitution closure and can not bespecified with an SL-definition.4.2 Locally Testable StringsetsIn order to distinguish stringsets like Some-B it is necessary to differentiatebetween strings on the basis of the whole set of k-factors that they contain,not just on the basis of the individual k-factors in isolation. Descriptions, atthis level, are k-expressions, formulae in a propositional language in whichthe atomic formulae are k-factors which are taken to be true of a string w iffthey occur in the augmented string ⋊w⋉. More complicated k-expressionsare built up from k-factors using the usual logical connectives, e.g., for conjunction(∧), disjunction (∨) and negation (¬). Stringsets are Locally Testable(LT) (McNaughton and Papert 1971) iff they are definable by a k-expression,for some k. As an example, Some-B is defined by the following 2-expression,which is true of strings that either start with B or include AB:k−1(4) ϕ Some-Bdef= ⋊B ∨ ABThe relation between strings and the k-expressions which holds iff the stringsatisfies the k-expression is denoted |=; the set of strings licensed by a k-expression ϕ, then, is L(ϕ) def = {w | w |= ϕ}. For the k-expression ϕ Some-B ,L(ϕ Some-B ) = Some-B. As this witnesses, k-expressions have more descriptivepower than SL k definitions. 9Because membership in an LT k stringset depends on the whole set ofk-factors that occur in a string k-factors can be required to occur as wellas prohibited from occurring and they can be required to occur in arbitraryk−110
combinations—to occur together, to occur only if some other combination ofk-factors does not occur, etc.On the other hand, membership in an LT k stringset depends only on theset of k-factors which occur in the string. Any mechanism which can distinguishstrings on this basis is capable of recognizing (some) LT k stringset.Conversely, any mechanism that can distinguish members of a stringset thatis LT k (but not SL) must be able to distinguish strings on this basis. Cognitively,this corresponds to being sensitive to the set of all k-factors that occuranywhere in the input. If the strings are presented sequentially, it amountsto being able to remember which k-factors have and which have not beenencountered in the stimulus.An example of a stringset that is not LT is the set of strings over {A, B}in which exactly one B occurs:(5) One-B def = {w ∈ {A, B} ∗ | |w| B= 1}To see that this is not LT note that, for any k, A k BA k is in One-B whileA k BA k BA k is not. Since these have exactly the same set of k-factors, however,there is no k-expression that can distinguish them.4.3 FO(+1) Definable StringsetsThe next step in extending the complexity of the stringsets we can distinguishis to add the power to discriminate between strings on the basis of the specificpositions in which blocks of symbols occur rather than simply on the basisof blocks of symbols occurring somewhere in the string. Descriptions at thislevel are first-order logical sentences (formulae with no free variables) over arestricted string-oriented signature. Atomic formulae assert relationships betweenvariables (x, y, . . .) ranging over positions in the string: x ⊳ y (meaningthat y is the next position following x), x ≈ y (x and y are the same position)and A(x), B(x), . . . (A occurs in position x, etc.). Larger formulae arebuilt from these using the logical connectives and existential (∃) and universal(∀) quantification. A string w satisfies an existential (sub)formula (∃x)[ϕ(x)](i.e., w |= (∃x)[ϕ(x)]) iff some assignment of a position for x makes it true.It satisfies a universal (sub)formula iff all such assignments make it true. Theclass of stringsets definable in this way is denoted FO(+1).As an example, One-B is FO(+1) definable:(6) One-B = {w ∈ {A, B} ∗ | w |= (∃x)[B(x) ∧ (∀y)[B(y) → x ≈ y] ]},11
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