Asveld, P.R.J.

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University of Twente Research Information

On grammatical errors and other imperfections

On Grammatical Errors and Other ImperfectionsPeter R.J. AsveldDepartment of Computer Science, Twente University of TechnologyP.O. Box 217, 7500 AE Enschede, the Netherlandse-mail: infprja@cs.utwente.nlWhen we extend a grammar G = (V,Σ, P, S) with additional productions to model grammat-ical errors we end up with some strings over Σ that do not belong completely to either L(G)or to Σ⋆−L(G). A similar situation occurs when an automaton or a recognition/parsing algo-rithm A has been augmented with certain transitions in order to deal with erroneous inputs:A processes strings over Σ that neither fully are in L(A) nor in Σ⋆ − L(A).To model these phenomena adequately the notions of fuzzy language, fuzzy grammar andfuzzy automaton turn out to be appropriate. A fuzzy language L over Σ is defined by itsmembership function µL : Σ⋆ → L where L is a structure that is a bit more complex then thetwo-element set {0, 1} used in case of ordinary or crisp languages. A fuzzy automaton [fuzzygrammar] is a weighted automaton [weighted grammar] where the weights are taken from L.Originally [10], L was taken equal to the real closed interval [0, 1], but nowadays we usea structure somewhat more complex [9, 12, 8] than the semirings usually encountered in thestudy of weighted automata. More precisely, L is a completely distributive complete latticeprovided with an additional operator ⋆, i.e., L = (L,∧,∨, 0, 1, ⋆) where (L,∧,∨, 0, 1) is acompletely distributive complete lattice and (L, ⋆, 1) is a commutative monoid; these twosubstructures are coupled by distributivity laws, and 0 ⋆ a = 0 holds for all a in L; cf. [1, 2].Imperfections can be small (“tiny mistakes”: weights are close to, but unequal to 1) or big(“capital blunders”: weights are close to, but unequal to 0). Multiple errors in a derivation orcomputation should be accumulated, the result of which determines the quality (the degreeof perfection) of the string to be generated or accepted ultimately.The three operations of L reflect several aspects of the generation or computation process:• The ∨-operation plays an important rôle in the presence of nondeterminism or ambiguity.• We need the ∧-operation for imposing regular restrictions, in particular for “intersectionwith the set of words over a terminal alphabet”.• The ⋆-operation is essential in describing the accumulation of errors properly (“Making anerror twice is worse than making it once.” “A long sequence of tiny mistakes may result insomething that looks like a capital blunder.”)The ∨-operation is needed for another reason; viz. the occurrence of an error may becompensated by another error to obtain a perfectly correct string w in the language L(G) orL(A). In order to yield µL(G)(w) = 1 or µL(A)(w) = 1, we need to take the supremum overall possible derivations or computations, respectively, including the completely correct ones.In this framework some properties of context-free [3], regular [2, 6] and Lindenmayer [1, 6]languages have been generalized to their “fuzzified” counterparts. For instance, context-freegrammars augmented with rules that model grammatical errors give rise to modifications ofwell-known parsing algorithms [4] which, apart from their usual job, are able to deal witherroneous inputs as well.Although many of these generalizations hold for arbitrary L, some results have been onlyestablished for the case in which L is linearly ordered (of which the real closed interval [0, 1]2 Peter R.J. Asveldwith operations min, max and multiplication is the most frequently used, practical example).As an illustration we consider the problem of (in)dependency of operations on fuzzy lan-guages. An operation o from a class C of operations on fuzzy languages is called independentif o cannot be expressed as a polynomial over C−{o}. A major tool in studying independencyis the set of algebraic closure operators that correspond to the subsets of C. This set turnsout to be a bounded partially ordered monoid, the structure of which (i.e., its Cayley tableand its partial order) answers many (in)dependency questions. In determining the structureof this monoid, it is crucial to establish (i) defining relations, and (ii) inequalities. Now (i) isusually possible for arbitrary L, whereas (ii) relies on translating a proper inclusion betweenfamilies of crisp languages to a proper inclusion between the corresponding families of fuzzylanguages. In this translation process the linear order of L is probably indispensable [5, 7].On the other hand, when L is not linearly ordered —even in the simplest case (i.e., L is thefour-element distributive lattice that is not a chain, while ⋆ coincides with ∧)— unexpectedmatters will occur: there exists two fuzzy context-free languages L1 and L2 over Σ with theircrisp parts, defined by c(Li) = {w ∈ Σ⋆ | µLi(w) = 1}, being empty (i.e., c(Li) = ∅ fori = 1, 2), whereas the crisp part of their union c(L1 ∪ L2) is not context-free; cf. [11, 4].1. P.R.J. Asveld, Controlled fuzzy parallel rewriting, pp. 49–70 in: Gh. Păun & A. Salomaa(eds.), New Trends in Formal Languages — Control, Cooperation, and Combinatorics(1997), Lect. Notes. in Comp. Sci. 1218, Springer, Berlin, etc.2. P.R.J. Asveld, The non-self-embedding property for generalized fuzzy context-free gram-mars, Publ. Math. Debrecen 54 Suppl. (1999) 553–573.3. P.R.J. Asveld, Fuzzy context-free languages — Part I: Generalized fuzzy context-freegrammars (2000), CTIT Tech. Rep. No. 00-03, Enschede, the Netherlands.4. P.R.J. Asveld, Fuzzy context-free languages — Part II: Recognition and parsing algorithms(2000), CTIT Techn. Rep. No. 00-25, Enschede, the Netherlands.5. P.R.J. Asveld, Some operators on families of fuzzy languages and their monoids, pp. 1–14in: M. Ito, Gh. Păun & Sh. Yu (eds.), Words, Semigroups, and Translations (2001), WorldScientific, Singapore.6. P.R.J. Asveld, Algebraic aspects of families of fuzzy languages, Theor. Comp. Sci. (toappear).7. P.R.J. Asveld, Independency of operations on fuzzy languages, [in preparation].8. G. Gerla, Fuzzy grammars and recursively enumerable fuzzy languages, Inform. Sci. 60(1992) 137–143.9. H.H. Kim, M. Mizumoto, J. Toyoda & K. Tanaka, L-fuzzy grammars, Inform. Sci. 8(1975) 123–140.10. E.T. Lee & L.A. Zadeh, Note on fuzzy languages, Inform. Sci. 1 (1969) 421–434.11. M. Mizumoto, J. Toyoda & K. Tanaka, Examples of formal grammars with weights, In-form. Process. Lett. 2 (1973) 74–78.12. W. Wechler: The Concept of Fuzziness in Automata and Language Theory (1978),Akademie-Verlag, Berlin.

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On grammatical errors and other imperfections

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