Lindström quantifier

Abstract: We describe a general way of building logics with Lindstrom quantifiers, which capture regular complexity classes on ordered structures with polysize reductions. We then extend this method so as to accommodate complexity classes based on oracle Turing machines. Our main result shows an equivalence between enhancing a logic with a Lindstrom quantifier and enhancing a complexity class with an oracle such that, if K is a set of structures, Q K the associated Lindstrom quantifier and L a logic that captures a complexity class D, then the enhanced logic L[K] captures D K — the complexity class of machines in D using oracles for K. Our results are sensitive to the oracle computation model and hold in a natural modification of the unbounded model introduced by Buss [Bus88]. They do not hold in the, so called, space bounded oracle models or those that violate the ‘relativization thesis’ of Buss. Our results generalize and extend previous results of Stewart [Ste93a, Ste93b] and Makowsky and Pnueli [MP93].

Abstract: Lindstrom (Theoria, 1966; 32:186–195) introduced a very powerful notion of quantifiers, which permits multiplace quantification and the simultaneous binding of several variables. “Branching” quantification was found to be useful by linguists, e.g., for modeling reciprocal constructions like “Most men and most women admire each other.” Westerstahl (In Gardenfors P., editor. Generalized Quantifiers. New York: Reidel; 1987. pp 269–298) showed how to compute the three-place Lindstrom quantifier for “Q1 As and Q2 Bs R each other” from the binary quantifiers Q1 and Q2, assuming crisp quantifiers and arguments. In the paper, this basic method will be generalized to vague quantifiers like “many” and fuzzy arguments like “young.” A consistent interpretation is achieved by extending the DFS theory of fuzzy quantification (Glockner, TR97-06, 1997; TR2002-07, (Int J Approx Reason) 2003; 2004; 37(2):93–126), which introduces fuzzy generalized quantifiers in conformance with the original linguistic notion, and controls their interpretation by formal adequacy criteria. The new analysis is important to linguistic data summarization because the full meaning of reciprocal summarizers (e.g., describing factors, which are “correlated” or “associated” with each other), can only be captured by branching quantification. The proofs of all theorems cited in the paper are listed in (Glockner, TR2002-07, 2003). © 2009 Wiley Periodicals, Inc.

Abstract: The connection between measure once quantum finite automata (MO-QFA) and logic is studied in this paper. The language class recognized by MO-QFA is compared to languages described by the first order logics and modular logics. And the equivalence between languages accepted by MO-QFA and languages described by formulas using Lindstrom quantifier is shown.

Abstract: Lindstrom [1] introduced a very powerful notion of quantifiers, which permits multi-place quantification and the simultaneous binding of several variables. 'Branching' quantifification was found to be useful by linguists e.g. for modelling reciprocal constructions like "Most men and most women admire each other".Westerstahl [2] showed how to compute the three-place Lindstrom quantifier for "Q1 A's and Q2 B'sR each other" from the binary quantifiers Q1 and Q2, assuming crisp quantifiers and arguments. In the paper, I generalize his method to approximate quantifiers like "many" and fuzzy arguments like "young". A consistent interpretation is achieved by extending the DFS theory of fuzzy quantification [3, 4], which rests on a system of formal adequacy criteria. The new analysis is important to linguistic data summarization because the full meaning of reciprocal summarizers (e.g. describing factors which are "correlated" or "associated" with each other), can only be captured by branching quantification.