By Jouko Väänänen (auth.), Jouko Väänänen (eds.)
This e-book includes revised complete models of lectures given through the ninth ecu summer season tuition in good judgment, Languages, and knowledge, ESSLLI'97, held in Aix-en-Provence, France, in August 1997. The six lectures provided introduce the reader to the state-of-the-art within the sector of generalized quantifiers and computation. along with an introductory survey by way of the quantity editor quite a few elements of generalized quantifiers are studied in depth.
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Extra resources for Generalized Quantifiers and Computation: 9th European Summer School in Logic, Language, and Information ESSLLI’97Workshop Aix-en-Provence, France, August 11–22, 1997 Revised Lectures
Note that there are other well-known examples of close connection between separation of complexity classes and expressivity bounds of query languages, see . One can go further and add more operations to D. For example, one can assume that D is the field of real numbers with the usual operations +, ∗, −, or perhaps more complex such as ex . The results of  show that for the class of generic  queries (those that commute with permutations of D), these extra operations do not add expressive power, beyond a linear order.
2. Hanf’s technique proves that connectivity is not first-order As we noticed before, there is only one type of a d-neighborhood that these graphs realize, namely a chain on 2d + 1 points. Thus, A d B, since they have the same number of vertices. At the same time, A is connected, but B is not, proving that connectivity cannot be expressed by Φ. 30 L. Libkin and J. Nurmonen Other examples, proved previously with games, can be shown to be derivable from Hanf’s technique. For example, with the concept of (d, m)-equivalence, it is now easy to show that the majority quantifier (or Rescher and H¨ artig quantifiers) are not definable in first-order logic.
In [21,36] it was conjectured that relational queries in AGGRQ have the BDP. By relational queries we mean those whose inputs and outputs only contain elements of D, but no numbers, although rational numbers can be used in the process of evaluating a query. An SQL example of finding nodes with equal in- and out-degrees shows that there are relational queries definable in SQL but not in first-order. Clearly, proving the above conjecture would resolve the problem for queries such as transitive closure or deterministic transitive closure.