0000000000302854
AUTHOR
Nicole Schweikardt
The Natural Order-Generic Collapse for ω-Representable Databases over the Rational and the Real Ordered Group
We consider order-generic queries, i.e., queries which commute with every order-preserving automorphism of a structure's universe. It is well-known that first-order logic has the natural order-generic collapse over the rational and the real ordered group for the class of dense order constraint databases (also known as finitely representable databases). I.e., on this class of databases over 〈Q, <〉 or 〈R, <〉, addition does not add to the expressive power of first-order logic for defining order-generic queries. In the present paper we develop a natural generalization of the notion of finitely representable databases, where an arbitrary (i.e. possibly infinite) number of regions is allowed. We …
The Monadic Quantifier Alternation Hierarchy over Grids and Graphs
AbstractThe monadic second-order quantifier alternation hierarchy over the class of finite graphs is shown to be strict. The proof is based on automata theoretic ideas and starts from a restricted class of graph-like structures, namely finite two-dimensional grids. Considering grids where the width is a function of the height, we prove that the difference between the levels k+1 and k of the monadic hierarchy is witnessed by a set of grids where this function is (k+1)-fold exponential. We then transfer the hierarchy result to the class of directed (or undirected) graphs, using an encoding technique called strong reduction. It is notable that one can obtain sets of graphs which occur arbitrar…
The Crane Beach Conjecture
A language L over an alphabet A is said to have a neutral letter if there is a letter e/spl isin/A such that inserting or deleting e's from any word in A* does not change its membership (or non-membership) in L. The presence of a neutral letter affects the definability of a language in first-order logic. It was conjectured that it renders all numerical predicates apart from the order predicate useless, i.e., that if a language L with a neutral letter is not definable in first-order logic with linear order then it is not definable in first-order. Logic with any set /spl Nscr/ of numerical predicates. We investigate this conjecture in detail, showing that it fails already for /spl Nscr/={+, *…
An Ehrenfeucht-Fraïssé Approach to Collapse Results for First-Order Queries over Embedded Databases
We present a new proof technique for collapse results for first-order queries on databases which are embedded in N or R>o. Our proofs are by means of an explicitly constructed winning strategy for Duplicator in an Ehrenfeucht-FraissE game, and can deal with certain infinite databases where previous, highly involved methods fail. Our main result is that first-order logic has the natural-generic collapse over {N,≤ ,+} for arbitrary (i.e., possibly infinite) databases. Furthermore, a first application of this result shows the natural-generic collapse of first-order logic over {R>o,≤,+} for a certain kind of databases over R>o which consist of a possibly infinite number of regions.
A Logical Characterisation of Linear Time on Nondeterministic Turing Machines
The paper gives a logical characterisation of the class NTIME(n) of problems that can be solved on a nondeterministic Turing machine in linear time. It is shown that a set L of strings is in this class if and only if there is a formula of the form ∃f1..∃fk∃R1..∃Rm∀xφv; that is true exactly for all strings in L. In this formula the fi are unary function symbols, the Ri are unary relation symbols and φv; is a quantifierfree formula. Furthermore, the quantification of functions is restricted to non-crossing, decreasing functions and in φv; no equations in which different functions occur are allowed. There are a number of variations of this statement, e.g., it holds also for k = 3. From these r…
The monadic quantifier alternation hierarchy over grids and pictures
The subject of this paper is the expressive power of monadic second-order logic over two-dimensional grids. We give a new, self-contained game-theoretical proof of the nonexpressibility results of Matz and Thomas. As we show, this implies the strictness of the monadic second-order quantifier alternation hierarchy over grids.
First-order expressibility of languages with neutral letters or: The Crane Beach conjecture
A language L over an alphabet A is said to have a neutral letter if there is a letter [email protected]?A such that inserting or deleting e's from any word in A^* does not change its membership or non-membership in L. The presence of a neutral letter affects the definability of a language in first-order logic. It was conjectured that it renders all numerical predicates apart from the order predicate useless, i.e., that if a language L with a neutral letter is not definable in first-order logic with linear order, then it is not definable in first-order logic with any set N of numerical predicates. Named after the location of its first, flawed, proof this conjecture is called the Crane Beach …