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On Existentially First-Order Definable Languages and Their Relation to NP

Bernd Borchert, Dietrich Kuske, Frank Stephan (2010)

RAIRO - Theoretical Informatics and Applications

Under the assumption that the Polynomial-Time Hierarchy does not collapse we show for a regular language L: the unbalanced polynomial-time leaf language class determined by L equals  iff L is existentially but not quantifierfree definable in FO[<, min, max, +1, −1]. Furthermore, no such class lies properly between NP and co-1-NP or NP⊕co-NP. The proofs rely on a result of Pin and Weil characterizing the automata of existentially first-order definable languages.

On L 1 Space Formed by Complex-Valued Partial Functions

Yasushige Watase, Noboru Endou, Yasunari Shidama (2012)

Formalized Mathematics

In this article, we formalized L1 space formed by complexvalued partial functions [11], [15]. The real-valued case was formalized in [22] and this article is its generalization.

On L 1 Space Formed by Real-Valued Partial Functions

Yasushige Watase, Noboru Endou, Yasunari Shidama (2008)

Formalized Mathematics

This article contains some definitions and properties refering to function spaces formed by partial functions defined over a measurable space. We formalized a function space, the so-called L1 space and proved that the space turns out to be a normed space. The formalization of a real function space was given in [16]. The set of all function forms additive group. Here addition is defined by point-wise addition of two functions. However it is not true for partial functions. The set of partial functions...

On L p Space Formed by Real-Valued Partial Functions

Yasushige Watase, Noboru Endou, Yasunari Shidama (2010)

Formalized Mathematics

This article is the continuation of [31]. We define the set of Lp integrable functions - the set of all partial functions whose absolute value raised to the p-th power is integrable. We show that Lp integrable functions form the Lp space. We also prove Minkowski's inequality, Hölder's inequality and that Lp space is Banach space ([15], [27]).

On problems of databases over a fixed infinite universe

Oleg Belegradek, Alexei Stolboushkin, Michael Taitslin (1999)

Banach Center Publications

In the relational model of databases a database state is thought of as a finite collection of relations between elements. For many applications it is convenient to pre-fix an infinite domain where the finite relations are going to be defined. Often, we also fix a set of domain functions and/or relations. These functions/relations are infinite by their nature. Some special problems arise if we use such an approach. In the paper we discuss some of the problems. We show that there exists a recursive...

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