Search results for "predicate"
showing 10 items of 216 documents
On certain extension theorems in the mixed Borel setting
2004
Abstract Given two sequences M 1 and M 2 of positive numbers, we give necessary and sufficient conditions under which the inclusions Λ { M 1 } ⊂ f (j) (0) j∈ N 0 : f∈ D { M 2 } [−1,1] , Λ ( M 1 ) ⊂ f (j) (0) j∈ N 0 : f∈ D ( M 2 ) [−1,1] hold, by means of explicit constructions. This answers a question raised by Chaumat and Chollet (Math. Ann. 298 (1994) 7–40). We also consider the case when [−1,1] is replaced by [−1,1]m as well as the possibility to get ultraholomorphic extensions.
Analytic Extension of Non Quasi - Analytic Whitney Jets of Beurling Type
1998
Let (Mr)r∈ℕ0 be a logarithmically convex sequence of positive numbers which verifies M0 = 1 as well as Mr ≥ 1 for every r ∈ ℕ and defines a non quasi - analytic class. Let moreover F be a closed proper subset of ℝn. Then for every function f on ℝn belonging to the non quasi - analytic (Mr)-class of Beurling type, there is an element g of the same class which is analytic on ℝ,nF and such that Dαf(x) = Dαg(x) for every α ∈ ℕn0 and x ∈ F.
Graph connectivity and monadic NP
2002
Ehrenfeucht games are a useful tool in proving that certain properties of finite structures are not expressible by formulas of a certain type. In this paper a new method is introduced that allows the extension of a local winning strategy for Duplicator, one of the two players in Ehrenfeucht games, to a global winning strategy. As an application it is shown that graph connectivity cannot be expressed by existential second-order formulas, where the second-order quantification is restricted to unary relations (monadic NP), even, in the presence of a built-in linear order. As a second application it is stated, that, on the other hand, the presence of a linear order increases the power of monadi…
First-order expressibility of languages with neutral letters or: The Crane Beach conjecture
2005
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 …
Logical definability of NP-optimisation problems with monadic auxiliary predicates
1993
Given a first-order formula ϕ with predicate symbols e1...el, so,...,sr, an NP-optimisation problem on -structures can be defined as follows: for every -structure G, a sequence of relations on G is a feasible solution iff satisfies ϕ, and the value of such a solution is defined to be ¦S0¦. In a strong sense, every polynomially bounded NP-optimisation problem has such a representation, however, it is shown here that this is no longer true if the predicates s1, ...,sr are restricted to be monadic. The result is proved by an Ehrenfeucht-Fraisse game and remains true in several more general situations.
Absolutely Convergent Extensions of Nonclosable Positive Linear Functionals
2010
The existence of extensions of a positive linear functional ω defined on a dense *-subalgebra \({\mathfrak{A}_0}\) of a topological *-algebra \({\mathfrak{A}}\), satisfying certain regularity conditions, is examined. The main interest is focused on the case where ω is nonclosable and sufficient conditions for the existence of an absolutely convergent extension of ω are given.
Single-valued extension property at the points of the approximate point spectrum
2003
Abstract A localized version of the single-valued extension property is studied at the points which are not limit points of the approximate point spectrum, as well as of the surjectivity spectrum. In particular, we shall characterize the single-valued extension property at a point λ o ∈ C in the case that λoI−T is of Kato type. From this characterizations we shall deduce several results on cluster points of some distinguished parts of the spectrum.
Operators Which Do Not Have the Single Valued Extension Property
2000
Abstract In this paper we shall consider the relationships between a local version of the single valued extension property of a bounded operator T ∈ L ( X ) on a Banach space X and some quantities associated with T which play an important role in Fredholm theory. In particular, we shall consider some conditions for which T does not have the single valued extension property at a point λ o ∈ C .
Extensions and intentions in the rough set theory
1998
Abstract The approach to rough set theory proposed in this paper is based on the mutual correspondence of the concepts of extension and intension. It is different from the well-known approaches in the literature in that the upper approximations and the lower approximations of ‘unknown’ sets are considered as certain families of ‘known’ sets. This approach makes it possible to formulate necessary and sufficient conditions for the existence of operations on rough sets, which are analogous to classical operations on sets. The basic results presented in this paper, based on certain ideas of the second author, were formulated by the first author in his doctoral dissertation prepared under the su…
Logics with counting and equivalence
2014
We consider the two-variable fragment of first-order logic with counting, subject to the stipulation that a single distinguished binary predicate be interpreted as an equivalence. We show that the satisfiability and finite satisfiability problems for this logic are both NEXPTIME-complete. We further show that the corresponding problems for two-variable first-order logic with counting and two equivalences are both undecidable.