Search results for "predicate"

Time-Efficient Quantum Walks for 3-Distinctness

2013

We present two quantum walk algorithms for 3-Distinctness. Both algorithms have time complexity $\tilde{O}(n^{5/7})$, improving the previous $\tilde{O}(n^{3/4})$ and matching the best known upper bound for query complexity (obtained via learning graphs) up to log factors. The first algorithm is based on a connection between quantum walks and electric networks. The second algorithm uses an extension of the quantum walk search framework that facilitates quantum walks with nested updates.

On the best Lipschitz extension problem for a discrete distance and the discrete ∞-Laplacian

2012

Abstract This paper concerns the best Lipschitz extension problem for a discrete distance that counts the number of steps. We relate this absolutely minimizing Lipschitz extension with a discrete ∞-Laplacian problem, which arises as the dynamic programming formula for the value function of some e -tug-of-war games. As in the classical case, we obtain the absolutely minimizing Lipschitz extension of a datum f by taking the limit as p → ∞ in a nonlocal p -Laplacian problem.

General aggregation operators based on a fuzzy equivalence relation in the context of approximate systems

2016

Our paper deals with special constructions of general aggregation operators, which are based on a fuzzy equivalence relation and provide upper and lower approximations of the pointwise extension of an ordinary aggregation operator. We consider properties of these approximations and explore their role in the context of extensional fuzzy sets with respect to the corresponding equivalence relation. We consider also upper and lower approximations of a t-norm extension of an ordinary aggregation operator. Finally, we describe an approximate system, considering the lattice of all general aggregation operators and the lattice of all fuzzy equivalence relations.

The Monadic Quantifier Alternation Hierarchy over Grids and Graphs

2002

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 Riesz Representation Theorem and Extension of Vector Valued Additive Measures

2001

Extensions and Imbeddings

1998

AbstractWe establish a connection between the Sobolev imbedding theorem and the extendability of Sobolev functions. As applications we give geometric criteria for extendability and give a result on the dependence of the extension property on the exponentp.

The Gupta-Belnap Fixed-Point Problem and the Theory of Clones of Functions

2003

This paper presents the Gupta-Belnap Fixed-Point problem: to characterize the truth-functional schemes of the first-order logic such that, for every interpreted language L, a truth predicate for L can be defined in L using a Kripkean fixed-point. A propositional version of the problem is defined using the stipulation logic of A. Visser and then the strategy adopted for the solution to the three-valued case is presented, using the theory of clones of functions.

The monadic quantifier alternation hierarchy over grids and pictures

1998

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.

Counting in the Two Variable Guarded Logic with Transitivity

2005

We show that the extension of the two-variable guarded fragment with transitive guards (GF+TG) by functionality statements is undecidable. This gives immediately undecidability of the extension of GF+TG by counting quantifiers. The result is optimal, since both the three-variable fragment of the guarded fragment with counting quantifiers and the two-variable guarded fragment with transitivity are undecidable. We also show that the extension of GF+TG with functionality, where functional predicate letters appear in guards only, is decidable and of the same complexity as GF+TG. This fragment captures many expressive modal and description logics.

The Descriptive Complexity Approach to LOGCFL

1999

Building upon the known generalized-quantifier-based firstorder characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC1 and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's "hardest contextfree language" is LOGCFL-complete under quantifier-free BIT-free interpre…