149,229 research outputs found
Formalization of a normalization theorem in simplicial topology
In this paper we present a complete formalization of the Normalization
Theorem, a result in Algebraic Simplicial Topology stating that there exists a
homotopy equivalence between the chain complex of a simplicial set, and a smaller
chain complex for the same simplicial set, called the normalized chain complex.
Even if the Normalization Theorem is usually stated as a higher-order result (with
a Category Theory flavor) we manage to give a first-order proof of it. To this aim
it is instrumental the introduction of an algebraic data structure called simplicial
polynomial. As a demonstration of the validity of our techniques we developed a
formal proof in the ACL2 theorem prover.Ministerio de Ciencia e Innovación MTM2009-13842European Commission FP7 STREP project ForMath n. 24384
A Normalizing Intuitionistic Set Theory with Inaccessible Sets
We propose a set theory strong enough to interpret powerful type theories
underlying proof assistants such as LEGO and also possibly Coq, which at the
same time enables program extraction from its constructive proofs. For this
purpose, we axiomatize an impredicative constructive version of
Zermelo-Fraenkel set theory IZF with Replacement and -many
inaccessibles, which we call \izfio. Our axiomatization utilizes set terms, an
inductive definition of inaccessible sets and the mutually recursive nature of
equality and membership relations. It allows us to define a weakly-normalizing
typed lambda calculus corresponding to proofs in \izfio according to the
Curry-Howard isomorphism principle. We use realizability to prove the
normalization theorem, which provides a basis for program extraction
capability.Comment: To be published in Logical Methods in Computer Scienc
Braid Matrices and Quantum Gates for Ising Anyons Topological Quantum Computation
We study various aspects of the topological quantum computation scheme based
on the non-Abelian anyons corresponding to fractional quantum hall effect
states at filling fraction 5/2 using the Temperley-Lieb recoupling theory.
Unitary braiding matrices are obtained by a normalization of the degenerate
ground states of a system of anyons, which is equivalent to a modification of
the definition of the 3-vertices in the Temperley-Lieb recoupling theory as
proposed by Kauffman and Lomonaco. With the braid matrices available, we
discuss the problems of encoding of qubit states and construction of quantum
gates from the elementary braiding operation matrices for the Ising anyons
model. In the encoding scheme where 2 qubits are represented by 8 Ising anyons,
we give an alternative proof of the no-entanglement theorem given by Bravyi and
compare it to the case of Fibonacci anyons model. In the encoding scheme where
2 qubits are represented by 6 Ising anyons, we construct a set of quantum gates
which is equivalent to the construction of Georgiev.Comment: 25 pages, 13 figure
Efficient Data Structures for Automated Theorem Proving in Expressive Higher-Order Logics
Church's Simple Theory of Types (STT), also referred to as classical higher-order logik, is an elegant and expressive formal system built on top of the simply typed λ-calculus. Its mechanisms of explicit binding and quantification over arbitrary sets and functions allow the representation of complex mathematical concepts and formulae in a concise and unambiguous manner. Higher-order automated theorem proving (ATP) has recently made major progress and several sophisticated ATP systems for higher-order logic have been developed, including Satallax, Osabelle/HOL and LEO-II. Still, higher-order theorem proving is not as mature as its first-order counterpart, and robust implementation techniques for efficient data structures are scarce.
In this thesis, a higher-order term representation based upon the polymorphically typed λ-calculus is presented. This term representation employs spine notation, explicit substitutions and perfect term sharing for efficient term traversal, fast β-normalization and reuse of already constructed terms, respectively. An evaluation of the term representation is performed on the basis of a heterogeneous benchmark set. It shows that while the presented term data structure performs quite well in general, the normalization results indicate that a context dependent choice of reduction strategies is beneficial.
A term indexing data structure for fast term retrieval based on various low-level criteria is presented and discussed. It supports symbol-based term retrieval, indexing of terms via structural properties, and subterm indexing
Severi Varieties and Brill-Noether theory of curves on abelian surfaces
Severi varieties and Brill-Noether theory of curves on K3 surfaces are well
understood. Yet, quite little is known for curves on abelian surfaces. Given a
general abelian surface with polarization of type , we prove
nonemptiness and regularity of the Severi variety parametrizing -nodal
curves in the linear system for (here is
the arithmetic genus of any curve in ). We also show that a general genus
curve having as nodal model a hyperplane section of some -polarized
abelian surface admits only finitely many such models up to translation;
moreover, any such model lies on finitely many -polarized abelian
surfaces. Under certain assumptions, a conjecture of Dedieu and Sernesi is
proved concerning the possibility of deforming a genus curve in
equigenerically to a nodal curve. The rest of the paper deals with the
Brill-Noether theory of curves in . It turns out that a general curve in
is Brill-Noether general. However, as soon as the Brill-Noether number is
negative and some other inequalities are satisfied, the locus of
smooth curves in possessing a is nonempty and has a component of
the expected dimension. As an application, we obtain the existence of a
component of the Brill-Noether locus having the expected
codimension in the moduli space of curves . For , the
results are generalized to nodal curves.Comment: 29 pages, 3 figures. Comments are welcome. 2nd version: added some
references in Rem. 7.1
- …