579 research outputs found
Essential Incompleteness of Arithmetic Verified by Coq
A constructive proof of the Goedel-Rosser incompleteness theorem has been
completed using the Coq proof assistant. Some theory of classical first-order
logic over an arbitrary language is formalized. A development of primitive
recursive functions is given, and all primitive recursive functions are proved
to be representable in a weak axiom system. Formulas and proofs are encoded as
natural numbers, and functions operating on these codes are proved to be
primitive recursive. The weak axiom system is proved to be essentially
incomplete. In particular, Peano arithmetic is proved to be consistent in Coq's
type theory and therefore is incomplete.Comment: This paper is part of the proceedings of the 18th International
Conference on Theorem Proving in Higher Order Logics (TPHOLs 2005). For the
associated Coq source files see the TeX sources, or see
<http://r6.ca/Goedel20050512.tar.gz
On effective sigma-boundedness and sigma-compactness
We prove several theorems on sigma-bounded and sigma-compact pointsets. We
start with a known theorem by Kechris, saying that any lightface \Sigma^1_1 set
of the Baire space either is effectively sigma-bounded (that is, covered by a
countable union of compact lightface \Delta^1_1 sets), or contains a
superperfect subset (and then the set is not sigma-bounded, of course). We add
different generalizations of this result, in particular, 1) such that the
boundedness property involved includes covering by compact sets and equivalence
classes of a given finite collection of lightface \Delta^1_1 equivalence
relations, 2) generalizations to lightface \Sigma^1_2 sets, 3) generalizations
true in the Solovay model.
As for effective sigma-compactness, we start with a theorem by Louveau,
saying that any lightface \Delta^1_1 set of the Baire space either is
effectively sigma-compact (that is, is equal to a countable union of compact
lightface \Delta^1_1 sets), or it contains a relatively closed superperfect
subset. Then we prove a generalization of this result to lightface \Sigma^1_1
sets.Comment: arXiv admin note: substantial text overlap with arXiv:1103.106
A thread calculus with molecular dynamics
We present a theory of threads, interleaving of threads, and interaction
between threads and services with features of molecular dynamics, a model of
computation that bears on computations in which dynamic data structures are
involved. Threads can interact with services of which the states consist of
structured data objects and computations take place by means of actions which
may change the structure of the data objects. The features introduced include
restriction of the scope of names used in threads to refer to data objects.
Because that feature makes it troublesome to provide a model based on
structural operational semantics and bisimulation, we construct a projective
limit model for the theory.Comment: 47 pages; examples and results added, phrasing improved, references
replace
Hybrid Rules with Well-Founded Semantics
A general framework is proposed for integration of rules and external first
order theories. It is based on the well-founded semantics of normal logic
programs and inspired by ideas of Constraint Logic Programming (CLP) and
constructive negation for logic programs. Hybrid rules are normal clauses
extended with constraints in the bodies; constraints are certain formulae in
the language of the external theory. A hybrid program is a pair of a set of
hybrid rules and an external theory. Instances of the framework are obtained by
specifying the class of external theories, and the class of constraints. An
example instance is integration of (non-disjunctive) Datalog with ontologies
formalized as description logics.
The paper defines a declarative semantics of hybrid programs and a
goal-driven formal operational semantics. The latter can be seen as a
generalization of SLS-resolution. It provides a basis for hybrid
implementations combining Prolog with constraint solvers. Soundness of the
operational semantics is proven. Sufficient conditions for decidability of the
declarative semantics, and for completeness of the operational semantics are
given
Efficient Implementation and the Product State Representation of Numbers
The relation between the requirement of efficient implementability and the
product state representation of numbers is examined. Numbers are defined to be
any model of the axioms of number theory or arithmetic. Efficient
implementability (EI) means that the basic arithmetic operations are physically
implementable and the space-time and thermodynamic resources needed to carry
out the implementations are polynomial in the range of numbers considered.
Different models of numbers are described to show the independence of both EI
and the product state representation from the axioms. The relation between EI
and the product state representation is examined. It is seen that the condition
of a product state representation does not imply EI. Arguments used to refute
the converse implication, EI implies a product state representation, seem
reasonable; but they are not conclusive. Thus this implication remains an open
question.Comment: Paragraph in page proof for Phys. Rev. A revise
The Representation of Natural Numbers in Quantum Mechanics
This paper represents one approach to making explicit some of the assumptions
and conditions implied in the widespread representation of numbers by composite
quantum systems. Any nonempty set and associated operations is a set of natural
numbers or a model of arithmetic if the set and operations satisfy the axioms
of number theory or arithmetic. This work is limited to k-ary representations
of length L and to the axioms for arithmetic modulo k^{L}. A model of the
axioms is described based on states in and operators on an abstract L fold
tensor product Hilbert space H^{arith}. Unitary maps of this space onto a
physical parameter based product space H^{phy} are then described. Each of
these maps makes states in H^{phy}, and the induced operators, a model of the
axioms. Consequences of the existence of many of these maps are discussed along
with the dependence of Grover's and Shor's Algorithms on these maps. The
importance of the main physical requirement, that the basic arithmetic
operations are efficiently implementable, is discussed. This conditions states
that there exist physically realizable Hamiltonians that can implement the
basic arithmetic operations and that the space-time and thermodynamic resources
required are polynomial in L.Comment: Much rewrite, including response to comments. To Appear in Phys. Rev.
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