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A modular, efficient formalisation of real algebraic numbers
This paper presents a construction of the real algebraic numbers with executable arithmetic operations in Isabelle/HOL. Instead of verified resultants, arithmetic operations on real algebraic numbers are based on a decision procedure to decide the sign of a bivariate polynomial (with rational coefficients) at a real algebraic point. The modular design allows the safe use of fast external code. This work can be the basis for decision procedures that rely on real algebraic numbers.The CSC Cambridge International Scholarship is generously funding Wenda Li’s Ph.D. course.This is the author accepted manuscript. The final version is available from the Association for Computing Machinery via http://dx.doi.org/10.1145/2854065.285407
Formalization of the fundamental group in untyped set theory using auto2
We present a new framework for formalizing mathematics in untyped set theory
using auto2. Using this framework, we formalize in Isabelle/FOL the entire
chain of development from the axioms of set theory to the definition of the
fundamental group for an arbitrary topological space. The auto2 prover is used
as the sole automation tool, and enables succinct proof scripts throughout the
project.Comment: 17 pages, accepted for ITP 201
Matching concepts across HOL libraries
Many proof assistant libraries contain formalizations of the same
mathematical concepts. The concepts are often introduced (defined) in different
ways, but the properties that they have, and are in turn formalized, are the
same. For the basic concepts, like natural numbers, matching them between
libraries is often straightforward, because of mathematical naming conventions.
However, for more advanced concepts, finding similar formalizations in
different libraries is a non-trivial task even for an expert.
In this paper we investigate automatic discovery of similar concepts across
libraries of proof assistants. We propose an approach for normalizing
properties of concepts in formal libraries and a number of similarity measures.
We evaluate the approach on HOL based proof assistants HOL4, HOL Light and
Isabelle/HOL, discovering 398 pairs of isomorphic constants and types
A mechanized proof of loop freedom of the (untimed) AODV routing protocol
The Ad hoc On-demand Distance Vector (AODV) routing protocol allows the nodes
in a Mobile Ad hoc Network (MANET) or a Wireless Mesh Network (WMN) to know
where to forward data packets. Such a protocol is 'loop free' if it never leads
to routing decisions that forward packets in circles. This paper describes the
mechanization of an existing pen-and-paper proof of loop freedom of AODV in the
interactive theorem prover Isabelle/HOL. The mechanization relies on a novel
compositional approach for lifting invariants to networks of nodes. We exploit
the mechanization to analyse several improvements of AODV and show that
Isabelle/HOL can re-establish most proof obligations automatically and identify
exactly the steps that are no longer valid.Comment: The Isabelle/HOL source files, and a full proof document, are
available in the Archive of Formal Proofs, at
http://afp.sourceforge.net/entries/AODV.shtm
Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)
Oxford, UK, 26 August 200
Proving soundness of combinatorial Vickrey auctions and generating verified executable code
Using mechanised reasoning we prove that combinatorial Vickrey auctions are
soundly specified in that they associate a unique outcome (allocation and
transfers) to any valid input (bids). Having done so, we auto-generate verified
executable code from the formally defined auction. This removes a source of
error in implementing the auction design. We intend to use formal methods to
verify new auction designs. Here, our contribution is to introduce and
demonstrate the use of formal methods for auction verification in the familiar
setting of a well-known auction
Towards MKM in the Large: Modular Representation and Scalable Software Architecture
MKM has been defined as the quest for technologies to manage mathematical
knowledge. MKM "in the small" is well-studied, so the real problem is to scale
up to large, highly interconnected corpora: "MKM in the large". We contend that
advances in two areas are needed to reach this goal. We need representation
languages that support incremental processing of all primitive MKM operations,
and we need software architectures and implementations that implement these
operations scalably on large knowledge bases.
We present instances of both in this paper: the MMT framework for modular
theory-graphs that integrates meta-logical foundations, which forms the base of
the next OMDoc version; and TNTBase, a versioned storage system for XML-based
document formats. TNTBase becomes an MMT database by instantiating it with
special MKM operations for MMT.Comment: To appear in The 9th International Conference on Mathematical
Knowledge Management: MKM 201
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