123 research outputs found
Stateless HOL
We present a version of the HOL Light system that supports undoing
definitions in such a way that this does not compromise the soundness of the
logic. In our system the code that keeps track of the constants that have been
defined thus far has been moved out of the kernel. This means that the kernel
now is purely functional.
The changes to the system are small. All existing HOL Light developments can
be run by the stateless system with only minor changes.
The basic principle behind the system is not to name constants by strings,
but by pairs consisting of a string and a definition. This means that the data
structures for the terms are all merged into one big graph. OCaml - the
implementation language of the system - can use pointer equality to establish
equality of data structures fast. This allows the system to run at acceptable
speeds. Our system runs at about 85% of the speed of the stateful version of
HOL Light.Comment: In Proceedings TYPES 2009, arXiv:1103.311
Proviola: A Tool for Proof Re-animation
To improve on existing models of interaction with a proof assistant (PA), in
particular for storage and replay of proofs, we in- troduce three related
concepts, those of: a proof movie, consisting of frames which record both user
input and the corresponding PA response; a camera, which films a user's
interactive session with a PA as a movie; and a proviola, which replays a movie
frame-by-frame to a third party. In this paper we describe the movie data
structure and we discuss a proto- type implementation of the camera and
proviola based on the ProofWeb system. ProofWeb uncouples the interaction with
a PA via a web- interface (the client) from the actual PA that resides on the
server. Our camera films a movie by "listening" to the ProofWeb communication.
The first reason for developing movies is to uncouple the reviewing of a formal
proof from the PA used to develop it: the movie concept enables users to
discuss small code fragments without the need to install the PA or to load a
whole library into it. Other advantages include the possibility to develop a
separate com- mentary track to discuss or explain the PA interaction. We assert
that a combined camera+proviola provides a generic layer between a client
(user) and a server (PA). Finally we claim that movies are the right type of
data to be stored in an encyclopedia of formalized mathematics, based on our
experience in filming the Coq standard library.Comment: Accepted for the 9th International Conference on Mathematical
Knowledge Management (MKM 2010), 15 page
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
SEPIA: Search for Proofs Using Inferred Automata
This paper describes SEPIA, a tool for automated proof generation in Coq.
SEPIA combines model inference with interactive theorem proving. Existing proof
corpora are modelled using state-based models inferred from tactic sequences.
These can then be traversed automatically to identify proofs. The SEPIA system
is described and its performance evaluated on three Coq datasets. Our results
show that SEPIA provides a useful complement to existing automated tactics in
Coq.Comment: To appear at 25th International Conference on Automated Deductio
Univalent Foundations and the UniMath Library
We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3). On the way our odyssey from the foundations to the "horizon" of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander
Decidability of Univariate Real Algebra with Predicates for Rational and Integer Powers
We prove decidability of univariate real algebra extended with predicates for
rational and integer powers, i.e., and . Our decision procedure combines computation over real algebraic
cells with the rational root theorem and witness construction via algebraic
number density arguments.Comment: To appear in CADE-25: 25th International Conference on Automated
Deduction, 2015. Proceedings to be published by Springer-Verla
Adding an Abstraction Barrier to ZF Set Theory
Much mathematical writing exists that is, explicitly or implicitly, based on
set theory, often Zermelo-Fraenkel set theory (ZF) or one of its variants. In
ZF, the domain of discourse contains only sets, and hence every mathematical
object must be a set. Consequently, in ZF, with the usual encoding of an
ordered pair , formulas like have truth values, and operations like have results that are sets. Such 'accidental theorems' do not match
how people think about the mathematics and also cause practical difficulties
when using set theory in machine-assisted theorem proving. In contrast, in a
number of proof assistants, mathematical objects and concepts can be built of
type-theoretic stuff so that many mathematical objects can be, in essence,
terms of an extended typed -calculus. However, dilemmas and
frustration arise when formalizing mathematics in type theory.
Motivated by problems of formalizing mathematics with (1) purely
set-theoretic and (2) type-theoretic approaches, we explore an option with much
of the flexibility of set theory and some of the useful features of type
theory. We present ZFP: a modification of ZF that has ordered pairs as
primitive, non-set objects. ZFP has a more natural and abstract axiomatic
definition of ordered pairs free of any notion of representation. This paper
presents axioms for ZFP, and a proof in ZF (machine-checked in Isabelle/ZF) of
the existence of a model for ZFP, which implies that ZFP is consistent if ZF
is. We discuss the approach used to add this abstraction barrier to ZF
A Paraconsistent Higher Order Logic
Classical logic predicts that everything (thus nothing useful at all) follows
from inconsistency. A paraconsistent logic is a logic where an inconsistency
does not lead to such an explosion, and since in practice consistency is
difficult to achieve there are many potential applications of paraconsistent
logics in knowledge-based systems, logical semantics of natural language, etc.
Higher order logics have the advantages of being expressive and with several
automated theorem provers available. Also the type system can be helpful. We
present a concise description of a paraconsistent higher order logic with
countable infinite indeterminacy, where each basic formula can get its own
indeterminate truth value (or as we prefer: truth code). The meaning of the
logical operators is new and rather different from traditional many-valued
logics as well as from logics based on bilattices. The adequacy of the logic is
examined by a case study in the domain of medicine. Thus we try to build a
bridge between the HOL and MVL communities. A sequent calculus is proposed
based on recent work by Muskens.Comment: Originally in the proceedings of PCL 2002, editors Hendrik Decker,
Joergen Villadsen, Toshiharu Waragai (http://floc02.diku.dk/PCL/). Correcte
Mining State-Based Models from Proof Corpora
Interactive theorem provers have been used extensively to reason about
various software/hardware systems and mathematical theorems. The key challenge
when using an interactive prover is finding a suitable sequence of proof steps
that will lead to a successful proof requires a significant amount of human
intervention. This paper presents an automated technique that takes as input
examples of successful proofs and infers an Extended Finite State Machine as
output. This can in turn be used to generate proofs of new conjectures. Our
preliminary experiments show that the inferred models are generally accurate
(contain few false-positive sequences) and that representing existing proofs in
such a way can be very useful when guiding new ones.Comment: To Appear at Conferences on Intelligent Computer Mathematics 201
A Foundational View on Integration Problems
The integration of reasoning and computation services across system and
language boundaries is a challenging problem of computer science. In this
paper, we use integration for the scenario where we have two systems that we
integrate by moving problems and solutions between them. While this scenario is
often approached from an engineering perspective, we take a foundational view.
Based on the generic declarative language MMT, we develop a theoretical
framework for system integration using theories and partial theory morphisms.
Because MMT permits representations of the meta-logical foundations themselves,
this includes integration across logics. We discuss safe and unsafe integration
schemes and devise a general form of safe integration
- …