3,275 research outputs found
An Open Challenge Problem Repository for Systems Supporting Binders
A variety of logical frameworks support the use of higher-order abstract
syntax in representing formal systems; however, each system has its own set of
benchmarks. Even worse, general proof assistants that provide special libraries
for dealing with binders offer a very limited evaluation of such libraries, and
the examples given often do not exercise and stress-test key aspects that arise
in the presence of binders. In this paper we design an open repository ORBI
(Open challenge problem Repository for systems supporting reasoning with
BInders). We believe the field of reasoning about languages with binders has
matured, and a common set of benchmarks provides an important basis for
evaluation and qualitative comparison of different systems and libraries that
support binders, and it will help to advance the field.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759
A proof-theoretic analysis of the classical propositional matrix method
The matrix method, due to Bibel and Andrews, is a proof procedure designed for automated theorem-proving. We show that underlying this method is a fully structured combinatorial model of conventional classical proof theory. © 2012 The Author, 2012. Published by Oxford University Press
Informal proof, formal proof, formalism
Increases in the use of automated theorem-provers have renewed focus on the relationship between the informal proofs normally found in mathematical research and fully formalised derivations. Whereas some claim that any correct proof will be underwritten by a fully formal proof, sceptics demur. In this paper I look at the relevance of these issues for formalism, construed as an anti-platonistic metaphysical doctrine. I argue that there are strong reasons to doubt that all proofs are fully formalisable, if formal proofs are required to be finitary, but that, on a proper view of the way in which formal proofs idealise actual practice, this restriction is unjustified and formalism is not threatened
Coinductive Formal Reasoning in Exact Real Arithmetic
In this article we present a method for formally proving the correctness of
the lazy algorithms for computing homographic and quadratic transformations --
of which field operations are special cases-- on a representation of real
numbers by coinductive streams. The algorithms work on coinductive stream of
M\"{o}bius maps and form the basis of the Edalat--Potts exact real arithmetic.
We use the machinery of the Coq proof assistant for the coinductive types to
present the formalisation. The formalised algorithms are only partially
productive, i.e., they do not output provably infinite streams for all possible
inputs. We show how to deal with this partiality in the presence of syntactic
restrictions posed by the constructive type theory of Coq. Furthermore we show
that the type theoretic techniques that we develop are compatible with the
semantics of the algorithms as continuous maps on real numbers. The resulting
Coq formalisation is available for public download.Comment: 40 page
Experience Implementing a Performant Category-Theory Library in Coq
We describe our experience implementing a broad category-theory library in
Coq. Category theory and computational performance are not usually mentioned in
the same breath, but we have needed substantial engineering effort to teach Coq
to cope with large categorical constructions without slowing proof script
processing unacceptably. In this paper, we share the lessons we have learned
about how to represent very abstract mathematical objects and arguments in Coq
and how future proof assistants might be designed to better support such
reasoning. One particular encoding trick to which we draw attention allows
category-theoretic arguments involving duality to be internalized in Coq's
logic with definitional equality. Ours may be the largest Coq development to
date that uses the relatively new Coq version developed by homotopy type
theorists, and we reflect on which new features were especially helpful.Comment: The final publication will be available at link.springer.com. This
version includes a full bibliography which does not fit in the Springer
version; other than the more complete references, this is the version
submitted as a final copy to ITP 201
An Improved Implementation and Abstract Interface for Hybrid
Hybrid is a formal theory implemented in Isabelle/HOL that provides an
interface for representing and reasoning about object languages using
higher-order abstract syntax (HOAS). This interface is built around an HOAS
variable-binding operator that is constructed definitionally from a de Bruijn
index representation. In this paper we make a variety of improvements to
Hybrid, culminating in an abstract interface that on one hand makes Hybrid a
more mathematically satisfactory theory, and on the other hand has important
practical benefits. We start with a modification of Hybrid's type of terms that
better hides its implementation in terms of de Bruijn indices, by excluding at
the type level terms with dangling indices. We present an improved set of
definitions, and a series of new lemmas that provide a complete
characterization of Hybrid's primitives in terms of properties stated at the
HOAS level. Benefits of this new package include a new proof of adequacy and
improvements to reasoning about object logics. Such proofs are carried out at
the higher level with no involvement of the lower level de Bruijn syntax.Comment: In Proceedings LFMTP 2011, arXiv:1110.668
Mathematics and language
This essay considers the special character of mathematical reasoning, and
draws on observations from interactive theorem proving and the history of
mathematics to clarify the nature of formal and informal mathematical language.
It proposes that we view mathematics as a system of conventions and norms that
is designed to help us make sense of the world and reason efficiently. Like any
designed system, it can perform well or poorly, and the philosophy of
mathematics has a role to play in helping us understand the general principles
by which it serves its purposes well
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