139 research outputs found
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
Introduction to Univalent Foundations of Mathematics with Agda
We introduce Voevodsky's univalent foundations and univalent mathematics, and
explain how to develop them with the computer system Agda, which is based on
Martin-L\"of type theory. Agda allows us to write mathematical definitions,
constructions, theorems and proofs, for example in number theory, analysis,
group theory, topology, category theory or programming language theory,
checking them for logical and mathematical correctness.
Agda is a constructive mathematical system by default, which amounts to
saying that it can also be considered as a programming language for
manipulating mathematical objects. But we can assume the axiom of choice or the
principle of excluded middle for pieces of mathematics that require them, at
the cost of losing the implicit programming-language character of the system.
For a fully constructive development of univalent mathematics in Agda, we would
need to use its new cubical flavour, and we hope these notes provide a base for
researchers interested in learning cubical type theory and cubical Agda as the
next step.
Compared to most expositions of the subject, we work with explicit universe
levels.Comment: 200 pages, extended version of Midlands Graduate School course
(2019), includes Agda-verified mathematics. Sources available at github (as
explained in the pdf file), but not in LaTe
Injective types in univalent mathematics
We investigate the injective types and the algebraically injective types in
univalent mathematics, both in the absence and in the presence of propositional
resizing. Injectivity is defined by the surjectivity of the restriction map
along any embedding, and algebraic injectivity is defined by a given section of
the restriction map along any embedding. Under propositional resizing axioms,
the main results are easy to state: (1) Injectivity is equivalent to the
propositional truncation of algebraic injectivity. (2) The algebraically
injective types are precisely the retracts of exponential powers of universes.
(2a) The algebraically injective sets are precisely the retracts of powersets.
(2b) The algebraically injective -types are precisely the retracts of
exponential powers of universes of -types. (3) The algebraically injective
types are also precisely the retracts of algebras of the partial-map
classifier. From (2) it follows that any universe is embedded as a retract of
any larger universe. In the absence of propositional resizing, we have similar
results which have subtler statements that need to keep track of universe
levels rather explicitly, and are applied to get the results that require
resizing.Comment: Includes revisions after review proces
A rich hierarchy of functionals of finite types
We are considering typed hierarchies of total, continuous functionals using
complete, separable metric spaces at the base types. We pay special attention
to the so called Urysohn space constructed by P. Urysohn. One of the properties
of the Urysohn space is that every other separable metric space can be
isometrically embedded into it. We discuss why the Urysohn space may be
considered as the universal model of possibly infinitary outputs of algorithms.
The main result is that all our typed hierarchies may be topologically
embedded, type by type, into the corresponding hierarchy over the Urysohn
space. As a preparation for this, we prove an effective density theorem that is
also of independent interest.Comment: 21 page
Higher-order Games with Dependent Types
In previous work on higher-order games, we accounted for finite games of
unbounded length by working with continuous outcome functions, which carry
implicit game trees. In this work we make such trees explicit. We use concepts
from dependent type theory to capture history-dependent games, where the set of
available moves at a given position in the game depends on the moves played up
to that point. In particular, games are modelled by a W-type, which is
essentially the same type used by Aczel to model constructive Zermelo-Frankel
set theory (CZF). We have also implemented all our definitions, constructions,
results and proofs in the dependently-typed programming language Agda, which,
in particular, allows us to run concrete examples of computations of optimal
strategies, that is, strategies in subgame perfect equilibrium.Comment: 20 page
Notions of Anonymous Existence in Martin-L\"of Type Theory
As the groupoid model of Hofmann and Streicher proves, identity proofs in
intensional Martin-L\"of type theory cannot generally be shown to be unique.
Inspired by a theorem by Hedberg, we give some simple characterizations of
types that do have unique identity proofs. A key ingredient in these
constructions are weakly constant endofunctions on identity types. We study
such endofunctions on arbitrary types and show that they always factor through
a propositional type, the truncated or squashed domain. Such a factorization is
impossible for weakly constant functions in general (a result by Shulman), but
we present several non-trivial cases in which it can be done. Based on these
results, we define a new notion of anonymous existence in type theory and
compare different forms of existence carefully. In addition, we show possibly
surprising consequences of the judgmental computation rule of the truncation,
in particular in the context of homotopy type theory. All the results have been
formalized and verified in the dependently typed programming language Agda.Comment: 36 pages, to appear in the special issue of TLCA'13 (LMCS
Higher-order games with dependent types
In previous work on higher-order games, we accounted for finite games of unbounded length by working with continuous outcome functions, which carry implicit game trees. In this work we make such trees explicit. We use concepts from dependent type theory to capture history-dependent games, where the set of available moves at a given position in the game depends on the moves played up to that point. In particular, games are modelled by a W-type, which is essentially the same type used by Aczel to model constructive Zermelo-Frankel set theory (CZF). We have also implemented all our definitions, constructions, results and proofs in the dependently-typed programming language Agda, which, in particular, allows us to run concrete examples of computations of optimal strategies, that is, strategies in subgame perfect equillibrium.</p
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