17,166 research outputs found
Mass problems and intuitionistic higher-order logic
In this paper we study a model of intuitionistic higher-order logic which we
call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the
category of sheaves of sets over the topological space consisting of the Turing
degrees, where the Turing cones form a base for the topology. We note that our
Muchnik topos interpretation of intuitionistic mathematics is an extension of
the well known Kolmogorov/Muchnik interpretation of intuitionistic
propositional calculus via Muchnik degrees, i.e., mass problems under weak
reducibility. We introduce a new sheaf representation of the intuitionistic
real numbers, \emph{the Muchnik reals}, which are different from the Cauchy
reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice
principle} and a \emph{bounding principle} where range over Muchnik
reals, ranges over functions from Muchnik reals to Muchnik reals, and
is a formula not containing or . For the convenience of the
reader, we explain all of the essential background material on intuitionism,
sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems,
Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an
English translation of Muchnik's 1963 paper on Muchnik degrees.Comment: 44 page
Signatures and Induction Principles for Higher Inductive-Inductive Types
Higher inductive-inductive types (HIITs) generalize inductive types of
dependent type theories in two ways. On the one hand they allow the
simultaneous definition of multiple sorts that can be indexed over each other.
On the other hand they support equality constructors, thus generalizing higher
inductive types of homotopy type theory. Examples that make use of both
features are the Cauchy real numbers and the well-typed syntax of type theory
where conversion rules are given as equality constructors. In this paper we
propose a general definition of HIITs using a small type theory, named the
theory of signatures. A context in this theory encodes a HIIT by listing the
constructors. We also compute notions of induction and recursion for HIITs, by
using variants of syntactic logical relation translations. Building full
categorical semantics and constructing initial algebras is left for future
work. The theory of HIIT signatures was formalised in Agda together with the
syntactic translations. We also provide a Haskell implementation, which takes
signatures as input and outputs translation results as valid Agda code
Impredicative Encodings of (Higher) Inductive Types
Postulating an impredicative universe in dependent type theory allows System
F style encodings of finitary inductive types, but these fail to satisfy the
relevant {\eta}-equalities and consequently do not admit dependent eliminators.
To recover {\eta} and dependent elimination, we present a method to construct
refinements of these impredicative encodings, using ideas from homotopy type
theory. We then extend our method to construct impredicative encodings of some
higher inductive types, such as 1-truncation and the unit circle S1
Functions out of Higher Truncations
In homotopy type theory, the truncation operator ||-||n (for a number n > -2)
is often useful if one does not care about the higher structure of a type and
wants to avoid coherence problems. However, its elimination principle only
allows to eliminate into n-types, which makes it hard to construct functions
||A||n -> B if B is not an n-type. This makes it desirable to derive more
powerful elimination theorems. We show a first general result: If B is an
(n+1)-type, then functions ||A||n -> B correspond exactly to functions A -> B
which are constant on all (n+1)-st loop spaces. We give one "elementary" proof
and one proof that uses a higher inductive type, both of which require some
effort. As a sample application of our result, we show that we can construct
"set-based" representations of 1-types, as long as they have "braided" loop
spaces. The main result with one of its proofs and the application have been
formalised in Agda.Comment: 15 pages; to appear at CSL'1
Lower Bounds for RAMs and Quantifier Elimination
We are considering RAMs , with wordlength , whose arithmetic
instructions are the arithmetic operations multiplication and addition modulo
, the unary function , the binary
functions (with ), ,
, and the boolean vector operations defined on
sequences of length . It also has the other RAM instructions. The size
of the memory is restricted only by the address space, that is, it is
words. The RAMs has a finite instruction set, each instruction is encoded by a
fixed natural number independently of . Therefore a program can run on
each machine , if is sufficiently large. We show that there
exists an and a program , such that it satisfies the following
two conditions.
(i) For all sufficiently large , if running on gets an
input consisting of two words and , then, in constant time, it gives a
output .
(ii) Suppose that is a program such that for each sufficiently large
, if , running on , gets a word of length as an
input, then it decides whether there exists a word of length such that
. Then, for infinitely many positive integers , there exists a
word of length , such that the running time of on at
input is at least
Two-Level Type Theory and Applications
We define and develop two-level type theory (2LTT), a version of Martin-L\"of
type theory which combines two different type theories. We refer to them as the
inner and the outer type theory. In our case of interest, the inner theory is
homotopy type theory (HoTT) which may include univalent universes and higher
inductive types. The outer theory is a traditional form of type theory
validating uniqueness of identity proofs (UIP). One point of view on it is as
internalised meta-theory of the inner type theory.
There are two motivations for 2LTT. Firstly, there are certain results about
HoTT which are of meta-theoretic nature, such as the statement that
semisimplicial types up to level can be constructed in HoTT for any
externally fixed natural number . Such results cannot be expressed in HoTT
itself, but they can be formalised and proved in 2LTT, where will be a
variable in the outer theory. This point of view is inspired by observations
about conservativity of presheaf models.
Secondly, 2LTT is a framework which is suitable for formulating additional
axioms that one might want to add to HoTT. This idea is heavily inspired by
Voevodsky's Homotopy Type System (HTS), which constitutes one specific instance
of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves
like the external natural numbers, which allows the construction of a universe
of semisimplicial types. In 2LTT, this axiom can be stated simply be asking the
inner and outer natural numbers to be isomorphic.
After defining 2LTT, we set up a collection of tools with the goal of making
2LTT a convenient language for future developments. As a first such
application, we develop the theory of Reedy fibrant diagrams in the style of
Shulman. Continuing this line of thought, we suggest a definition of
(infinity,1)-category and give some examples.Comment: 53 page
Higher Homotopies in a Hierarchy of Univalent Universes
For Martin-Lof type theory with a hierarchy U(0): U(1): U(2): ... of
univalent universes, we show that U(n) is not an n-type. Our construction also
solves the problem of finding a type that strictly has some high truncation
level without using higher inductive types. In particular, U(n) is such a type
if we restrict it to n-types. We have fully formalized and verified our results
within the dependently typed language and proof assistant Agda.Comment: v1: 30 pages, main results and a connectedness construction; v2: 14
pages, only main results, improved presentation, final journal version,
ancillary files with electronic appendix; v3: content unchanged, different
documentclass reduced the number of pages to 1
A realizability semantics for inductive formal topologies, Church's Thesis and Axiom of Choice
We present a Kleene realizability semantics for the intensional level of the
Minimalist Foundation, for short mtt, extended with inductively generated
formal topologies, Church's thesis and axiom of choice. This semantics is an
extension of the one used to show consistency of the intensional level of the
Minimalist Foundation with the axiom of choice and formal Church's thesis in
previous work. A main novelty here is that such a semantics is formalized in a
constructive theory represented by Aczel's constructive set theory CZF extended
with the regular extension axiom
- …