142 research outputs found
Undecidability of Equality in the Free Locally Cartesian Closed Category (Extended version)
We show that a version of Martin-L\"of type theory with an extensional
identity type former I, a unit type N1 , Sigma-types, Pi-types, and a base type
is a free category with families (supporting these type formers) both in a 1-
and a 2-categorical sense. It follows that the underlying category of contexts
is a free locally cartesian closed category in a 2-categorical sense because of
a previously proved biequivalence. We show that equality in this category is
undecidable by reducing it to the undecidability of convertibility in
combinatory logic. Essentially the same construction also shows a slightly
strengthened form of the result that equality in extensional Martin-L\"of type
theory with one universe is undecidable
The extended predicative Mahlo universe in Martin-Lof type theory
This paper addresses the long-standing question of the predicativity of the Mahlo universe. A solution, called the extended predicative Mahlo universe, has been proposed by Kahle and Setzer in the context of explicit mathematics. It makes use of the collection of untyped terms (denoting partial functions) which are directly available in explicit mathematics but not in Martin-Lof type theory. In this paper, we overcome the obstacle of not having direct access to untyped terms in Martin-Lof type theory by formalizing explicit mathematics with an extended predicative Mahlo universe in Martin-Lof type theory with certain indexed inductive-recursive definitions. In this way, we can relate the predicativity question to the fundamental semantics of Martin-Lof type theory in terms of computation to canonical form. As a result, we get the first extended predicative definition of a Mahlo universe in Martin-Lof type theory. To this end, we first define an external variant of Kahle and Setzer\u27s internal extended predicative universe in explicit mathematics. This is then formalized in Martin-Lof type theory, where it becomes an internal extended predicative Mahlo universe. Although we make use of indexed inductive-recursive definitions that go beyond the type theory of indexed inductive-recursive definitions defined in previous work by the authors, we argue that they are constructive and predicative in Martin-Lof\u27s sense. The model construction has been type-checked in the proof assistant Agda
Type Theory with Explicit Universe Polymorphism
The aim of this paper is to refine and extend proposals by Sozeau and Tabareau and by Voevodsky for universe polymorphism in type theory. In those systems judgments can depend on explicit constraints between universe levels. We here present a system where we also have products indexed by universe levels and by constraints. Our theory has judgments for internal universe levels, built up from level variables by a successor operation and a binary supremum operation, and also judgments for equality of universe levels
Finitary Higher Inductive Types in the Groupoid Model
A higher inductive type of level 1 (a 1-hit) has constructors for points and paths only, whereas a higher inductive type of level 2 (a 2-hit) has constructors for surfaces too. We restrict attention to finitary higher inductive types and present general schemata for the types of their point, path, and surface constructors. We also derive the elimination and equality rules from the types of constructors for 1-hits and 2-hits. Moreover, we construct a groupoid model for dependent type theory with 2-hits and point out that we obtain a setoid model for dependent type theory with 1-hits by truncating the groupoid model
The Biequivalence of Locally Cartesian Closed Categories and Martin-L\"of Type Theories
Seely's paper "Locally cartesian closed categories and type theory" contains
a well-known result in categorical type theory: that the category of locally
cartesian closed categories is equivalent to the category of Martin-L\"of type
theories with Pi-types, Sigma-types and extensional identity types. However,
Seely's proof relies on the problematic assumption that substitution in types
can be interpreted by pullbacks. Here we prove a corrected version of Seely's
theorem: that the B\'enabou-Hofmann interpretation of Martin-L\"of type theory
in locally cartesian closed categories yields a biequivalence of 2-categories.
To facilitate the technical development we employ categories with families as a
substitute for syntactic Martin-L\"of type theories. As a second result we
prove that if we remove Pi-types the resulting categories with families are
biequivalent to left exact categories.Comment: TLCA 2011 - 10th Typed Lambda Calculi and Applications, Novi Sad :
Serbia (2011
Type Theories with Explicit Universe Polymorphism
The aim of this paper is to refine and extend proposals by Sozeau and
Tabareau and by Voevodsky for universe polymorphism in type theory. In those
systems judgments can depend on explicit constraints between universe levels.
We here present a system where we also have products indexed by universe levels
and by constraints. Our theory has judgments for internal universe levels,
built up from level variables by a successor operation and a binary supremum
operation, and also judgments for equality of universe levels.Comment: This paper was presented at Types'2022 and has been submitted to its
postconference proceeding
On generalized algebraic theories and categories with families
We give a syntax independent formulation of finitely presented generalized algebraic theories as initial objects in categories of categories with families (cwfs) with extra structure. To this end, we simultaneously define the notion of a presentation ÎŁ of a generalized algebraic theory and the associated category CwFÎŁ of small cwfs with a ÎŁ-structure and cwf-morphisms that preserve ÎŁ-structure on the nose. Our definition refers to the purely semantic notion of uniform family of contexts, types, and terms in CwFÎŁ. Furthermore, we show how to syntactically construct an initial cwf with a ÎŁ-structure. This result can be viewed as a generalization of Birkhoffâs completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjerâs construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of Martin-Löf type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a small category with families. Finally, we show how to extend our definition to some generalized algebraic theories that are not finitely presented, such as the theory of contextual cwfs.publishedVersio
Indexed inductionârecursion
AbstractAn indexed inductive definition (IID) is a simultaneous inductive definition of an indexed family of sets. An inductiveârecursive definition (IRD) is a simultaneous inductive definition of a set and a recursive definition of a function on that set. An indexed inductiveârecursive definition (IIRD) is a combination of both.We present a closed theory which allows us to introduce all IIRD in a natural way without much encoding. By specialising it we also get a closed theory of IID. Our theory of IIRD includes essentially all definitions of sets which occur in MartinâLöf type theory. We show in particular that MartinâLöfâs computability predicates for dependent types and Palmgrenâs higher order universes are special kinds of IIRD and thereby clarify why they are constructively acceptable notions.We give two axiomatisations. The first formalises a principle for introducing meaningful IIRD by using the data-construct in the original version of the proof assistant Agda for MartinâLöf type theory. The second one admits a more general form of introduction rule, including the introduction rule for the intensional identity relation, which is not covered by the first axiomatisation. If we add an extensional identity relation to our logical framework, we show that the theories of restricted and general IIRD are equivalent by interpreting them in each other.Finally, we show the consistency of our theories by constructing a model in classical set theory extended by a Mahlo cardinal
Partial Univalence in n-truncated Type Theory
It is well known that univalence is incompatible with uniqueness of identity
proofs (UIP), the axiom that all types are h-sets. This is due to finite h-sets
having non-trivial automorphisms as soon as they are not h-propositions.
A natural question is then whether univalence restricted to h-propositions is
compatible with UIP. We answer this affirmatively by constructing a model where
types are elements of a closed universe defined as a higher inductive type in
homotopy type theory. This universe has a path constructor for simultaneous
"partial" univalent completion, i.e., restricted to h-propositions.
More generally, we show that univalence restricted to -types is
consistent with the assumption that all types are -truncated. Moreover we
parametrize our construction by a suitably well-behaved container, to abstract
from a concrete choice of type formers for the universe.Comment: 21 pages, long version of paper accepted at LICS 202
Anisotropic intrinsic lattice thermal conductivity of phosphorene from first principles
Phosphorene, the single layer counterpart of black phosphorus, is a novel
two-dimensional semiconductor with high carrier mobility and a large
fundamental direct band gap, which has attracted tremendous interest recently.
Its potential applications in nano-electronics and thermoelectrics call for a
fundamental study of the phonon transport. Here, we calculate the intrinsic
lattice thermal conductivity of phosphorene by solving the phonon Boltzmann
transport equation (BTE) based on first-principles calculations. The thermal
conductivity of phosphorene at is
(zigzag) and
(armchair), showing an obvious anisotropy along different directions. The
calculated thermal conductivity fits perfectly to the inverse relation with
temperature when the temperature is higher than Debye temperature (). In comparison to graphene, the minor contribution around
of the ZA mode is responsible for the low thermal conductivity of
phosphorene. In addition, the representative mean free path (MFP), a critical
size for phonon transport, is also obtained.Comment: 5 pages and 6 figures, Supplemental Material available as
http://www.rsc.org/suppdata/cp/c4/c4cp04858j/c4cp04858j1.pd
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