90 research outputs found
Inductive-Inductive Definitions
This article presents a new extension of inductive definitions, namely inductive-inductive definitions
A Categorical Semantics for Inductive-Inductive Definitions
Induction-induction is a principle for defining data types in Martin-Löf Type Theory. An inductive-inductive definition consists of a set A, together with an A-indexed family B : AâââSet, where both A and B are inductively defined in such a way that the constructors for A can refer to B and vice versa. In addition, the constructors for B can refer to the constructors for A. We extend the usual initial algebra semantics for ordinary inductive data types to the inductive-inductive setting by considering dialgebras instead of ordinary algebras. This gives a new and compact formalisation of inductive-inductive definitions, which we prove is equivalent to the usual formulation with elimination rules
Homotopy Type Theory in Lean
We discuss the homotopy type theory library in the Lean proof assistant. The
library is especially geared toward synthetic homotopy theory. Of particular
interest is the use of just a few primitive notions of higher inductive types,
namely quotients and truncations, and the use of cubical methods.Comment: 17 pages, accepted for ITP 201
Quotient inductive-inductive types
Higher inductive types (HITs) in Homotopy Type Theory (HoTT) allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types and allow to define types which are not sets in the sense of HoTT (i.e. do not satisfy uniqueness of equality proofs) such as spheres, suspensions and the torus. However, there are also interesting uses of HITs to define sets, such as the Cauchy reals, the partiality monad, and the internal, total syntax of type theory. In each of these examples we define several types that depend on each other mutually, i.e. they are inductive-inductive definitions. We call those HITs quotient inductive-inductive types (QIITs).
Although there has been recent progress on the general theory of HITs, there isn't yet a theoretical foundation of the combination of equality constructors and induction-induction, despite having many interesting applications. In the present paper we present a first step towards a semantic definition of QIITs. In particular, we give an initial-algebra semantics and show that this is equivalent to the section induction principle, which justifies the intuitively expected elimination rules
Indexed induction and coinduction, fibrationally.
This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a sound coinduction rule for any data type arising as the final coalgebra of a functor, thus relaxing Hermida and Jacobsâ restriction to polynomial data types. For this we introduce the notion of a quotient category with equality (QCE), which both abstracts the standard notion of a fibration of relations constructed from a given fibration, and plays a role in the theory of coinduction dual to that of a comprehension category with unit (CCU) in the theory of induction. Second, we show that indexed inductive and coinductive types also admit sound induction and coinduction rules. Indexed data types often arise as initial algebras and final coalgebras of functors on slice categories, so our key technical results give sufficent conditions under which we can construct, from a CCU (QCE) U : E -> B, a fibration with base B/I that models indexing by I and is also a CCU (QCE)
Refining Inductive Types
Dependently typed programming languages allow sophisticated properties of
data to be expressed within the type system. Of particular use in dependently
typed programming are indexed types that refine data by computationally useful
information. For example, the N-indexed type of vectors refines lists by their
lengths. Other data types may be refined in similar ways, but programmers must
produce purpose-specific refinements on an ad hoc basis, developers must
anticipate which refinements to include in libraries, and implementations must
often store redundant information about data and their refinements. In this
paper we show how to generically derive inductive characterisations of
refinements of inductive types, and argue that these characterisations can
alleviate some of the aforementioned difficulties associated with ad hoc
refinements. Our characterisations also ensure that standard techniques for
programming with and reasoning about inductive types are applicable to
refinements, and that refinements can themselves be further refined
Algebraic totality, towards completeness
Finiteness spaces constitute a categorical model of Linear Logic (LL) whose
objects can be seen as linearly topologised spaces, (a class of topological
vector spaces introduced by Lefschetz in 1942) and morphisms as continuous
linear maps. First, we recall definitions of finiteness spaces and describe
their basic properties deduced from the general theory of linearly topologised
spaces. Then we give an interpretation of LL based on linear algebra. Second,
thanks to separation properties, we can introduce an algebraic notion of
totality candidate in the framework of linearly topologised spaces: a totality
candidate is a closed affine subspace which does not contain 0. We show that
finiteness spaces with totality candidates constitute a model of classical LL.
Finally, we give a barycentric simply typed lambda-calculus, with booleans
and a conditional operator, which can be interpreted in this
model. We prove completeness at type for
every n by an algebraic method
Code Generation for Higher Inductive Types
Higher inductive types are inductive types that include nontrivial
higher-dimensional structure, represented as identifications that are not
reflexivity. While work proceeds on type theories with a computational
interpretation of univalence and higher inductive types, it is convenient to
encode these structures in more traditional type theories with mature
implementations. However, these encodings involve a great deal of error-prone
additional syntax. We present a library that uses Agda's metaprogramming
facilities to automate this process, allowing higher inductive types to be
specified with minimal additional syntax.Comment: 16 pages, Accepted for presentation in WFLP 201
Your Proof Fails? Testing Helps to Find the Reason
Applying deductive verification to formally prove that a program respects its
formal specification is a very complex and time-consuming task due in
particular to the lack of feedback in case of proof failures. Along with a
non-compliance between the code and its specification (due to an error in at
least one of them), possible reasons of a proof failure include a missing or
too weak specification for a called function or a loop, and lack of time or
simply incapacity of the prover to finish a particular proof. This work
proposes a new methodology where test generation helps to identify the reason
of a proof failure and to exhibit a counter-example clearly illustrating the
issue. We describe how to transform an annotated C program into C code suitable
for testing and illustrate the benefits of the method on comprehensive
examples. The method has been implemented in STADY, a plugin of the software
analysis platform FRAMA-C. Initial experiments show that detecting
non-compliances and contract weaknesses allows to precisely diagnose most proof
failures.Comment: 11 pages, 10 figure
Generic Fibrational Induction
This paper provides an induction rule that can be used to prove properties of
data structures whose types are inductive, i.e., are carriers of initial
algebras of functors. Our results are semantic in nature and are inspired by
Hermida and Jacobs' elegant algebraic formulation of induction for polynomial
data types. Our contribution is to derive, under slightly different
assumptions, a sound induction rule that is generic over all inductive types,
polynomial or not. Our induction rule is generic over the kinds of properties
to be proved as well: like Hermida and Jacobs, we work in a general fibrational
setting and so can accommodate very general notions of properties on inductive
types rather than just those of a particular syntactic form. We establish the
soundness of our generic induction rule by reducing induction to iteration. We
then show how our generic induction rule can be instantiated to give induction
rules for the data types of rose trees, finite hereditary sets, and
hyperfunctions. The first of these lies outside the scope of Hermida and
Jacobs' work because it is not polynomial, and as far as we are aware, no
induction rules have been known to exist for the second and third in a general
fibrational framework. Our instantiation for hyperfunctions underscores the
value of working in the general fibrational setting since this data type cannot
be interpreted as a set.Comment: For Special Issue from CSL 201
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