885 research outputs found
Extensionality of simply typed logic programs
We set up a framework for the study of extensionality in the context of higher-order logic programming. For simply typed logic programs we propose a novel declarative semantics, consisting of a model class with a semi-computable initial model, and a notion of extensionality. We show that the initial model of a simply typed logic program, in case the program is extensional, collapses into a simple, set-theoretic representation. Given the undecidability of extensionality in general, we develop a decidable, syntactic criterion which is sufficient for extensionality. Some typical examples of higher-order logic programs are shown to be extensional
Relational parametricity for higher kinds
Reynolds’ notion of relational parametricity has been extremely influential and well studied for polymorphic programming languages and type theories based on System F. The extension of relational parametricity to higher kinded polymorphism, which allows quantification over type operators as well as types, has not received as much attention. We present a model of relational parametricity for System Fω, within the impredicative Calculus of Inductive Constructions, and show how it forms an instance of a general class of models defined by Hasegawa. We investigate some of the consequences of our model and show that it supports the definition of inductive types, indexed by an arbitrary kind, and with reasoning principles provided by initiality
On Equivalence and Canonical Forms in the LF Type Theory
Decidability of definitional equality and conversion of terms into canonical
form play a central role in the meta-theory of a type-theoretic logical
framework. Most studies of definitional equality are based on a confluent,
strongly-normalizing notion of reduction. Coquand has considered a different
approach, directly proving the correctness of a practical equivalance algorithm
based on the shape of terms. Neither approach appears to scale well to richer
languages with unit types or subtyping, and neither directly addresses the
problem of conversion to canonical.
In this paper we present a new, type-directed equivalence algorithm for the
LF type theory that overcomes the weaknesses of previous approaches. The
algorithm is practical, scales to richer languages, and yields a new notion of
canonical form sufficient for adequate encodings of logical systems. The
algorithm is proved complete by a Kripke-style logical relations argument
similar to that suggested by Coquand. Crucially, both the algorithm itself and
the logical relations rely only on the shapes of types, ignoring dependencies
on terms.Comment: 41 page
On Irrelevance and Algorithmic Equality in Predicative Type Theory
Dependently typed programs contain an excessive amount of static terms which
are necessary to please the type checker but irrelevant for computation. To
separate static and dynamic code, several static analyses and type systems have
been put forward. We consider Pfenning's type theory with irrelevant
quantification which is compatible with a type-based notion of equality that
respects eta-laws. We extend Pfenning's theory to universes and large
eliminations and develop its meta-theory. Subject reduction, normalization and
consistency are obtained by a Kripke model over the typed equality judgement.
Finally, a type-directed equality algorithm is described whose completeness is
proven by a second Kripke model.Comment: 36 pages, superseds the FoSSaCS 2011 paper of the first author,
titled "Irrelevance in Type Theory with a Heterogeneous Equality Judgement
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
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
Call-by-name Gradual Type Theory
We present gradual type theory, a logic and type theory for call-by-name
gradual typing. We define the central constructions of gradual typing (the
dynamic type, type casts and type error) in a novel way, by universal
properties relative to new judgments for gradual type and term dynamism, which
were developed in blame calculi and to state the "gradual guarantee" theorem of
gradual typing. Combined with the ordinary extensionality () principles
that type theory provides, we show that most of the standard operational
behavior of casts is uniquely determined by the gradual guarantee. This
provides a semantic justification for the definitions of casts, and shows that
non-standard definitions of casts must violate these principles. Our type
theory is the internal language of a certain class of preorder categories
called equipments. We give a general construction of an equipment interpreting
gradual type theory from a 2-category representing non-gradual types and
programs, which is a semantic analogue of Findler and Felleisen's definitions
of contracts, and use it to build some concrete domain-theoretic models of
gradual typing
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
A Case Study on Logical Relations using Contextual Types
Proofs by logical relations play a key role to establish rich properties such
as normalization or contextual equivalence. They are also challenging to
mechanize. In this paper, we describe the completeness proof of algorithmic
equality for simply typed lambda-terms by Crary where we reason about logically
equivalent terms in the proof environment Beluga. There are three key aspects
we rely upon: 1) we encode lambda-terms together with their operational
semantics and algorithmic equality using higher-order abstract syntax 2) we
directly encode the corresponding logical equivalence of well-typed
lambda-terms using recursive types and higher-order functions 3) we exploit
Beluga's support for contexts and the equational theory of simultaneous
substitutions. This leads to a direct and compact mechanization, demonstrating
Beluga's strength at formalizing logical relations proofs.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759
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