1,173 research outputs found
A Relationally Parametric Model of Dependent Type Theory
Reynolds’ theory of relational parametricity captures the invariance of polymorphically typed programs under change of data representation. Reynolds’ original work exploited the typing discipline of the polymorphically typed -calculus System F, but there is now considerable interest in extending relational parametricity to type systems that are richer and more expressive than that of System F.This paper constructs parametric models of predicative and impredicative dependent type theory. The significance of our models is twofold. Firstly, in the impredicative variant we are able to deduce the existence of initial algebras for all indexed functors. To our knowledge, ours is the first account of parametricity for dependent types that is able to lift the useful deduction of the existence of initial algebras in parametric models of System F to the dependently typed setting. Secondly, our models offer conceptual clarity by uniformly expressing relational parametricity for dependent types in terms of reflexive graphs, which allows us to unify the interpretations of types and kinds, instead of taking the relational interpretation of types as a primitive notion. Expressing our model in terms of reflexive graphs ensures that it has canonical choices for the interpretations of the standard type constructors of dependent type theory, except for the interpretation of the universe of small types, where we formulate a refined interpretation tailored for relational parametricity. Moreover, our reflexive graph model opens the door to generalizations of relational parametricity, for example to higher-dimensional relational parametricity
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
Relational Parametricity and Separation Logic
Separation logic is a recent extension of Hoare logic for reasoning about
programs with references to shared mutable data structures. In this paper, we
provide a new interpretation of the logic for a programming language with
higher types. Our interpretation is based on Reynolds's relational
parametricity, and it provides a formal connection between separation logic and
data abstraction
Fibred Fibration Categories
We introduce fibred type-theoretic fibration categories which are fibred
categories between categorical models of Martin-L\"{o}f type theory. Fibred
type-theoretic fibration categories give a categorical description of logical
predicates for identity types. As an application, we show a relational
parametricity result for homotopy type theory. As a corollary, it follows that
every closed term of type of polymorphic endofunctions on a loop space is
homotopic to some iterated concatenation of a loop
From parametricity to conservation laws, via Noether's Theorem
Invariance is of paramount importance in programming languages and in physics. In programming languages, John Reynolds' theory of relational parametricity demonstrates that parametric polymorphic programs are invariant under change of data representation, a property that yields "free" theorems about programs just from their types. In physics, Emmy Noether showed that if the action of a physical system is invariant under change of coordinates, then the physical system has a conserved quantity: a quantity that remains constant for all time. Knowledge of conserved quantities can reveal deep properties of physical systems. For example, the conservation of energy is by Noether's theorem a consequence of a system's invariance under time-shifting. In this paper, we link Reynolds' relational parametricity with Noether's theorem for deriving conserved quantities. We propose an extension of System Fω with new kinds, types and term constants for writing programs that describe classical mechanical systems in terms of their Lagrangians. We show, by constructing a relationally parametric model of our extension of Fω, that relational parametricity is enough to satisfy the hypotheses of Noether's theorem, and so to derive conserved quantities for free, directly from the polymorphic types of Lagrangians expressed in our system
Internal Parametricity for Cubical Type Theory
We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we call relativity. We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, and we give an account of the identity extension lemma for internal parametricity
The Structure of Cluster Knowledge Networks Uneven, not Pervasive and Collective
This study focuses on the relationship between industrial clustering and innovation. It contributes to this literature by showing two empirical properties of the cluster learning process: first, that the structure of the knowledge network in a cluster is related with the heterogeneous distribution of firm knowledge bases and, second, that business interactions and inter-firm knowledge flows are not highly co-occurring phenomena. In particular, this paper highlights how the heterogeneity of firms’ knowledge bases generates uneven distribution of knowledge and selective inter-firm learning. This study has been based on empirical evidence collected at firm level in three wine clusters in Italy and Chile. Methods of social network analysis have been applied to process the data.Industrial clusters, knowledge flows, business interactions, networks.
Media Interaction on Relationally Aggressive Behaviors of Middle School Girls
Using a quantitative approach, this study investigates media interaction on relationally aggressive behaviors of middle-school girls by examining television consumption and an individual\u27s proclivity to engage in relational aggression. It also investigates whether participation in a workshop explaining relational aggression assisted participants in recognizing the behavior and its consequences on aggressors and victims in the Disney Channel\u27s Suite Life of Zach & Cody. Results indicate that the amount of television watched does not correlate with participation in the behavior generally, but that the use of sarcasm to hurt a friend decreases as television viewing increases. Results also indicate that knowledge about the behavior is associated with awareness of occurrences and consequences to the aggressor, but not with consequences to the victim. Together, these results should have implications for regulations regarding television violence and mediation of relational aggressio
Relational Cost Analysis for Functional-Imperative Programs
Relational cost analysis aims at formally establishing bounds on the
difference in the evaluation costs of two programs. As a particular case, one
can also use relational cost analysis to establish bounds on the difference in
the evaluation cost of the same program on two different inputs. One way to
perform relational cost analysis is to use a relational type-and-effect system
that supports reasoning about relations between two executions of two programs.
Building on this basic idea, we present a type-and-effect system, called
ARel, for reasoning about the relative cost of array-manipulating, higher-order
functional-imperative programs. The key ingredient of our approach is a new
lightweight type refinement discipline that we use to track relations
(differences) between two arrays. This discipline combined with Hoare-style
triples built into the types allows us to express and establish precise
relative costs of several interesting programs which imperatively update their
data.Comment: 14 page
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