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12
A Semantics for Static Type Inference
 Information and Computation
, 1993
"... Curry's system for Fdeducibility is the basis for static type inference algorithms for programming languages such as ML. If a natural "preservation of types by conversion" rule is added to Curry's system, it becomes undecidable, but complete relative to a variety of model classes. We show compl ..."
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Curry's system for Fdeducibility is the basis for static type inference algorithms for programming languages such as ML. If a natural "preservation of types by conversion" rule is added to Curry's system, it becomes undecidable, but complete relative to a variety of model classes. We show completeness for Curry's system itself, relative to an extended notion of model that validates reduction but not conversion.
Aspect Oriented Programming: a language for 2categories
"... AspectOriented Programming (AOP) started ten years ago with the remark that modularization of socalled crosscutting functionalities is a fundamental problem for the engineering of largescale applications. Originating at Xerox PARC, this observation has sparked the development of a new style of pr ..."
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AspectOriented Programming (AOP) started ten years ago with the remark that modularization of socalled crosscutting functionalities is a fundamental problem for the engineering of largescale applications. Originating at Xerox PARC, this observation has sparked the development of a new style of programming featured that is gradually gaining traction, as it is the case for the related concept of code injection, in the guise of frameworks such as Swing and Google Guice. However, AOP lacks theoretical foundations to clarify this new idea. This paper proposes to put a bridge between AOP and the notion of 2category to enhance the conceptual understanding of AOP. Starting from the connection between the λcalculus and the theory of categories, we propose to see an aspect as a morphism between morphisms—that is as a program that transforms the execution of a program. To make this connection precise, we develop an advised λcalculus that provides an internal language for 2categories and show how it can be used as a base for the definition of the weaving mechanism of a realistic functional AOP language, called MinAML.
2Dimensional Directed Type Theory
"... Recent work on higherdimensional type theory has explored connections between MartinLöf type theory, higherdimensional category theory, and homotopy theory. These connections suggest a generalization of dependent type theory to account for computationally relevant proofs of propositional equality ..."
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Recent work on higherdimensional type theory has explored connections between MartinLöf type theory, higherdimensional category theory, and homotopy theory. These connections suggest a generalization of dependent type theory to account for computationally relevant proofs of propositional equality—for example, taking IdSet A B to be the isomorphisms between A and B. The crucial observation is that all of the familiar type and term constructors can be equipped with a functorial action that describes how they preserve such proofs. The key benefit of higherdimensional type theory is that programmers and mathematicians may work up to isomorphism and higher equivalence, such as equivalence of categories. In this paper, we consider a further generalization of higherdimensional type theory, which associates each type with a directed notion of transformation between its elements. Directed type theory accounts for phenomena not expressible in symmetric higherdimensional type theory, such as a universe set of sets and functions, and a type Ctx used in functorial abstract syntax. Our formulation requires two main ingredients: First, the types themselves must be reinterpreted to take account of variance; for example, a Π type is contravariant in its domain, but covariant in its range. Second, whereas in symmetric type theory proofs of equivalence can be internalized using the MartinLöf identity type, in directed type theory the twodimensional structure must be made explicit at the judgemental level. We describe a 2dimensional directed type theory, or 2DTT, which is validated by an interpretation into the strict 2category Cat of categories, functors, and natural transformations. We also discuss applications of 2DTT for programming with abstract syntax, generalizing the functorial approach to syntax to the dependently typed and mixedvariance case. 1
2Categorical Specification of Partial Algebras
 Recent Trends in Data Type Specification, Proc. 9th Workshop on Specification of Abstract Data Types, Caldes de Malavella
, 1992
"... The purpose of this paper is to present a short survey of possible results of an application of general concepts from categorical algebra to the specification of partial algebras with conditional existence equations. The general concept, which models theories (= formulas and equivalence classes of t ..."
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The purpose of this paper is to present a short survey of possible results of an application of general concepts from categorical algebra to the specification of partial algebras with conditional existence equations. The general concept, which models theories (= formulas and equivalence classes of terms) as categories, is extended to 2categories, such that rewriting between terms can be made explicit. To make clear the benefits of such an approach the results are presented in the usual terminology of algebraic specifications. 1 Introduction The aim of this paper is to use concepts from categorical algebra to obtain a clear and extendable description of partial algebras and their specification. For the more general case, categorical algebra provides such a clear and extendable methodology, which could briefly be described as follows. ffl A theory is a category with a certain structure, called the syntactic category. (E.g. an equational theory of total many sorted operations is a cat...
On Double Categories and Multiplicative Linear Logic
, 1999
"... this article, we attack the converse problem of explaining semantics as an artifact of syntax, in other words, of extracting the meaning of a program from syntactical considerations on its dynamics, or the way it interacts with the environment. We start the analysis with a very simple slogan, where ..."
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this article, we attack the converse problem of explaining semantics as an artifact of syntax, in other words, of extracting the meaning of a program from syntactical considerations on its dynamics, or the way it interacts with the environment. We start the analysis with a very simple slogan, where we use module to mean procedure, in the fashion of (Girard 1987b):
betaetaEquality for Coproducts
 In Typed calculus and Applications, number 902 in Lecture Notes in Computer Science
, 1995
"... . Recently several researchers have investigated fijequality for the simply typed calculus with exponentials, products and unit types. In these works, jconversion was interpreted as an expansion with syntactic restrictions imposed to prevent the expansion of introduction terms or terms which for ..."
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. Recently several researchers have investigated fijequality for the simply typed calculus with exponentials, products and unit types. In these works, jconversion was interpreted as an expansion with syntactic restrictions imposed to prevent the expansion of introduction terms or terms which form the major premise of elimination rules. The resulting rewrite relation was shown confluent and strongly normalising to the long fijnormal forms. Thus reduction to normal form provides a decision procedure for fijequality. This paper extends these methods to sum types. Although this extension was originally thought to be straight forward, the proposed jrule for the sum is substantially more complex than that for the exponent or product and contains features not present in the previous systems. Not only is there a facility for expanding terms of sum type analogous to that for product and exponential, but also the ability to permute the order in which different subterms of sum type are e...
Categorifying Fundamental Physics
"... Despite the incredible progress over most of the 20th century, and a continuing flow of new observational discoveries — neutrino oscillations, dark matter, dark energy, and evidence for inflation — theoretical research in fundamental physics seems to be in a ‘stuck ’ phase. So, now more than ever, i ..."
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Despite the incredible progress over most of the 20th century, and a continuing flow of new observational discoveries — neutrino oscillations, dark matter, dark energy, and evidence for inflation — theoretical research in fundamental physics seems to be in a ‘stuck ’ phase. So, now more than ever, it seems important to reexamine basic assumptions and seek fundamentally new ideas. Work along these lines is inherently risky: many different directions must be explored, since while few will lead to important new insights, it is hard to know in advance which these will be. For this reason, I have ceased for now to work on loop quantum gravity, and begun to rethink basic mathematical structures in physics. The unifying idea behind this multipronged project is ‘categorification’, or in simple terms: giving up the naive concept of equality. While equations play an utterly fundamental role in physics, and this is unlikely to change, equations between elements of a set often arise as a shorthand for something deeper: isomorphisms between objects in a category.
Foundations and Applications of HigherDimensional Directed Type Theory
"... Intuitionistic type theory [43] is an expressive formalism that unifies mathematics and computation. A central concept is the propositionsastypes principle, according to which propositions are interpreted as types, and proofs of a proposition are interpreted as programs of the associated type. Mat ..."
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Intuitionistic type theory [43] is an expressive formalism that unifies mathematics and computation. A central concept is the propositionsastypes principle, according to which propositions are interpreted as types, and proofs of a proposition are interpreted as programs of the associated type. Mathematical propositions are thereby to be understood as specifications, or problem descriptions, that are solved by providing a program that meets the specification. Conversely, a program can, by the same token, be understood as a proof of its type viewed as a proposition. Over the last quartercentury type theory has emerged as the central organizing principle of programming language research, through the identification of the informal concept of language features with type structure. Numerous benefits accrue from the identification of proofs and programs in type theory. First, it provides the foundation for integrating types and verification, the two most successful formal methods used to ensure the correctness of software. Second, it provides a language for the mechanization of mathematics in which proof checking is equivalent to type checking, and proof search is equivalent to writing a program to meet a specification.
2Dimensional Directed Dependent Type Theory
 SUBMITTED TO POPL 2011
, 2011
"... The groupoid interpretation of dependent type theory given by Hofmann and Streicher associates to each closed type a category whose objects represent the elements of that type and whose maps represent proofs of equality of elements. The categorial structure ensures that equality is reflexive (identi ..."
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The groupoid interpretation of dependent type theory given by Hofmann and Streicher associates to each closed type a category whose objects represent the elements of that type and whose maps represent proofs of equality of elements. The categorial structure ensures that equality is reflexive (identity maps) and transitive (closure under composition); the groupoid structure, which demands that every map be invertible, ensures symmetry. Families of types are interpreted as functors; the action on maps (equality proofs) ensures that families respect equality of elements of the index type. The functorial action of a type family is computationally nontrivial in the case that the groupoid associated to the index type is nontrivial. For example, one may identity elements of a universe of sets up to isomorphism, in which case the action of a family of types indexed by sets must respect set isomorphism. The groupoid interpretation is 2dimensional in that the coherence requirements on proofs of equality are required to hold “on the