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Categorical Logic of Names and Abstraction in Action Calculi
, 1993
"... ion elimination Definition 3.1. A monoidal category where every object has a commutative comonoid structure is said to be semicartesian. An action category is a K\Omega category with a distinguished admissible commutative comonoid structure on every object. A semicartesian category is cartesi ..."
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Cited by 21 (9 self)
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ion elimination Definition 3.1. A monoidal category where every object has a commutative comonoid structure is said to be semicartesian. An action category is a K\Omega category with a distinguished admissible commutative comonoid structure on every object. A semicartesian category is cartesian if and only if each object carries a unique comonoid structure, and such structures form two natural families, \Delta and !. The naturality means that all morphisms of the category must be comonoid homomorphisms. In action categories, the property of semicartesianness is fixed as structure: on each object, a particular comonoid structure is chosen. This choice may be constrained by some given graphic operations, with respect to which the structures must be admissible. The proof of proposition 2.6 shows that such structures determine the abstraction operators, and are determined by them. This is the essence of the equivalence of action categories and action calculi. As the embodiment of 2...
Generality of proofs and its Brauerian representation
 J. Symbolic Logic
"... The generality of a derivation is an equivalence relation on the set of occurrences of variables in its premises and conclusion such that two occurrences of the same variable are in this relation if and only if they must remain occurrences of the same variable in every generalization of the derivati ..."
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Cited by 14 (4 self)
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The generality of a derivation is an equivalence relation on the set of occurrences of variables in its premises and conclusion such that two occurrences of the same variable are in this relation if and only if they must remain occurrences of the same variable in every generalization of the derivation. The variables in question are propositional or of another type. A generalization of the derivation consists in diversifying variables without changing the rules of inference. This paper examines in the setting of categorial proof theory the conjecture that two derivations with the same premises and conclusions stand for the same proof if and only if they have the same generality. For that purpose generality is defined within a category of equivalence relations on finite ordinals, where composition is rather complicated. Several examples are given of deductive systems of derivations covering fragments of
Decomposing Typed Lambda Calculus Into a Couple of Categorical Programming Languages
 Proc. CTCS , Lect. Notes in Computer Science 953
, 1995
"... . We give two categorical programming languages with variable arrows and associated abstraction/reduction mechanisms, which extend the possibility of categorical programming [Hag87, CF92] in practice. These languages are complementary to each other  one of them provides a firstorder programming s ..."
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Cited by 8 (0 self)
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. We give two categorical programming languages with variable arrows and associated abstraction/reduction mechanisms, which extend the possibility of categorical programming [Hag87, CF92] in practice. These languages are complementary to each other  one of them provides a firstorder programming style whereas the other does higherorder  and are "children" of the simply typed lambda calculus in the sense that we can decompose typed lambda calculus into them and, conversely, the combination of them is equivalent to typed lambda calculus. This decomposition is a consequence of a semantic analysis on typed lambda calculus due to C. Hermida and B. Jacobs [HJ94]. 1 Introduction There have been several attempts applying category theory to designing programming languages directly, especially to typed functional programming languages, since category theory itself has been a typed functional language for various mathematics. If one (possibly a programmer or a mathematician) can regard a cat...
Parameterizations and FixedPoint Operators on Control Categories
 Fundam. Inform
, 2005
"... The #calculus features both variables and names, together with their binding mechanisms. This means that constructions on open terms are necessarily parameterized in two di#erent ways for both variables and names. Semantically, such a construction must be modeled by a biparameterized family of ope ..."
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Cited by 3 (3 self)
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The #calculus features both variables and names, together with their binding mechanisms. This means that constructions on open terms are necessarily parameterized in two di#erent ways for both variables and names. Semantically, such a construction must be modeled by a biparameterized family of operators. In this paper, we study these biparameterized operators on Selinger's categorical models of the # calculus called control categories. The overall development is analogous to that of Lambek's functional completeness of cartesian closed categories via polynomial categories. As a particular and important case, we study parameterizations of uniform fixedpoint operators on control categories, and show bijective correspondences between parameterized fixedpoint operators and nonparameterized ones under uniformity conditions.
LambdaCalculus and Functional Programming
"... This paper deals with the problem of a program that is essentially the same over any of several types but which, in the older imperative languages must be rewritten for each separate type. For example, a sort routine may be written with essentially the same code except for the types for integers, bo ..."
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This paper deals with the problem of a program that is essentially the same over any of several types but which, in the older imperative languages must be rewritten for each separate type. For example, a sort routine may be written with essentially the same code except for the types for integers, booleans, and strings. It is clearly desirable to have a method of writing a piece of code that can accept the specific type as an argument. Milner developed his ideas in terms of type assignment to lambdaterms. It is based on a result due originally to Curry (Curry 1969) and Hindley (Hindley 1969) known as the principal typescheme theorem, which says that (assuming that the typing assumptions are sufficiently wellbehaved) every term has a principal typescheme, which is a typescheme such that every other typescheme which can be proved for the given term is obtained by a substitution of types for type variables. This use of type schemes allows a kind of generality over all types, which is known as polymorphism.
Abstraction and Application in
, 2001
"... The postulates of comprehension and extensionality in set theory are based on an inversion principle connecting settheoretic abstraction and the property of having a member. An exactly analogous inversion principle connects functional abstraction and application to an argument in the postulates of ..."
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The postulates of comprehension and extensionality in set theory are based on an inversion principle connecting settheoretic abstraction and the property of having a member. An exactly analogous inversion principle connects functional abstraction and application to an argument in the postulates of the lambda calculus. Such an inversion principle arises also in two adjoint situations involving a cartesian closed category and its polynomial extension. Composing these two adjunctions, which stem from the deduction theorem of logic, produces the adjunction connecting product and exponentiation, i.e. conjunction and implication. Mathematics Subject Classification: 18A15, 18A40, 18D15 1
TYPES, SETS AND CATEGORIES
"... This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category t ..."
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This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. Since it is effectively impossible to describe these relationships (especially in regard to the latter) with any pretensions to completeness within the space of a comparatively short article, I have elected to offer detailed technical presentations of just a few important instances. 1 THE ORIGINS OF TYPE THEORY The roots of type theory lie in set theory, to be precise, in Bertrand Russell’s efforts to resolve the paradoxes besetting set theory at the end of the 19 th century. In analyzing these paradoxes Russell had come to find the set, or class, concept itself philosophically perplexing, and the theory of types can be seen as the outcome of his struggle to resolve these perplexities. But at first he seems to have regarded type theory as little more than a faute de mieux.