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Some Lambda Calculi With Categorical Sums and Products
, 1993
"... . We consider the simply typed calculus with primitive recursion operators and types corresponding to categorical products and coproducts.. The standard equations corresponding to extensionality and to surjectivity of pairing and its dual are oriented as expansion rules. Strong normalization an ..."
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Cited by 20 (1 self)
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. We consider the simply typed calculus with primitive recursion operators and types corresponding to categorical products and coproducts.. The standard equations corresponding to extensionality and to surjectivity of pairing and its dual are oriented as expansion rules. Strong normalization and ground (basetype) confluence is proved for the full calculus; full confluence is proved for the calculus omitting the rule for strong sums. In the latter case, fixedpoint constructors may be added while retaining confluence. 1 Introduction The systems investigated here are simply typed caluli whose types include pairs, unit, sums, an empty type, and a type of natural numbers supporting constructions by primitive recursion. In the core system the types behave as categorical product and coproducts, so the subject at hand is equivalently ([LS86]) the equational theory of the free bicartesian closed category (generated by objects for the base types) with weak natural numbers object. Su...
The maximality of the typed lambda calculus and of cartesian closed categories
 Publ. Inst. Math. (N.S
"... From the analogue of Böhm’s Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here ..."
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Cited by 17 (2 self)
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From the analogue of Böhm’s Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here of these results, which were obtained previously by Richard Statman and Alex K. Simpson.
Equality Between Functionals in the Presence of Coproducts
 Information and Computation
, 1995
"... We consider the lambdacalculus obtained from the simplytyped calculus by adding products, coproducts, and a terminal type. We prove the following theorem: The equations provable in this calculus are precisely those true in any settheoretic model with an infinite base type. 1 Introduction The mod ..."
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Cited by 9 (1 self)
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We consider the lambdacalculus obtained from the simplytyped calculus by adding products, coproducts, and a terminal type. We prove the following theorem: The equations provable in this calculus are precisely those true in any settheoretic model with an infinite base type. 1 Introduction The model theory of the simplytyped lambda calculus, ! , has been well developed in the last two decades. For the most part, techniques and results generalize readily to the calculus when product types are added. Indeed, a categorical treatment goes more smoothly in the presence of products. But very little is known about the model theory of the simplytyped lambda calculus with coproducts for two chief reasons. First, techniques in the model theory of ! often rely heavily on the strong syntactic properties of the calculus; many of these properties fail in the presence of coproducts. Second, the natural generalizations of several key theorems in the model theory of ! fail in the setting wi...
Categorical Completeness Results for the SimplyTyped LambdaCalculus
 Proceedings of TLCA '95, Springer LNCS 902
, 1995
"... . We investigate, in a categorical setting, some completeness properties of betaeta conversion between closed terms of the simplytyped lambda calculus. A cartesianclosed category is said to be complete if, for any two unconvertible terms, there is some interpretation of the calculus in the catego ..."
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Cited by 9 (0 self)
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. We investigate, in a categorical setting, some completeness properties of betaeta conversion between closed terms of the simplytyped lambda calculus. A cartesianclosed category is said to be complete if, for any two unconvertible terms, there is some interpretation of the calculus in the category that distinguishes them. It is said to have a complete interpretation if there is some interpretation that equates only interconvertible terms. We give simple necessary and sufficient conditions on the category for each of the two forms of completeness to hold. The classic completeness results of, e.g., Friedman and Plotkin are immediate consequences. As another application, we derive a syntactic theorem of Statman characterizing betaeta conversion as a maximum consistent congruence relation satisfying a property known as typical ambiguity. 1 Introduction In 1970 Friedman proved that betaeta conversion is complete for deriving all equalities between the (simplytyped) lambdadefinable...
Functionality, polymorphism, and concurrency: a mathematical investigation of programming paradigms
, 1997
"... ii COPYRIGHT ..."
Topological Representation of the &ambda;Calculus
, 1998
"... The calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of conversion is shown to be deductively complete with respect to such topological semantics. It is al ..."
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The calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of conversion is shown to be deductively complete with respect to such topological semantics. It is also shown to be functionally complete, in the sense that there is always a "minimal" topological model, in which every continuous function is definable. These results subsume earlier ones using cartesian closed categories, as well as those employing socalled Henkin and Kripke models. Introduction The calculus originates with Church [6]; it is intended as a formal calculus of functional application and specification. In this paper, we are mainly interested in the version known as simply typed calculus ; as is now wellknown, the untyped version can be treated as a special case of this ([17]). We present here a topological representation of the calculus: types are represented by cert...
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