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On understanding types, data abstraction, and polymorphism
ACM Computing Surveys (CSUR) - The MIT Press scientific computation series
Three approaches to type structure
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The Expressiveness of Simple and Second-Order Type Structures
Journal of the ACM (JACM)
POPL '86 Proceedings of the 13th ACM SIGACT-SIGPLAN symposium on Principles of programming languages
Constructions: A Higher Order Proof System for Mechanizing Mathematics
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Towards a theory of type structure
Programming Symposium, Proceedings Colloque sur la Programmation
An investigation of a programming language with a polymorphic type structure.
An investigation of a programming language with a polymorphic type structure.
Completeness of many-sorted equational logic
ACM SIGPLAN Notices
Relating models of polymorphism
POPL '89 Proceedings of the 16th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
On the type structure of standard ML
ACM Transactions on Programming Languages and Systems (TOPLAS)
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The model theory of simply typed and polymorphic (second-order) lambda calculus changes when types are allowed to be empty. For example, the “polymorphic Boolean” type really has exactly two elements in a polymorphic model only if the “absurd” type ∀t.t is empty. The standard &bgr;-&egr; axioms and equational inference rules which are complete when all types are nonempty are not complete for models with empty types. Without a little care about variable elimination, the standard rules are not even sound for empty types. We extend the standard system to obtain a complete proof system for models with empty types. The completeness proof is complicated by the fact that equational “term models” are not so easily obtained: in contrast to the nonempty case, not every theory with empty types is the theory of a single model.