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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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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.
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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. 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...
Algebraic Reasoning and Completeness in Typed Languages
 In Proc. 20th ACM Symposium on Principles of Programming Languages
, 1992
"... : We consider the following problem in proving observational congruences in functional languages: given a callbyname language based on the simplytyped calculus with algebraic operations axiomatized by algebraic equations E, is the set of observational congruences between terms exactly those prov ..."
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: We consider the following problem in proving observational congruences in functional languages: given a callbyname language based on the simplytyped calculus with algebraic operations axiomatized by algebraic equations E, is the set of observational congruences between terms exactly those provable from (fi), (j), and E? We find conditions for determining whether fijEequational reasoning is complete for proving the observational congruences between such terms. We demonstrate the power and generality of the theorems by presenting a number of easy corollaries for particular algebras. 1 Introduction The (fi) and (j) axioms form the basis for proving equations in callbyname functional languages. In these languages, (fi) and (j) yield sound program optimizations. For example, consider a version of the callbyname language PCF [11, 15] which is described in Appendix A. Our version of PCF includes simplytyped calculus, numerals 0; 1; 2; : : :, successor and predecessor, addition, ...
SPCF: Its Model, Calculus, and Computational Power (Preliminary Version)
, 1992
"... SPCF is an idealized sequential programming language, based on Plotkin's language PCF, that permits programmers and programs to observe the evaluation order of procedures. In this paper, we construct a fully abstract model of SPCF using a new mathematical framework suitable for defining fully abstra ..."
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SPCF is an idealized sequential programming language, based on Plotkin's language PCF, that permits programmers and programs to observe the evaluation order of procedures. In this paper, we construct a fully abstract model of SPCF using a new mathematical framework suitable for defining fully abstract models of sequential functional languages. Then, we develop an extended typed calculus to specify the operational semantics of SPCF and show that the calculus is complete for the constantfree sublanguage. Finally, we prove that SPCF is computationally complete: it can express all the computable (recursively enumerable) elements in its fully abstract model. 1 SPCF: Observing Sequentiality Most contemporary programming languages, e.g., Scheme, Pascal, Fortran, C, and ML, are "sequential", that is, they impose a serial order on the evaluation of parts of programs. Unfortunately, the familiar mathematical models for sequential languages based on continuous functions do not capture this pro...
Completeness of Conversion between Reactive Programs for Ultrametric Models
"... Abstract. In 1970 Friedman proved completeness of beta eta conversion in the simplytyped lambda calculus for the settheoretical model. Recently Krishnaswami and Benton have captured the essence of Hudak’s reactive programs in an extension of simply typed lambda calculus with causal streams and a t ..."
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Abstract. In 1970 Friedman proved completeness of beta eta conversion in the simplytyped lambda calculus for the settheoretical model. Recently Krishnaswami and Benton have captured the essence of Hudak’s reactive programs in an extension of simply typed lambda calculus with causal streams and a temporal modality and provided this typed lambda calculus for reactive programs with a sound ultrametric semantics. We show that beta eta conversion in the typed lambda calculus of reactive programs is complete for the ultrametric model. 1