A confluent reduction for the λ−calculus with surjective pairing and terminal object
نویسندگان
چکیده
We exhibit confluent and effectively weakly normalizing (thus decidable) rewriting systems for the full equational theory underlying cartesian closed categories, and for polymorphic extensions of it. The λ-calculus extended with surjective pairing has been well-studied in the last two decades. It is not confluent in the untyped case, and confluent in the typed case. But to the best of our knowledge the present work is the first treatment of the lambda calculus extended with surjective pairing and terminal object via a confluent rewriting system, and is the first solution to the decidability problem of the full equational theory of Cartesian Closed Categories extended with polymorphic types. Our approach yields conservativity results as well. In separate papers we apply our results to the study of provable type isomorphisms, and to the decidability of equality in a typed λ-calculus with subtyping.
منابع مشابه
Confluence Results for the Pure Strong Categorical Logic CCL: lambda-Calculi as Subsystems of CCL
The Strong Categorical Combinatory Logic (CCL, CCLpqSP), developed by Curien (1986) is, when typed and augmented with a rule defining a terminal object, a presentation of Cartesian Closed Categories. Furthermore, it is equationally equivalent to the Lambda-calculus with explicit couples and Surjective Pairing. Here we study the confluence properties of (CCL, CCLpqSP) and of several of its subsy...
متن کاملUnique Normal Forms for Lambda Calculus with Surjective Pairing
We consider the equational theory 3iz of I-calculus extended with constants n, no, =I and axioms for surjective pairing: rr,(nXY) = X, n,(rrXY) = Y, n(n,X)(n,X) = X. Two reduction systems yielding the equality of In are introduced; the first is not confluent and, for the second, confluence is an open problem. It is shown, however, that in both systems each term possessing a normal form has a un...
متن کاملLife without the Terminal Type
We introduce a method of extending arbitrary categories by a terminal object and apply this method in various type theoretic settings. In particular, we show that categories that are cartesian closed except for the lack of a terminal object have a universal full extension to a cartesian closed category, and we characterize categories for which the latter category is a topos. Both the basic cons...
متن کاملExtending the Extensional Lambda Calculus with Surjective Pairing is Conservative
We answer Klop and de Vrijer’s question whether adding surjective-pairing axioms to the extensional lambda calculus yields a conservative extension. The answer is positive. As a byproduct we obtain a “syntactic” proof that the extensional lambda calculus with surjective pairing is consistent.
متن کاملA confluent λ-calculus with a catch/throw mechanism
We derive a confluent λ-calculus with a catch/throw mechanism (called λct-calculus) from M. Parigot’s λμ-calculus. We also present several translations from one calculus into the other which are morphisms for the reduction. We use them to show that the λct-calculus is a retract of λμ-calculus (these calculi are isomorphic if we consider only convertibility). As a by-product, we obtain the subje...
متن کامل