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Lambda Calculus: A Case for Inductive Definitions
, 2000
"... These lecture notes intend to introduce to the subject of lambda calculus and types. A special focus is on the use of inductive denitions. The ultimate goal of the course is an advanced treatment of inductive types. Contents 1 Overview 2 2 Introduction to Inductive Denitions 4 3 Lambda Calculus 13 ..."
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These lecture notes intend to introduce to the subject of lambda calculus and types. A special focus is on the use of inductive denitions. The ultimate goal of the course is an advanced treatment of inductive types. Contents 1 Overview 2 2 Introduction to Inductive Denitions 4 3 Lambda Calculus 13 3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Pure Untyped Lambda Calculus . . . . . . . . . . . . . . . . . . 15 4 Conuence 19 5 Weak and Strong Normalization 27 6 Simple and Intersection Types 33 6.1 SimplyTyped Lambda Calculus . . . . . . . . . . . . . . . . . . 34 6.2 Lambda Calculus with Intersection Types . . . . . . . . . . . . . 36 6.3 Strong Normalization of Typable Terms . . . . . . . . . . . . . . 39 6.4 Typability of Strongly Normalizing Terms . . . . . . . . . . . . . 41 7 Parametric Polymorphism 41 7.1 Strong Normalization of Typable Terms . . . . . . . . . . . . . . 44 7.1.1 Saturated Sets . . . . . . . . . . . . . . . . . . . . . ....
Monotone FixedPoint Types and Strong Normalization
 In Proceedings of CSL 1998, Lecture Notes in Computer Science. Submitted
, 1998
"... Several systems of fixedpoint types (also called retract types or recursive types with explicit isomorphisms) extending system F are considered. The seemingly strongest systems have monotonicity witnesses and use them in the definition of beta reduction for those types. A more naive approach lea ..."
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Several systems of fixedpoint types (also called retract types or recursive types with explicit isomorphisms) extending system F are considered. The seemingly strongest systems have monotonicity witnesses and use them in the definition of beta reduction for those types. A more naive approach leads to nonnormalizing terms. All the other systems are strongly normalizing because they embed in a reductionpreserving way into the system of noninterleaved positive fixedpoint types which can be shown to be strongly normalizing by an easy extension of some proof of strong normalization for system F. Due to the presence of F's impredicativity it is also possible to embed monotone inductive types (with full primitive recursion) into the system of noninterleaved positive fixedpoint types. In the author's view this gives the easiest way to proving strong normalization for systems of inductive types. 1 Definition of the systems of monotone fixedpoint types We consider extensions o...
Weak βηnormalization and normalization by evaluation for System F
 In LPAR’08, volume 5330 of LNAI
, 2008
"... Abstract. A general version of the fundamental theorem for System F is presented which can be instantiated to obtain proofs of weak β and βηnormalization and normalization by evaluation. 1 Introduction and Related Work Dependently typed lambdacalculi have been successfully used as proof languages ..."
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Abstract. A general version of the fundamental theorem for System F is presented which can be instantiated to obtain proofs of weak β and βηnormalization and normalization by evaluation. 1 Introduction and Related Work Dependently typed lambdacalculi have been successfully used as proof languages in proof assistants like Agda [Nor07], Coq [INR07], LEGO [Pol94], and NuPrl [Ct86]. Since types may depend on values in these type theories, checking equality of types, which is crucial for type and, thus, proof checking, is nontrivial for these
Characterizing Strongly Normalizing Terms of a lambdaCalculus with Generalized Applications via Intersection Types
"... An intersection type assignment system for the extension LJ of the untyped lcalculus, introduced by Joachimski and Matthes, is given and proven to characterize the strongly normalizing terms of LJ. Since LJ's generalized applications naturally allow permutative/commuting conversions, this is th ..."
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An intersection type assignment system for the extension LJ of the untyped lcalculus, introduced by Joachimski and Matthes, is given and proven to characterize the strongly normalizing terms of LJ. Since LJ's generalized applications naturally allow permutative/commuting conversions, this is the first analysis of a term rewrite system with permutative conversions by help of intersection types. Two proofs are given for the fact that the typable terms are strongly normalizing: One by the computability predicates method a la Tait and one showing directly that strongly normalizing typable terms are closed under (generalized) application and substitution. It is also shown that a straightforward extension of the type assignment for lcalculus fails to capture the strongly normalizing terms. Keywords Intersection Types, Strong Normalization, Permutative Conversions, Saturated Sets. 1 Introduction In [5] an extension LJ of lcalculus with generalized applications inspired by vo...
Parigot's Second Order λμCalculus and Inductive Types
, 2001
"... . A new proof of strong normalization of Parigot's (second order) calculus is given by a reductionpreserving embedding into system F (second order polymorphic calculus). The main idea is to use the least stable supertype for any type. These nonstrictly positive inductive types and their associat ..."
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. A new proof of strong normalization of Parigot's (second order) calculus is given by a reductionpreserving embedding into system F (second order polymorphic calculus). The main idea is to use the least stable supertype for any type. These nonstrictly positive inductive types and their associated iteration principle are available in system F, and allow to give a translation vaguely related to CPS translations (corresponding to the Kolmogorov embedding of classical logic into intuitionistic logic). However, they simulate Parigot's reductions whereas CPS translations hide them. As a major advantage, this embedding does not use the idea of reducing stability (:: ! ) to that for atomic formulae. Therefore, it even extends to noninterleaving positive xedpoint types. As a nontrivial application, strong normalization of calculus, extended by primitive recursion on monotone inductive types, is established. 1 Introduction calculus [12] essentially is the extension of nat...
Fair reactive programming
, 2013
"... Functional Reactive Programming (FRP) models reactive systems with events and signals, which have previously been observed to correspond to the “eventually ” and “always ” modalities of linear temporal logic (LTL). In this paper, we define a constructive variant of LTL with least fixed point and gre ..."
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Functional Reactive Programming (FRP) models reactive systems with events and signals, which have previously been observed to correspond to the “eventually ” and “always ” modalities of linear temporal logic (LTL). In this paper, we define a constructive variant of LTL with least fixed point and greatest fixed point operators in the spirit of the modal mucalculus, and give it a proofsasprograms interpretation in the realm of reactive programs. Previous work emphasized the propositionsastypes part of the correspondence between LTL and FRP; here we emphasize the proofsasprograms part by employing structural proof theory. We show that this type system is expressive enough to enforce liveness properties such as the fairness of schedulers and the eventual delivery of results. We illustrate programming in this language using (co)iteration operators. We prove type preservation of our operational semantics, which guarantees that our programs are causal. We give also a proof of strong normalization which provides justification that the language is productive and that our programs satisfy liveness properties derived from their types.
Monotone (Co)Inductive Types and Positive FixedPoint Types
, 1998
"... We study five extensions of the polymorphically typed lambdacalculus (system F) by type constructs intended to model fixedpoints of monotone operators. Building on work by H. Geuvers concerning the relation between term rewrite systems for least prefixedpoints and greatest postfixedpoints of po ..."
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We study five extensions of the polymorphically typed lambdacalculus (system F) by type constructs intended to model fixedpoints of monotone operators. Building on work by H. Geuvers concerning the relation between term rewrite systems for least prefixedpoints and greatest postfixedpoints of positive type schemes (i. e., nonnested positive inductive and coinductive types) and socalled retract types, we show that there are typerespecting and reductionpreserving embeddings even between systems of monotone (co)inductive types and noninterleaving positive fixedpoint types (which are essentially those retract types). The reduction relation considered is fi and jreduction for system F plus either (full) primitive recursion on the inductive types or (full) primitive corecursion on the coinductive types or an extremely simple rule for the fixedpoint types. Monotonicity is not reduced to the syntactic restriction of only positive occurrences of the type variable ff in ae when fo...
MendlerStyle Inductive Types, Categorically (Extended Abstract)
, 1998
"... Abstract We present a basis for a categorical account of Mendlerstyle inductive types by introducing a notion of initial Mendlerstyle algebras and use it for giving a reduction of (conventional) inductive types to Mendlerstyle inductive types and two reductions of Mendlerstyle inductive types to ..."
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Abstract We present a basis for a categorical account of Mendlerstyle inductive types by introducing a notion of initial Mendlerstyle algebras and use it for giving a reduction of (conventional) inductive types to Mendlerstyle inductive types and two reductions of Mendlerstyle inductive types to (conventional) inductive types. 1 Introduction For a categorytheoretically minded computing scientist, the phrase "inductive types" normally serves as a name for a construction that is usually conveniently modelled by means of the notion of initial algebras [Mal90,Fok92]. In typed lambda calculi (type theories), at the same time, one encounters both inductive types in this (call it conventional) sense and Mendlerstyle inductive types [Men...
Least and Greatest Fixed Points in Intuitionistic Natural Deduction
, 2002
"... This paper is a comparative study of a number of (intensionalsemantically distinct) least and greatest fixed point operators that naturaldeduction proof systems for intuitionistic logics can be extended with in a prooftheoretically defendable way. Eight pairs of such operators are analysed. The e ..."
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This paper is a comparative study of a number of (intensionalsemantically distinct) least and greatest fixed point operators that naturaldeduction proof systems for intuitionistic logics can be extended with in a prooftheoretically defendable way. Eight pairs of such operators are analysed. The exposition is centered around a cubeshaped classification where each node stands for an axiomatization of one pair of operators as logical constants by intended proof and reduction rules and each arc for a proof and reductionpreserving encoding of one pair in terms of another. The three dimensions of the cube reflect three orthogonal binary options: conventionalstyle vs. Mendlerstyle, basic (``[co]iterative'') vs. enhanced (``primitive[co]recursive''), simple vs. courseofvalue [co]induction. Some of the axiomatizations and encodings are wellknown; others, however, are novel; the classification into a cube is also new. The differences between the least fixed point operators considered are illustrated on the example of the corresponding natural number types.