Strong Normalization for the Parameter-Free Polymorphic Lambda Calculus Based on the Omega-Rule.

Following Aehlig, we consider a hierarchy F^p= { F^p_n }_{n in Nat} of parameter-free subsystems of System F, where each F^p_n corresponds to ID_n, the theory of n-times iterated inductive definitions (thus our F^p_n corresponds to the n+1th system of Aehlig). We here present two proofs of strong normalization for F^p_n, which are directly formalizable with inductive definitions. The first one, based on the Joachimski-Matthes method, can be fully formalized in ID_n+1. This provides a tight upper bound on the complexity of the normalization theorem for System F^p_n. The second one, based on the Godel-Tait method, can be locally formalized in ID_n. This provides a direct proof to the known result that the representable functions in F^p_n are provably total in ID_n. In both cases, Buchholz\u27 Omega-rule plays a central role

Two applications of analytic functors

AbstractWe apply the theory of analytic functors to two topics related to theoretical computer science. One is a mathematical foundation of certain syntactic well-quasi-orders and well-orders appearing in graph theory, the theory of term rewriting systems, and proof theory. The other is a new verification of the Lagrange–Good inversion formula using several ideas appearing in semantics of lambda calculi, especially the relation between categorical traces and fixpoint operators

Categories with New Foundations

While the interaction between set theory and category theory has been studied extensively, the set theories considered have remained almost entirely within the Zermelo family. Quine’s New Foundations has received limited attention, despite being the one-sorted version of a theory mentioned as a possible foundation for Category Theory by Mac Lane and Eilenberg in their seminal paper on the subject. The lack of attention given to NF is not without justification. The category of NF sets is not cartesian closed and the failure of choice is a theorem of NF. But those results should not obscure the aspects of NF that have foundational appeal, nor the value of studying category theory in the context of a universal set. The present research is not intended to “advocate” for the use of NF as a practical foundation for category theory. Instead, the work presents a broad survey of the interaction between the set theory and category theory of NF, examining the relationship in both directions. The abstract structure, of which both type restriction (in the category of NF sets) and size restriction (in the category of all categories) are specific cases, appears to be the study of relative algebra. In a number of cases, the existence of relative algebraic structures in NF can be proven more generally for a class of relative adjoints, (pseudo)monads, etc. Thus, where it seems appropriate to do so, this thesis seeks to contribute to the broader study of relative algebra

Nominal techniques

This is the author accepted manuscript. The final version is available from the Association for Computing Machinery via http://dx.doi.org/10.1145/2893582.2893594 Programming languages abound with features making use of names in various ways. There is a mathematical foundation for the semantics of such features which uses groups of permutations of names and the notion of the support of an object with respect to the action of such a group. The relevance of this kind of mathematics for the semantics of names is perhaps not immediately obvious. That it is relevant and useful has emerged over the last 15 years or so in a body of work that has acquired its own name: nominal techniques. At the same time, the application of these techniques has broadened from semantics to computation theory in general. This article introduces the subject and is based upon a tutorial at LICS-ICALP 2015 [Pitts 2015a]. </jats:p

12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser

This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto

A predicative variant of a realizability tripos for the Minimalist Foundation.

open2noHere we present a predicative variant of a realizability tripos validating the intensional level of the Minimalist Foundation extended with Formal Church thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel

Type-based termination of recursive definitions

This paper introduces "lambda-hat", a simply typed lambda calculus supporting inductive types and recursive function definitions with termination ensured by types. The system is shown to enjoy subject reduction, strong normalisation of typable terms and to be stronger than a related system "lambda-G" in which termination is ensured by a syntactic guard condition. The system can, at will, be extended to also support coinductive types and corecursive function definitions.Information Society Technologies (IST) - Fifth Framework Programm (FP5) - TYPES.Fundação para a Ciência e a Tecnologia (FCT) – PRAXIS XXI/C/EEI/14172/98.INRIA-ICCTI.Estonian Science Foundation (ETF) - grant no. 4155