Related papers: General Recursion and Formal Topology
We show that an algorithmic construction of sequences of recursive trees leads to a direct proof of the convergence of random recursive trees in an associated Doob-Martin compactification; it also gives a representation of the limit in…
Contrary to several other families of lambda terms, no closed formula or generating function is known and none of the sophisticated techniques devised in analytic combinatorics can currently help with counting or generating the set of {\em…
Researchers and developers increasingly rely on toxicity scoring to moderate generative language model outputs, in settings such as customer service, information retrieval, and content generation. However, toxicity scoring may render…
We define the syntax and reduction relation of a recursively typed lambda calculus with a parallel case-function (a parallel conditional). The reduction is shown to be confluent. We interpret the recursive types as information systems in a…
While large language models (LLMs) have demonstrated impressive performance in question-answering tasks, their performance is limited when the questions require knowledge that is not included in the model's training data and can only be…
In the article 'Ordinal Logics and the Characterizations of the Informal Concept of Proof', Georg Kreisel poses the problem of assigning unique notations to recursive ordinals, and additionally suggests that the methods which are developed…
This paper introduces Relational Type Theory (RelTT), a new approach to type theory with extensionality principles, based on a relational semantics for types. The type constructs of the theory are those of System F plus relational…
In this chapter we propose Generic Ontology Design Patterns, GODPs, as a methodology for representing and instantiating ontology design patterns in a way that is adaptable, and allows domain experts (and other users) to safely use them…
We prove a recursive identity involving formal iterated logarithms and formal iterated exponentials. These iterated logarithms and exponentials appear in a natural extension of the logarithmic formal calculus used in the study of…
We prove normalization for MTT, a general multimodal dependent type theory capable of expressing modal type theories for guarded recursion, internalized parametricity, and various other prototypical modal situations. We prove that deciding…
Many formal languages of contemporary mathematical music theory -- particularly those employing category theory -- are powerful but cumbersome: ideas that are conceptually simple frequently require expression through elaborate categorical…
User defined recursive types are a fundamental feature of modern functional programming languages like Haskell, Clean, and the ML family of languages. Properties of programs defined by recursion on the structure of recursive types are…
The method of exhaustion is generalized to a simple formula that can be used to integrate functions under very general conditions, provided that the integral exists. Both a geometric proof (following the usual procedure for the method of…
This paper provides a new, decidable definition of the higher- order recursive path ordering in which type comparisons are made only when needed, therefore eliminating the need for the computability clo- sure, and bound variables are…
Within dependent type theory, we provide a topological counterpart of well-founded trees (for short, W-types) by using a proof-relevant version of the notion of inductively generated suplattices introduced in the context of formal topology…
This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing the Hofmann-Streicher groupoid model of…
Global-type formalisms enable to describe the overall behaviour of distributed systems and at the same time to enforce safety properties for communications between system components. Our goal is that of amending a weakness of such…
Topology offers a means to formally generalize digital filtering methods based on digital linear translation-invariant (LTI) filters while also, in principle, incorporating translation-variant and nonlinear methods as well as studying large…
Designing programming languages that enable intuitive and safe manipulation of data structures is a critical research challenge. Conventional destructive memory operations using pointers are complex and prone to errors. Existing type…
We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms…