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

On the Semantics of Intensionality and Intensional Recursion

Intensionality is a phenomenon that occurs in logic and computation. In the most general sense, a function is intensional if it operates at a level finer than (extensional) equality. This is a familiar setting for computer scientists, who often study different programs or processes that are interchangeable, i.e. extensionally equal, even though they are not implemented in the same way, so intensionally distinct. Concomitant with intensionality is the phenomenon of intensional recursion, which refers to the ability of a program to have access to its own code. In computability theory, intensional recursion is enabled by Kleene's Second Recursion Theorem. This thesis is concerned with the crafting of a logical toolkit through which these phenomena can be studied. Our main contribution is a framework in which mathematical and computational constructions can be considered either extensionally, i.e. as abstract values, or intensionally, i.e. as fine-grained descriptions of their construction. Once this is achieved, it may be used to analyse intensional recursion.Comment: DPhil thesis, Department of Computer Science & St John's College, University of Oxfor

In Search of Effectful Dependent Types

Real world programming languages crucially depend on the availability of computational effects to achieve programming convenience and expressive power as well as program efficiency. Logical frameworks rely on predicates, or dependent types, to express detailed logical properties about entities. According to the Curry-Howard correspondence, programming languages and logical frameworks should be very closely related. However, a language that has both good support for real programming and serious proving is still missing from the programming languages zoo. We believe this is due to a fundamental lack of understanding of how dependent types should interact with computational effects. In this thesis, we make a contribution towards such an understanding, with a focus on semantic methods.Comment: PhD thesis, Version submitted to Exam School

Categories with Foundation

We develop the theory of categories from foundations up. The thesis culminates in a theorem in which we assert that any concrete functor between categories of models of algebraic theories, where the codomain categories' alphabet does not contain relational information, has a left adjoint functor. This theorem is based on The General Adjoint Functor Theorem by Peter Freyd. The first chapter is about the set theoretic foundations of category theory. We present the needed ideas about recursion so that we may define what is meant by first order predicate logic. The first chapter ends in the exposition of the connection between the Grothendieck universes and the inaccessible cardinals. The second chapter starts our conversation about categories and functors between categories. We define properties of morphisms, subobjects, quotient objects and Cartesian closed categories. Furthermore, we talk about embedding and identification morphisms of concrete categories. Much of the third chapter is to show that the category of small categories is a Cartesian closed category. This leads us to talk about natural transformation and canonical constructions relating to functors. To define equivalences and their generalizations, adjoint functors, natural transformations are needed. The fourth chapter enlarges our knowledge about hom-functors and their adjacent functors, representable functors. The study of representable functors yields a profound lemma called Yoneda lemma. Yoneda lemma implies the fully faithfulness of Yoneda embedding. The fifth chapter concentrates to limit operations in a category, which leads us to talk about completeness. We find out how limit procedures are preserved in constructions and how they behave when functors pass them forward. The last chapter is about adjoint functors. The general and the special adjoint functor theorems, due to Peter Freyd, are proven. Using The General Adjoint Functor Theorem, we prove the existence of a left adjoint functor for all suitable forgetful functors among algebraic categories

Advances in Proof-Theoretic Semantics

Logic; Mathematical Logic and Foundations; Mathematical Logic and Formal Language

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum