A Normalizing Computation Rule for Propositional Extensionality in Higher-Order Minimal Logic

The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a canonical form. A computation becomes "stuck" when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the computations are expressed using a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension. As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over Omega. There are proofs which inhabit propositions, which are the terms of type Omega. The canonical propositions are those constructed from false by implication. Thirdly, there are paths which inhabit equations M =_A N, where M and N are terms of type A. There are two ways to prove an equality: reflexivity, and propositional extensionality - logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity. We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda

Engineering formal systems in constructive type theory

This thesis presents a practical methodology for formalizing the meta-theory of formal systems with binders and coinductive relations in constructive type theory. While constructive type theory offers support for reasoning about formal systems built out of inductive definitions, support for syntax with binders and coinductive relations is lacking. We provide this support. We implement syntax with binders using well-scoped de Bruijn terms and parallel substitutions. We solve substitution lemmas automatically using the rewriting theory of the -calculus. We present the Autosubst library to automate our approach in the proof assistant Coq. Our approach to coinductive relations is based on an inductive tower construction, which is a type-theoretic form of transfinite induction. The tower construction allows us to reduce coinduction to induction. This leads to a symmetric treatment of induction and coinduction and allows us to give a novel construction of the companion of a monotone function on a complete lattice. We demonstrate our methods with a series of case studies. In particular, we present a proof of type preservation for CC!, a proof of weak and strong normalization for System F, a proof that systems of weakly guarded equations have unique solutions in CCS, and a compiler verification for a compiler from a non-deterministic language into a deterministic language. All technical results in the thesis are formalized in Coq.In dieser Dissertation beschreiben wir praktische Techniken um Formale Systeme mit Bindern und koinduktiven Relationen in Konstruktiver Typtheorie zu implementieren. Während Konstruktive Typtheorie bereits gute Unterstützung für Induktive Definition bietet, gibt es momentan kaum Unterstützung für syntaktische Systeme mit Bindern, oder koinduktiven Definitionen. Wir kodieren Syntax mit Bindern in Typtheorie mit einer de Bruijn Darstellung und zeigen alle Substitutionslemmas durch Termersetzung mit dem -Kalkül. Wir präsentieren die Autosubst Bibliothek, die unseren Ansatz im Beweisassistenten Coq implementiert. Für koinduktive Relationen verwenden wir eine induktive Turmkonstruktion, welche das typtheoretische Analog zur Transfiniten Induktion darstellt. Auf diese Art erhalten wir neue Beweisprinzipien für Koinduktion und eine neue Konstruktion von Pous’ “companion” einer monotonen Funktion auf einem vollständigen Verband. Wir validieren unsere Methoden an einer Reihe von Fallstudien. Alle technischen Ergebnisse in dieser Dissertation sind mit Coq formalisiert

Relating Justification Logic Modality and Type Theory in Curry–Howard Fashion

This dissertation is a work in the intersection of Justification Logic and Curry--Howard Isomorphism. Justification logic is an umbrella of modal logics of knowledge with explicit evidence. Justification logics have been used to tackle traditional problems in proof theory (in relation to Godel\u27s provability) and philosophy (Gettier examples, Russel\u27s barn paradox). The Curry--Howard Isomorphism or proofs-as-programs is an understanding of logic that places logical studies in conjunction with type theory and -- in current developments -- category theory. The point being that understanding a system as a logic, a typed calculus and, a language of a class of categories constitutes a useful discovery that can have many applications. The applications we will be mainly concerned with are type systems for useful programming language constructs. This work is structured in three parts: The first part is a a bird\u27s eye view into my research topics: intuitionistic logic, justified modality and type theory. The relevant systems are introduced syntactically together with main metatheoretic proof techniques which will be useful in the rest of the thesis. The second part features my main contributions. I will propose a modal type system that extends simple type theory (or, isomorphically, intuitionistic propositional logic) with elements of justification logic and will argue about its computational significance. More specifically, I will show that the obtained calculus characterizes certain computational phenomena related to linking (e.g. module mechanisms, foreign function interfaces) that abound in semantics of modern programming languages. I will present full metatheoretic results obtained for this logic/ calculus utilizing techniques from the first part and will provide proofs in the Appendix. The Appendix contains also information about an implementation of our calculus in the metaprogramming framework Makam. Finally, I conclude this work with a small ``outro\u27\u27, where I informally show that the ideas underlying my contributions can be extended in interesting ways

An inverse of the evaluation functional for typed Lambda-calculus

In any model of typed λ-calculus conianing some basic arithmetic, a functional p - * (procedure—* expression) will be defined which inverts the evaluation functional for typed X-terms, Combined with the evaluation functional, p-e yields an efficient normalization algorithm. The method is extended to X-calculi with constants and is used to normalize (the X-representations of) natural deduction proofs of (higher order) arithmetic. A consequence of theoretical interest is a strong completeness theorem for βη-reduction, generalizing results of Friedman [1] and Statman [31: If two Xterms have the same value in some model containing representations of the primitive recursive functions (of level 1) then they are provably equal in the βη- calculus

Refinement kinds: type-safe programming with practical type-level computation

UID/CEC/04516/2019 PTDC/EEICTP/4293/2014This work introduces the novel concept of kind refinement, which we develop in the context of an explicitly polymorphic ML-like language with type-level computation. Just as type refinements embed rich specifications by means of comprehension principles expressed by predicates over values in the type domain, kind refinements provide rich kind specifications by means of predicates over types in the kind domain. By leveraging our powerful refinement kind discipline, types in our language are not just used to statically classify program expressions and values, but also conveniently manipulated as tree-like data structures, with their kinds refined by logical constraints on such structures. Remarkably, the resulting typing and kinding disciplines allow for powerful forms of type reflection, ad-hoc polymorphism and type-directed meta-programming, which are often found in modern software development, but not typically expressible in a type-safe manner in general purpose languages. We validate our approach both formally and pragmatically by establishing the standard meta-theoretical results of type safety and via a prototype implementation of a kind checker, type checker and interpreter for our language.publishersversionpublishe

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 24th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 28 regular papers presented in this volume were carefully reviewed and selected from 88 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

A continuous computational interpretation of type theories

This thesis provides a computational interpretation of type theory validating Brouwer’s uniform-continuity principle that all functions from the Cantor space to natural numbers are uniformly continuous, so that type-theoretic proofs with the principle as an assumption have computational content. For this, we develop a variation of Johnstone’s topological topos, which consists of sheaves on a certain uniform-continuity site that is suitable for predicative, constructive reasoning. Our concrete sheaves can be described as sets equipped with a suitable continuity structure, which we call C-spaces, and their natural transformations can be regarded as continuous maps. The Kleene-Kreisel continuous functional can be calculated within the category of C-spaces. Our C-spaces form a locally cartesian closed category with a natural numbers object, and hence give models of Gödel’s system T and of dependent type theory. Moreover, the category has a fan functional that continuously compute moduli of uniform continuity, which validates the uniform-continuity principle formulated as a skolemized formula in system T and as a type via the Curry-Howard interpretation in dependent type theory. We emphasize that the construction of C-spaces and the verification of the uniform-continuity principles have been formalized in intensional Martin-Löf type theory in Agda notation

On Induction, Coinduction and Equality in Martin-L\uf6f and Homotopy Type Theory

Martin L\uf6f Type Theory, having put computation at the center of logicalreasoning, has been shown to be an effective foundation for proof assistants,with applications both in computer science and constructive mathematics. Oneambition though is for MLTT to also double as a practical general purposeprogramming language. Datatypes in type theory come with an induction orcoinduction principle which gives a precise and concise specification of theirinterface. However, such principles can interfere with how we would like toexpress our programs. In this thesis, we investigate more flexible alternativesto direct uses of the (co)induction principles.As a first contribution, we consider the n-truncation of a type in Homo-topy Type Theory. We derive in HoTT an eliminator into (n+1)-truncatedtypes instead of n-truncated ones, assuming extra conditions on the underlyingfunction.As a second contribution, we improve on type-based criteria for terminationand productivity. By augmenting the types with well-foundedness information,such criteria allow function definitions in a style closer to general recursion.We consider two criteria: guarded types, and sized types.Guarded types introduce a modality ”later” to guard the availability ofrecursive calls provided by a general fixed-point combinator. In Guarded Cu-bical Type Theory we equip the fixed-point combinator with a propositionalequality to its one-step unfolding, instead of a definitional equality that wouldbreak normalization. The notion of path from Cubical Type Theory allows usto do so without losing canonicity or decidability of conversion.Sized types, on the other hand, explicitly index datatypes with size boundson the height or depth of their elements. The sizes however can get in theway of the reasoning principles we expect. Our approach is to introduce newquantifiers for ”irrelevant” size quantification. We present a type theory withparametric quantifiers where irrelevance arises as a “free theorem”. We alsodevelop a conversion checking algorithm for a more specific theory where thenew quantifiers are restricted to sizes.Finally, our third contribution is about the operational semantics of typetheory. For the extensions above we would like to devise a practical conversionchecking algorithm suitable for integration into a proof assistant. We formal-ized the correctness of such an algorithm for a small but challenging corecalculus, proving that conversion is decidable. We expect this development toform a good basis to verify more complex theories.The ideas discussed in this thesis are already influencing the developmentof Agda, a proof assistant based on type theory