Internal Parametricity for Cubical Type Theory

We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we call relativity. We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, and we give an account of the identity extension lemma for internal parametricity

Structure and Power: an emerging landscape

In this paper, we give an overview of some recent work on applying tools from category theory in finite model theory, descriptive complexity, constraint satisfaction, and combinatorics. The motivations for this work come from Computer Science, but there may also be something of interest for model theorists and other logicians. The basic setting involves studying the category of relational structures via a resource-indexed family of adjunctions with some process category - which unfolds relational structures into treelike forms, allowing natural resource parameters to be assigned to these unfoldings. One basic instance of this scheme allows us to recover, in a purely structural, syntax-free way: the Ehrenfeucht-Fraisse~game; the quantifier rank fragments of first-order logic; the equivalences on structures induced by (i) the quantifier rank fragments, (ii) the restriction of this fragment to the existential positive part, and (iii) the extension with counting quantifiers; and the combinatorial parameter of tree-depth (Nesetril and Ossona de Mendez). Another instance recovers the k-pebble game, the finite-variable fragments, the corresponding equivalences, and the combinatorial parameter of treewidth. Other instances cover modal, guarded and hybrid fragments, generalized quantifiers, and a wide range of combinatorial parameters. This whole scheme has been axiomatized in a very general setting, of arboreal categories and arboreal covers. Beyond this basic level, a landscape is beginning to emerge, in which structural features of the resource categories, adjunctions and comonads are reflected in degrees of logical and computational tractability of the corresponding languages. Examples include semantic characterisation and preservation theorems, and Lovasz-type results on counting homomorphisms.Comment: To appear in special issue for Trakhtenbrot centenary of Fundamenta Informaticae vol. 186 no 1-

Promonads and String Diagrams for Effectful Categories

Premonoidal and Freyd categories are both generalized by non-cartesian Freyd categories: effectful categories. We construct string diagrams for effectful categories in terms of the string diagrams for a monoidal category with a freely added object. We show that effectful categories are pseudomonoids in a monoidal bicategory of promonads with a suitable tensor product.Comment: In Proceedings ACT 2022, arXiv:2307.1551

High-level signatures and initial semantics

We present a device for specifying and reasoning about syntax for datatypes, programming languages, and logic calculi. More precisely, we study a notion of signature for specifying syntactic constructions. In the spirit of Initial Semantics, we define the syntax generated by a signature to be the initial object---if it exists---in a suitable category of models. In our framework, the existence of an associated syntax to a signature is not automatically guaranteed. We identify, via the notion of presentation of a signature, a large class of signatures that do generate a syntax. Our (presentable) signatures subsume classical algebraic signatures (i.e., signatures for languages with variable binding, such as the pure lambda calculus) and extend them to include several other significant examples of syntactic constructions. One key feature of our notions of signature, syntax, and presentation is that they are highly compositional, in the sense that complex examples can be obtained by assembling simpler ones. Moreover, through the Initial Semantics approach, our framework provides, beyond the desired algebra of terms, a well-behaved substitution and the induction and recursion principles associated to the syntax. This paper builds upon ideas from a previous attempt by Hirschowitz-Maggesi, which, in turn, was directly inspired by some earlier work of Ghani-Uustalu-Hamana and Matthes-Uustalu. The main results presented in the paper are computer-checked within the UniMath system.Comment: v2: extended version of the article as published in CSL 2018 (http://dx.doi.org/10.4230/LIPIcs.CSL.2018.4); list of changes given in Section 1.5 of the paper; v3: small corrections throughout the paper, no major change

Unifying Cubical Models of Univalent Type Theory

We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections. This is made possible by weakening the notion of fibration from the cartesian cubical set model, so that it is not necessary to assume that the diagonal on the interval is a cofibration. We have formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom. By applying the construction in the presence of diagonal cofibrations or connections and reversals, we recover the existing cartesian and De Morgan cubical set models as special cases. Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure