Initial Semantics for Reduction Rules

We give an algebraic characterization of the syntax and operational semantics of a class of simply-typed languages, such as the language PCF: we characterize simply-typed syntax with variable binding and equipped with reduction rules via a universal property, namely as the initial object of some category of models. For this purpose, we employ techniques developed in two previous works: in the first work we model syntactic translations between languages over different sets of types as initial morphisms in a category of models. In the second work we characterize untyped syntax with reduction rules as initial object in a category of models. In the present work, we combine the techniques used earlier in order to characterize simply-typed syntax with reduction rules as initial object in a category. The universal property yields an operator which allows to specify translations---that are semantically faithful by construction---between languages over possibly different sets of types. As an example, we upgrade a translation from PCF to the untyped lambda calculus, given in previous work, to account for reduction in the source and target. Specifically, we specify a reduction semantics in the source and target language through suitable rules. By equipping the untyped lambda calculus with the structure of a model of PCF, initiality yields a translation from PCF to the lambda calculus, that is faithful with respect to the reduction semantics specified by the rules. This paper is an extended version of an article published in the proceedings of WoLLIC 2012.Comment: Extended version of arXiv:1206.4547, proves a variant of a result of PhD thesis arXiv:1206.455

Terminal semantics for codata types in intensional Martin-L\"of type theory

In this work, we study the notions of relative comonad and comodule over a relative comonad, and use these notions to give a terminal coalgebra semantics for the coinductive type families of streams and of infinite triangular matrices, respectively, in intensional Martin-L\"of type theory. Our results are mechanized in the proof assistant Coq.Comment: 14 pages, ancillary files contain formalized proof in the proof assistant Coq; v2: 20 pages, title and abstract changed, give a terminal semantics for streams as well as for matrices, Coq proof files updated accordingl

Heterogeneous substitution systems revisited

Matthes and Uustalu (TCS 327(1-2):155-174, 2004) presented a categorical description of substitution systems capable of capturing syntax involving binding which is independent of whether the syntax is made up from least or greatest fixed points. We extend this work in two directions: we continue the analysis by creating more categorical structure, in particular by organizing substitution systems into a category and studying its properties, and we develop the proofs of the results of the cited paper and our new ones in UniMath, a recent library of univalent mathematics formalized in the Coq theorem prover.Comment: 24 page

Displayed Categories

We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but instead of having a single collection of "objects of D" with a map to the objects of C, the objects are given as a family indexed by objects of C, and similarly for the morphisms. This encapsulates a common way of building categories in practice, by starting with an existing category and adding extra data/properties to the objects and morphisms. The interest of this seemingly trivial reformulation is that various properties of functors are more naturally defined as properties of the corresponding displayed categories. Grothendieck fibrations, for example, when defined as certain functors, use equality on objects in their definition. When defined instead as certain displayed categories, no reference to equality on objects is required. Moreover, almost all examples of fibrations in nature are, in fact, categories whose standard construction can be seen as going via displayed categories. We therefore propose displayed categories as a basis for the development of fibrations in the type-theoretic setting, and similarly for various other notions whose classical definitions involve equality on objects. Besides giving a conceptual clarification of such issues, displayed categories also provide a powerful tool in computer formalisation, unifying and abstracting common constructions and proof techniques of category theory, and enabling modular reasoning about categories of multi-component structures. As such, most of the material of this article has been formalised in Coq over the UniMath library, with the aim of providing a practical library for use in further developments.Comment: v3: Revised and slightly expanded for publication in LMCS. Theorem numbering change

Non-wellfounded trees in Homotopy Type Theory

We prove a conjecture about the constructibility of coinductive types - in the principled form of indexed M-types - in Homotopy Type Theory. The conjecture says that in the presence of inductive types, coinductive types are derivable. Indeed, in this work, we construct coinductive types in a subsystem of Homotopy Type Theory; this subsystem is given by Intensional Martin-L\"of type theory with natural numbers and Voevodsky's Univalence Axiom. Our results are mechanized in the computer proof assistant Agda.Comment: 14 pages, to be published in proceedings of TLCA 2015; ancillary files contain Agda files with formalized proof

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

Non-Wellfounded Trees in Homotopy Type Theory

Coinductive data types are used in functional programming to represent infinite data struc-tures. Examples include the ubiquitous data type of streams over a given base type, but also more sophisticated types. From a categorical perspective, coinductive types are characterized by a universal property, which specifies the object with that property uniquely in a suitable sense. More precisely, a coinductive type is specified as the terminal coalgebra of a suitable endofunctor. In this category-theoretic viewpoint, coinductive types are dual to inductive types, which are defined as initial algebras. Inductive, resp. coinductive, types are usually considered in the principled form of the family of W-types, resp. M-types, parametrized by a type A and a dependent type family B over A, that is, a family of types (B(a))a:A. Intuitively, the elements of the coinductive type M(A,B) are trees with nodes labeled by elements of A such that a node labeled by a: A has B(a)-many subtrees, given by a map B(a) → M(A,B); see Figure 1 for an example. The inductive type W(A,B) contains only trees where any path within that tree eventually leads to a leaf, that is, to a node a: A such that B(a) is empty. a, b, c: A B(a) =

Initial Semantics for higher-order typed syntax in Coq

Initial Semantics aims at characterizing the syntax associated to a signature as the initial object of some category. We present an initial semantics result for typed higher-order syntax together with its formalization in the Coq proof assistant. The main theorem was first proved on paper in the second author's PhD thesis in 2010, and verified formally shortly afterwards. To a simply-typed binding signature S over a fixed set T of object types we associate a category called the category of representations of S. We show that this category has an initial object Sigma(S). From its construction it will be clear that the object Sigma(S) merits the name abstract syntax associated to S. Our theorem is implemented and proved correct in the proof assistant Coq through heavy use of dependent types. The approach through monads gives rise to an implementation of syntax where both terms and variables are intrinsically typed, i.e. where the object types are reflected in the meta-level types. This article is to be seen as a research article rather than about the formalization of a classical mathematical result. The nature of our theorem - involving lengthy, technical proofs and complicated algebraic structures - makes it particularly interesting for formal verification. Our goal is to promote the use of computer theorem provers as research tools, and, accordingly, a new way of publishing mathematical results: a parallel description of a theorem and its formalization should allow the verification of correct transcription of definitions and statements into the proof assistant, and straightforward but technical proofs should be well-hidden in a digital library. We argue that Coq's rich type theory, combined with its various features such as implicit arguments, allows a particularly readable formalization and is hence well-suited for communicating mathematics.Comment: Article as published in JFR (cf. Journal ref). Features some more example