Newton series, coinductively: a comparative study of composition

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Newton series, coinductively

We present a comparative study of four product operators on weighted languages: (i) the convolution, (ii) the shue, (iii) the inltration, and (iv) the Hadamard product. Exploiting the fact that the set of weighted languages is a nal coalgebra, we use coinduction to prove that a classical operator from dierence calculus in mathematics: the Newton transform, generalises (from innite sequences) to weighted lan- guages. We show that the Newton transform is an isomorphism of rings that transforms the Hadamard product of two weighted languages into an inltration product, and we develop various representations for the Newton transform of a language, together with concrete calculation rules for computing them

Session coalgebras: A coalgebraic view on session types and communication protocols

Compositional methods are central to the development and verification of software systems. They allow breaking down large systems into smaller components, while enabling reasoning about the behaviour of the composed system. For concurrent and communicating systems, compositional techniques based on behavioural type systems have received much attention. By abstracting communication protocols as types, these type systems can statically check that programs interact with channels according to a certain protocol, whether the intended messages are exchanged in a certain order. In this paper, we put on our coalgebraic spectacles to investigate session types, a widely studied class of behavioural type systems. We provide a syntax-free description of session-based concurrency as states of coalgebras. As a result, we rediscover type equivalence, duality, and subtyping rela

Code Generation for Higher Inductive Types

Higher inductive types are inductive types that include nontrivial higher-dimensional structure, represented as identifications that are not reflexivity. While work proceeds on type theories with a computational interpretation of univalence and higher inductive types, it is convenient to encode these structures in more traditional type theories with mature implementations. However, these encodings involve a great deal of error-prone additional syntax. We present a library that uses Agda's metaprogramming facilities to automate this process, allowing higher inductive types to be specified with minimal additional syntax.Comment: 16 pages, Accepted for presentation in WFLP 201

Quotient inductive-inductive types

Higher inductive types (HITs) in Homotopy Type Theory (HoTT) allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types and allow to define types which are not sets in the sense of HoTT (i.e. do not satisfy uniqueness of equality proofs) such as spheres, suspensions and the torus. However, there are also interesting uses of HITs to define sets, such as the Cauchy reals, the partiality monad, and the internal, total syntax of type theory. In each of these examples we define several types that depend on each other mutually, i.e. they are inductive-inductive definitions. We call those HITs quotient inductive-inductive types (QIITs). Although there has been recent progress on the general theory of HITs, there isn't yet a theoretical foundation of the combination of equality constructors and induction-induction, despite having many interesting applications. In the present paper we present a first step towards a semantic definition of QIITs. In particular, we give an initial-algebra semantics and show that this is equivalent to the section induction principle, which justifies the intuitively expected elimination rules

Mixed Inductive-Coinductive Reasoning Types, Programs and Logic

Contains fulltext : 190323.pdf (publisher's version ) (Open Access)Induction and coinduction are two complementary techniques used in mathematics and computer science. These techniques occur together, for example, in control systems: On the one hand, control systems are expected to run until turned off and to always react to their environment. This is what we call coinductive computations. On the other hand, they have to make internal computations. Restricting these computations to terminating, that is inductive, computations ensures that the systems continue to react to their environment. We develop in this thesis techniques for programming inductive-coinductive systems, and for describing their properties and proving these properties. The focus is on developing formal languages, in which proofsare written by humans and can be verified by a computer. This ensures the correctness of those proofs and thereby of the programmed systems. Due to their generality, the developed languages are also applicable to the formalisation of mathematics.Radboud University, 19 april 2018Promotores : Rutten, J., Geuvers, H. Co-promotor : Hansen, Helle H.ix, 330 p

Dependent Inductive and Coinductive Types Are Fibrational Dialgebras

In this paper, I establish the categorical structure necessary to interpret dependent inductive and coinductive types. It is well-known that dependent type theories à la Martin-Löf can be interpreted using fibrations. Modern theorem provers, however, are based on more sophisticated type systems that allow the definition of powerful inductive dependent types (known as inductive families) and, somewhat limited, coinductive dependent types. I define a class of functors on fibrations and show how data type definitions correspond to initial and final dialgebras for these functors. This description is also a proposal of how coinductive types should be treated in type theories, as they appear here simply as dual of inductive types. Finally, I show how dependent data types correspond to algebras and coalgebras, and give the correspondence to dependent polynomial functors.</blockquote

Well-definedness and observational equivalence for inductive-coinductive programs

We define notions of well-definedness and observational equivalence for programs of mixed inductive and coinductive types. These notions are defined by means of tests formulas which combine structural congruence for inductive types and modal logic for coinductive types. Tests also correspond to certain evaluation contexts. We define a program to be well-defined if it is strongly normalizing under all tests, and two programs are observationally equivalent if they satisfy the same tests.We show that observational equivalence is sufficiently coarse to ensure that least and greatest fixed point types are initial algebras and final coalgebras, respectively. This yields inductive and coinductive proof principles for reasoning about program behaviour. On the other hand, we argue that observational equivalence does not identify too many terms, by showing that tests induce a topology that, on streams, coincides with usual topology induced by the prefix metric. As one would expect, observational equivalence is, in general, undecidable, but in order to develop some practically useful heuristics we provide coinductive techniques for establishing observational normalization and observational equivalence, along with up-to techniques for enhancing these methods.</p