Context-free Coalgebras

In this article, we provide a coalgebraic account of parts of the mathematical theory underlying context-free languages. We characterize context-free languages, and power series and streams generalizing or corresponding to the context-free languages, by means of systems of behavioural differential equations; and prove a number of results, some of which are new, and some of which are new proofs of existing theorems, using the techniques of bisimulation and bisimulation up to linear combinations. Furthermore, we establish a link between automatic sequences and these systems of equations, allowing us to, given an automaton generating an automatic sequence, easily construct a system of behavioural differential equations yielding this sequence as a context-free stream

A Formal Study of Moessner's Sieve

In this dissertation, we present a new characterization of Moessner's sieve that brings a range of new results with it. As such, we present a dual to Moessner's sieve that generates a sequence of so-called Moessner triangles, instead of a traditional sequence of successive powers, where each triangle is generated column by column, instead of row by row. Furthermore, we present a new characteristic function of Moessner's sieve that calculates the entries of the Moessner triangles generated by Moessner's sieve, without having to calculate the prefix of the sequence.We prove Moessner's theorem adapted to our new dual sieve, called Moessner's idealized theorem, where we generalize the initial configuration from a sequence of natural numbers to a seed tuple containing just one non-zero entry. We discover a new property of Moessner's sieve that connects Moessner triangles of different rank, thus acting as a dual to the existing relation between Moessner triangles of different index, thereby suggesting the presence of a 2-dimensional grid of triangles, rather than the traditional 1-dimensional sequence of values.We adapt Long's theorem to the dual sieve and obtain a simplified initial configuration of Long's theorem, consisting of a seed tuple of two non-zero entries. We conjecture a new generalization of Long's theorem that has a seed tuple of arbitrary entries for its initial configuration and connects Moessner's sieve with polynomial evaluation. Lastly, we approach the connection between Moessner's sieve and polynomial evaluation from an alternative perspective and prove an equivalence relation between the triangle creation procedures of Moessner's sieve and the repeated application of Horner's method for polynomial division.All results presented in this dissertation have been formalized in the Coq proof assistant and proved using a minimal subset of the constructs and tactics available in the Coq language. As such, we demonstrate the potential of proof assistants to inspire new results while lowering the gap between programs (in computer science) and proofs (in mathematics)

Proving the unique fixed-point principle correct: an adventure with category theory

Abstract Say you want to prove something about an infinite data-structure, such as a stream or an infinite tree, but you would rather not subject yourself to coinduction. The unique fixed-point principle is an easyto-use, calculational alternative. The proof technique rests on the fact that certain recursion equations have unique solutions; if two elements of a coinductive type satisfy the same equation of this kind, then they are equal. In this paper we precisely characterize the conditions that guarantee a unique solution. Significantly, we do so not with a syntactic criterion, but with a semantic one that stems from the categorical notion of naturality. Our development is based on distributive laws and bialgebras, and draws heavily on Turi and Plotkin's pioneering work on mathematical operational semantics. Along the way, we break down the design space in two dimensions, leading to a total of nine points. Each gives rise to varying degrees of expressiveness, and we will discuss three in depth. Furthermore, our development is generic in the syntax of equations and in the behaviour they encode-we are not caged in the world of streams

Programming Languages and Systems

This open access book constitutes the proceedings of the 28th European Symposium on Programming, ESOP 2019, which took place in Prague, Czech Republic, in April 2019, held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2019

Programming Languages and Systems

This open access book constitutes the proceedings of the 30th European Symposium on Programming, ESOP 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 24 papers included in this volume were carefully reviewed and selected from 79 submissions. They deal with fundamental issues in the specification, design, analysis, and implementation of programming languages and systems

Programming Languages and Systems

This open access book constitutes the proceedings of the 29th European Symposium on Programming, ESOP 2020, which was planned to take place in Dublin, Ireland, in April 2020, as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The actual ETAPS 2020 meeting was postponed due to the Corona pandemic. The papers deal with fundamental issues in the specification, design, analysis, and implementation of programming languages and systems

Programming Languages and Systems

This open access book constitutes the proceedings of the 31st European Symposium on Programming, ESOP 2022, which was held during April 5-7, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 21 regular papers presented in this volume were carefully reviewed and selected from 64 submissions. They deal with fundamental issues in the specification, design, analysis, and implementation of programming languages and systems