Algebras for weighted search

Weighted search is an essential component of many fundamental and useful algorithms. Despite this, it is relatively under explored as a computational effect, receiving not nearly as much attention as either depth- or breadth-first search. This paper explores the algebraic underpinning of weighted search, and demonstrates how to implement it as a monad transformer. The development first explores breadth-first search, which can be expressed as a polynomial over semirings. These polynomials are generalised to the free semi module monad to capture a wide range of applications, including probability monads, polynomial monads, and monads for weighted search. Finally, a monad trans-former based on the free semi module monad is introduced. Applying optimisations to this type yields an implementation of pairing heaps, which is then used to implement Dijkstra’s algorithm and efficient probabilistic sampling. The construction is formalised in Cubical Agda and implemented in Haskell

Monads need not be endofunctors

We introduce a generalization of monads, called relative monads, allowing for underlying functors between different categories. Examples include finite-dimensional vector spaces, untyped and typed λ-calculus syntax and indexed containers. We show that the Kleisli and Eilenberg-Moore constructions carry over to relative monads and are related to relative adjunctions. Under reasonable assumptions, relative monads are monoids in the functor category concerned and extend to monads, giving rise to a coreflection between relative monads and monads. Arrows are also an instance of relative monads

List Objects with Algebraic Structure

We introduce and study the notion of list object with algebraic structure. The first key aspect of our development is that the notion of list object is considered in the context of monoidal structure; the second key aspect is that we further equip list objects with algebraic structure in this setting. Within our framework, we observe that list objects give rise to free monoids and moreover show that this remains so in the presence of algebraic structure. Furthermore, we provide a basic theory explicitly describing as an inductively defined object such free monoids with suitably compatible algebraic structure in common practical situations. This theory is accompanied with the study of two technical themes that, besides being of interest in their own right, are important for establishing applications. These themes are: parametrised initiality, central to the universal property defining list objects; and approaches to algebraic structure, in particular in the context of monoidal theories. The latter leads naturally to a notion of nsr (or near semiring) category of independent interest. With the theoretical development in place, we touch upon a variety of applications, considering Natural Numbers Objects in domain theory, giving a universal property for the monadic list transformer, providing free instances of algebraic extensions of the Haskell Monad type class, elucidating the algebraic character of the construction of opetopes in higher-dimensional algebra, and considering free models of second-order algebraic theories

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

Über logisch-funktionale Programmierung und deren Anwendung zum Testen

Die vorliegende Arbeit untersucht die Implementierung logisch-funktionaler Programmierung und deren Anwendung zur automatischen Generierung von Testdaten. Logisch-funktionale Programmierung vereint zwei deklarative Programmierparadigmen, funktionale Programmierung und Logikprogrammierung, in einem einheitlichen Programmiermodell. Es stellt sich heraus, dass diese Kombination zur Spezifikation und automatischen Generierung von Testdaten gut geeignet ist. Motiviert durch die erkannte Verwandtschaft von Testdatengenerierung und logisch-funktionaler Programmierung, stellt die Arbeit einen neuen Ansatz vor, logisch-funktionale Programme in rein funktionalen Programmiersprachen auszudrücken

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

Ordered geometry in Hilbert’s Grundlagen der Geometrie

The Grundlagen der Geometrie brought Euclid’s ancient axioms up to the standards of modern logic, anticipating a completely mechanical verification of their theorems. There are five groups of axioms, each focused on a logical feature of Euclidean geometry. The first two groups give us ordered geometry, a highly limited setting where there is no talk of measure or angle. From these, we mechanically verify the Polygonal Jordan Curve Theorem, a result of much generality given the setting, and subtle enough to warrant a full verification. Along the way, we describe and implement a general-purpose algebraic language for proof search, which we use to automate arguments from the first axiom group. We then follow Hilbert through the preliminary definitions and theorems that lead up to his statement of the Polygonal Jordan Curve Theorem. These, once formalised and verified, give us a final piece of automation. Suitably armed, we can then tackle the main theorem

Proceedings of the ACM SIGPLAN Workshop on Approaches and Applications of Inductive Programming (AAIP 2009)

Inductive programming is concerned with the automated construction of declarative, often functional, recursive programs from incomplete specifications such as input/output examples. The inferred program must be correct with respect to the provided examples in a generalising sense: it should be neither equivalent to them, nor inconsistent. Inductive programming algorithms are guided explicitly or implicitly by a language bias (the class of programs that can be induced) and a search bias (determining which generalised program is constructed first). Induction strategies are either generate-and-test or example-driven. In generate-and-test approaches, hypotheses about candidate programs are generated independently from the given specifications. Program candidates are tested against the given specification and one or more of the best evaluated candidates are developed further. In analytical approaches, candidate programs are constructed in an example-driven way. While generate-and-test approaches can -- in principle -- construct any kind of program, analytical approaches have a more limited scope. On the other hand, efficiency of induction is much higher in analytical approaches. Inductive programming is still mainly a topic of basic research, exploring how the intellectual ability of humans to infer generalised recursive procedures from incomplete evidence can be captured in the form of synthesis methods. Intended applications are mainly in the domain of programming assistance -- either to relieve professional programmers from routine tasks or to enable non-programmers to some limited form of end-user programming. Furthermore, in the future, inductive programming techniques might be applied to further areas such as supporting the inference of lemmata in theorem proving or learning grammar rules. Inductive automated program construction has been originally addressed by researchers in artificial intelligence and machine learning. During the last years, some work on exploiting induction techniques has been started also in the functional programming community. Therefore, the third workshop on |Approaches and Applications of Inductive Programming| took place for the first time in conjunction with the ACM SIGPLAN International Conference on Functional Programming (ICFP 2009). The first and second workshop were associated with the International Conference on Machine Learning (ICML 2005) and the European Conference on Machine Learning (ECML 2007). AAIP´09 aimed to bring together researchers from the functional programming and the artificial intelligence communities, working in the field of inductive functional programming, and advance fruitful interactions between these communities with respect to programming techniques for inductive programming algorithms, the identification of challenge problems and potential applications. For everybody interested in inductive programming we recommend to visit the website: www.inductive-programming.org