A Study In Monads

This thesis investigates both theory and applications of the monad in a functional setting. Monads are constructs simulating different types of computations, such as indeterminism, exception handling, stateful computations, IO, and much more, all in the functional paradigm. The thesis starts by defining the monad, and builds up the theory around it. Three large implementaional examples using monads are presented: One implementation of a small interpreter for lambda calculus with graph sharing, one framework for writing symmetric cryptographic algorithms, and one more theoretical construction of a catamorphism and comprehension based monadic query language. All of these examples uses combinations of several monads to form complex computations with a high level of expressiveness. Other related constructs are discussed, as well as a presentation of the foundation for monads in category theory. A new monad is also introduced, the Pair monad. This monad is a generalisation of the list monad, and simulates substitution in tree-like structures, such as lambda terms. Furthermore, the thesis makes an investigation into the Curry-Howard Correspondence with a first order lambda calculus extended with monads, called computational lambda calculus. This results in a sublogic of constructive S4 modal logic. The thesis also uses the Kripke semantic for this logic to deduce a Kripke semantic for type setting lambda terms in computational lambda calculus

Introducing a Calculus of Effects and Handlers for Natural Language Semantics

In compositional model-theoretic semantics, researchers assemble truth-conditions or other kinds of denotations using the lambda calculus. It was previously observed that the lambda terms and/or the denotations studied tend to follow the same pattern: they are instances of a monad. In this paper, we present an extension of the simply-typed lambda calculus that exploits this uniformity using the recently discovered technique of effect handlers. We prove that our calculus exhibits some of the key formal properties of the lambda calculus and we use it to construct a modular semantics for a small fragment that involves multiple distinct semantic phenomena

Semantics of a Typed Algebraic Lambda-Calculus

Algebraic lambda-calculi have been studied in various ways, but their semantics remain mostly untouched. In this paper we propose a semantic analysis of a general simply-typed lambda-calculus endowed with a structure of vector space. We sketch the relation with two established vectorial lambda-calculi. Then we study the problems arising from the addition of a fixed point combinator and how to modify the equational theory to solve them. We sketch an algebraic vectorial PCF and its possible denotational interpretations

Logical Relations for Monadic Types

Logical relations and their generalizations are a fundamental tool in proving properties of lambda-calculi, e.g., yielding sound principles for observational equivalence. We propose a natural notion of logical relations able to deal with the monadic types of Moggi's computational lambda-calculus. The treatment is categorical, and is based on notions of subsconing, mono factorization systems, and monad morphisms. Our approach has a number of interesting applications, including cases for lambda-calculi with non-determinism (where being in logical relation means being bisimilar), dynamic name creation, and probabilistic systems.Comment: 83 page

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

Intersection Types for the Computational lambda-Calculus

We study polymorphic type assignment systems for untyped lambda-calculi with effects, based on Moggi's monadic approach. Moving from the abstract definition of monads, we introduce a version of the call-by-value computational lambda-calculus based on Wadler's variant with unit and bind combinators, and without let. We define a notion of reduction for the calculus and prove it confluent, and also we relate our calculus to the original work by Moggi showing that his untyped metalanguage can be interpreted and simulated in our calculus. We then introduce an intersection type system inspired to Barendregt, Coppo and Dezani system for ordinary untyped lambda-calculus, establishing type invariance under conversion, and provide models of the calculus via inverse limit and filter model constructions and relate them. We prove soundness and completeness of the type system, together with subject reduction and expansion properties. Finally, we introduce a notion of convergence, which is precisely related to reduction, and characterize convergent terms via their types