No-Go Theorems for Distributive Laws

Monads are commonplace in computer science, and can be composed using Beck's distributive laws. Unfortunately, finding distributive laws can be extremely difficult and error-prone. The literature contains some general principles for constructing distributive laws. However, until now there have been no such techniques for establishing when no distributive law exists. We present three families of theorems for showing when there can be no distributive law between two monads. The first widely generalizes a counterexample attributed to Plotkin. It covers all the previous known no-go results for specific pairs of monads, and includes many new results. The second and third families are entirely novel, encompassing various new practical situations. For example, they negatively resolve the open question of whether the list monad distributes over itself, reveal a previously unobserved error in the literature, and confirm a conjecture made by Beck himself in his first paper on distributive laws. In addition, we establish conditions under which there can be at most one possible distributive law between two monads, proving various known distributive laws to be unique.Comment: arXiv admin note: text overlap with arXiv:1811.0646

Categorical semantics and composition of tree transducers

In this thesis we see two new approaches to compose tree transducers and more general to fuse functional programs. The first abroach is based on initial algebras. We prove a new variant of the acid rain theorem for mutually recursive functions where the build function is substituted by a concrete functor. Moreover, we give a symmetric form (i.e. consumer and producer have the same syntactic form) of our new acid rain theorem where fusion is composition in a category and thus in particular associative. Applying this to compose top-down tree transducers yields the same result (on a syntactic level) as the classical top-down tree transducer composition. The second approach is based on free monads and monad transformers. In the same way as monoids are used in the theory of character string automata, we use monads in the theory of tree transducers. We generalize the notion of a tree transducer defining the monadic transducer, and we prove an according fusion theorem. Moreover, we prove that homomorphic monadic transducers are semantically equivalent. The latter makes it possible to compose syntactic classes of tree transducers (or particular functional programs) by simply composing endofunctors

Monad Composition via Preservation of Algebras

Monads are a central object of category theory and constitute crucial tools for many areas of Computer Science, from semantics of computation to functional programming. An important aspect of monads is their correspondence with algebraic theories (their ‘presentation’). As demonstrated by the history of this field, composing monads is a challenging task: the literature contains numerous mistakes and features no general method. One categorical construct, named ‘distributive law’ allows this composition, but its existence is not guaranteed. This thesis addresses the question of monad composition by presenting a method for the construction of distributive laws. For this purpose, we introduce a notion of preservation of algebraic features: considering an arbitrary algebra for the theory presenting a monad S, we examine whether its structure is preserved when applying another monad T . In the case of success, it allows us to construct a distributive law and to compose our monads into T S. In order to develop a general framework, we focus on the class of monoidal monads. If T is monoidal, the algebraic operations presenting S are preserved in a canonical fashion; it remains to examine whether the equations presenting S are also preserved by T . As it turns out, the preservation of an equation depends on the layout of its variables: if each variable appears once on each side, the considered equation is automatically preserved by a monoidal monad. On the other hand, if a variable is duplicated or only appears on one side, preservation is not systematic. The main results of this thesis connect the preservation of such equations with structural properties of monads. In the case where T does not preserve an equation presenting S, our distributive law cannot be built; we provide a series of methods to slightly modify our monads and overcome this issue, and we investigate some less conventional distributive laws. Finally, we consider the presentations of both S and T and revisit our construction of distributive laws, this time with an algebraic point of view. Overall, this thesis presents a general approach to the problem of monad composition by relating categorical properties of monads with preservation of algebras

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established