Functional Programming Languages and Computer Architecture

In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much like a "cobordism". Similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository paper, we make some of these analogies precise using the concept of "closed symmetric monoidal category". We assume no prior knowledge of category theory, proof theory or computer science.

Physics, Topology, Logic and Computation: A Rosetta Stone

We introduce a generalisation 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 monads and relative monads. Arrows are also an instance of relative monads.

/pdf/monads-need-not-be-endofunctors-6rdhvesasi.pdf

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 lambda-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.

A compact closed bicategory is a symmetric monoidal bicategory where every object is equipped with a weak dual. The unit and counit satisfy the usual "zig-zag" identities of a compact closed category only up to natural isomorphism, and the isomorphism is subject to a coherence law. We give several examples of compact closed bicategories, then review previous work. In particular, Day and Street defined compact closed bicategories indirectly via Gray monoids and then appealed to a coherence theorem to extend the concept to bicategories; we restate the definition directly. 
We prove that given a 2-category T with finite products and weak pullbacks, the bicategory of objects of C, spans, and isomorphism classes of maps of spans is compact closed. As corollaries, the bicategory of spans of sets and certain bicategories of "resistor networks" are compact closed.

Compact Closed Bicategories

This paper introduces a variant of the resource calculus, the rigid resource calculus, in which a permutation of elements in a bag is distinct from but isomorphic to the original bag. It is designed so that the Taylor expansion within it coincides with the interpretation by generalised species of Fiore et al., which generalises both Joyal's combinatorial species and Girard's normal functors, and which can be seen as a proof-relevant extension of the relational model. As an application, we prove the commutation between computing Bohm trees and (standard) Taylor expansions for a particular nondeterministic calculus.

Generalised species of rigid resource terms

Motivated by a tight connection between Joyal's combinatorial species and quantitative models of linear logic, this paper introduces weighted generalised species (or weighted profunctors), where weights are morphisms of a given symmetric monoidal closed category (SMCC). For each SMCC W, we show that the category of W-weighted profunctors is a Lafont category, a categorical model of linear logic with exponential. As a model of programming languages, the construction of this paper gives a unified framework that induces adequate models of nondeterministic, probabilistic, algebraic and quantum programming languages by an appropriate choice of the weight SMCC.

Species, Profunctors and Taylor Expansion Weighted by SMCC: A Unified Framework for Modelling Nondeterministic, Probabilistic and Quantum Programs

Hughes' arrows were shown, by Jacobs et al., to be roughly monads in the bicategory Prof of profunctors (distributors, modules). However in their work as well as others', the categorical nature of the first operator was not pursued and its formulation remained rather ad hoc. In this paper, we identify first with strength for a monad, therefore: arrows are strong monads inProf. Strong monads have been widely used in the semantics of functional programming after Moggi's seminal work, therefore our observation establishes categorical canonicity of the notion of arrow.

Arrows are strong monads

We introduce a lambda calculus λTFG for transformations of finite graphs by generalizing and extending an existing calculus UnCAL. Whereas UnCAL can treat only unordered graphs, λTFG can treat a variety of graph models: directed edge-labeled graphs whose branch styles are represented by monads T. For example, λTFG can treat unordered graphs, ordered graphs, weighted graphs, probability graphs, and so on, by using the powerset monad, list monad, multiset monad, probability monad, respectively. In λTFG, graphs are considered as extension of tree data structures, i.e. as infinite (regular) trees, so the semantics is given with bisimilarity.A remarkable feature of UnCAL and λTFG is structural recursion for graphs, which gives a systematic programming basis like that for trees. Despite the non-well-foundedness of graphs, by suitably restricting the structural recursion, UnCAL and λTFG ensures that there is a termination property and that all transformations preserve the finiteness of the graphs. The structural recursion is defined in a "divide-and-aggregate" way; "aggregation" is done by connecting graphs with e-edges, which are similar to the e-transitions of automata. We give a suitable general definition of bisimilarity, taking account of e-edges; then we show that the structural recursion is well defined with respect to the bisimilarity.

https://research.nii.ac.jp/~hu/pub/ppdp13.pdf

A parameterized graph transformation calculus for finite graphs with monadic branches

Structural recursion, in the form of, for example, folds on lists and catamorphisms on algebraic data structures including trees, plays an important role in functional programming, by providing a systematic way for constructing and manipulating functional programs. It is, however, a challenge to define structural recursions for graph data structures, the most ubiquitous sort of data in computing. This is because unlike lists and trees, graphs are essentially not inductive and cannot be formalized as an initial algebra in general. In this paper, we borrow from the database community the idea of structural recursion on how to restrict recursions on infinite unordered regular trees so that they preserve the finiteness property and become terminating, which are desirable properties for query languages. We propose a new graph transformation language called lambdaFG for transforming and querying ordered graphs, based on the well-defined bisimulation relation on ordered graphs with special epsilon-edges. The language lambdaFG is a higher order graph transformation language that extends the simply typed lambda calculus with graph constructors and more powerful structural recursions, which is extended for transformations on the sibling dimension. It not only gives a general framework for manipulating graphs and reasoning about them, but also provides a solution to the open problem of how to define a structural recursion on ordered graphs, with the help of the bisimilarity for ordered graphs with epsilon-edges.

Kazuyuki Asada

Papers

Generalised species of rigid resource terms

Species, Profunctors and Taylor Expansion Weighted by SMCC: A Unified Framework for Modelling Nondeterministic, Probabilistic and Quantum Programs

Arrows are strong monads

A parameterized graph transformation calculus for finite graphs with monadic branches

Structural recursion for querying ordered graphs