Motivated by the problem of separating syntax from semantics in programming with algebraic effects and handlers, we propose a categorical model of abstract syntax with so-called scoped operations. As a building block of a term, a scoped operation is not merely a node in a tree, as it can also encompass a whole part of the term (a scope). Some examples from the area of programming are given by the operation catch for handling exceptions, in which the part in the scope is the code that may raise an exception, or the operation once, which selects a single solution from a nondeterministic computation. A distinctive feature of such operations is their behaviour under program composition, that is, syntactic substitution.

Abstract There are different notions of computation, the most popular being monads, applicative functors, and arrows. In this article, we show that these three notions can be seen as instances of a unifying abstract concept: monoids in monoidal categories. We demonstrate that even when working at this high level of generality, one can obtain useful results. In particular, we give conditions under which one can obtain free monoids and Cayley representations at the level of monoidal categories, and we show that their concretisation results in useful constructions for monads, applicative functors, and arrows. Moreover, by taking advantage of the uniform presentation of the three notions of computation, we introduce a principled approach to the analysis of the relation between them.

Left-nested list concatenations, left-nested binds on the free monad, and left-nested choices in many non-determinism monads have an algorithmically bad performance. Can we solve this problem without losing the ability to pattern-match on the computation? Surprisingly, there is a deceptively simple solution: use a smart view to pattern-match on the datatype. We introduce the notion of smart view and show how it solves the problem of slow left-nested operations. In particular, we use the technique to obtain fast and simple implementations of lists, of free monads, and of two non-determinism monads.

It is well-known that monads are monoids in the category of endofunctors, and in fact so are applicative functors. Unfortunately, the benefits of this unified view are lost when the additional nondeterminism structure of MonadPlus or Alternative is required.

Abstract Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a datatype-generic representation theorem. More precisely, we prove a representation theorem for a wide class of second-order functionals which are polymorphic over a class of functors. Types polymorphic over a class of functors are easily representable in languages such as Haskell, but are difficult to analyse and reason about. The concrete representation provided by the theorem is easier to analyse, but it might not be as convenient to implement. Therefore, depending on the task at hand, the change of representation may prove valuable in one direction or the other. We showcase the usefulness of the representation theorem with a range of examples. Concretely, we show how the representation theorem can be used to prove that traversable functors are finitary containers, how coalgebras of a parameterised store comonad relate to very well-behaved lenses, and how algebraic effects might be implemented in a functional language.

In the past years, researchers have been focusing on applying information flow security to web applications. These mechanisms should raise a minimum of false alarms in order to be applicable to millions of existing web pages. A promising technique to achieve this is secure multi-execution (SME). If a program is already secure, its secure multi- execution produces the same output events; otherwise, this correspondence is intentionally broken in order to preserve security. Thus, there is no way to know if unexpected results are due to bugs or due to semantics changes produced by SME. Moreover, SME provides no guarantees on the relative ordering of output events from different security levels. We argue that these shortcomings limit the applicability of SME. In this article, we propose a scheduler for secure multi- execution which makes it possible to preserve the order of output events. Using this scheduler, we introduce a novel com- bination between monitoring and SME, called multi-execution monitor, which raises alarms only for actions breaking the non-interference notion of ID-security for reactive systems. Additionally, we show that the monitor guarantees trans- parency even for CP-similarity, a progress-sensitive notion of observation.

Abstract The worker/wrapper transformation is a general technique for improving the performance of recursive programs by changing their types. The previous formalisation (A. Gill & G. Hutton, J. Funct. Program. , vol. 19, 2009, pp. 227–251) was based upon a simple fixed-point semantics of recursion. In this paper, we develop a more structured approach, based upon initial-algebra semantics. In particular, we show how the worker/wrapper transformation can be applied to programs defined using the structured pattern of recursion captured by fold operators, and illustrate our new technique with a number of examples.