The semantic marriage of monads and effects

Wadler and Thiemann unified type-and-effect systems with monadic semantics via a syntactic correspondence and soundness results with respect to an operational semantics. They conjecture that a general, "coherent" denotational semantics can be given to unify effect systems with a monadic-style semantics. We provide such a semantics based on the novel structure of an indexed monad, which we introduce. We redefine the semantics of Moggi's computational lambda-calculus in terms of (strong) indexed monads which gives a one-to-one correspondence between indices of the denotations and the effect annotations of traditional effect systems. Dually, this approach yields indexed comonads which gives a unified semantics and effect system to contextual notions of effect (called coeffects), which we have previously described

The semantic marriage of monads and effects

Wadler and Thiemann unified type-and-effect systems with monadic semantics via a syntactic correspondence and soundness results with respect to an operational semantics. They conjecture that a general, “coherent” denotational semantics can be given to unify effect systems with a monadic-style semantics. We provide such a semantics based on the novel structure of an indexed monad, which we introduce. We redefine the semantics of Moggi’s computational ?-calculus in terms of (strong) indexed monads which gives a oneto-one correspondence between indices of the denotations and the effect annotations of traditional effect systems. Dually, this approach yields indexed comonads which gives a unified semantics and effect system to contextual notions of effect (called coeffects), which we have previously describe

The semantic marriage of monads and effects

Wadler and Thiemann unified type-and-effect systems with monadic semantics via a syntactic correspondence and soundness results with respect to an operational semantics. They conjecture that a general, “coherent” denotational semantics can be given to unify effect systems with a monadic-style semantics. We provide such a semantics based on the novel structure of an indexed monad, which we introduce. We redefine the semantics of Moggi’s computational ?-calculus in terms of (strong) indexed monads which gives a oneto-one correspondence between indices of the denotations and the effect annotations of traditional effect systems. Dually, this approach yields indexed comonads which gives a unified semantics and effect system to contextual notions of effect (called coeffects), which we have previously describe

Comprehending Ringads for Phil Wadler, on the occasion of his 60th birthday

Abstract. List comprehensions are a widely used programming construct, in languages such as Haskell and Python and in technologies such as Microsoft's Language Integrated Query. They generalize from lists to arbitrary monads, yielding a lightweight idiom of imperative programming in a pure functional language. When the monad has the additional structure of a so-called ringad, corresponding to 'empty' and 'union' operations, then it can be seen as some kind of collection type, and the comprehension notation can also be extended to incorporate aggregations. Ringad comprehensions represent a convenient notation for expressing database queries. The ringad structure alone does not provide a good explanation or an efficient implementation of relational joins; but by allowing heterogeneous comprehensions, involving both bag and indexed table ringads, we show how to accommodate these too

Unifying graded and parameterised monads

Monads are a useful tool for structuring effectful features of computation such as state, non-determinism, and continuations. In the last decade, several generalisations of monads have been suggested which provide a more fine-grained model of effects by replacing the single type constructor of a monad with an indexed family of constructors. Most notably, graded monads (indexed by a monoid) model effect systems and parameterised monads (indexed by pairs of pre- and post-conditions) model program logics. This paper studies the relationship between these two generalisations of monads via a third generalisation. This third generalisation, which we call category-graded monads, arises by generalising a view of monads as a particular special case of lax functors. A category-graded monad provides a family of functors T f indexed by morphisms f of some other category. This allows certain compositions of effects to be ruled out (in the style of a program logic) as well as an abstract description of effects (in the style of an effect system). Using this as a basis, we show how graded and parameterised monads can be unified, studying their similarities and differences along the way

Effect Systems Revisited - Control-Flow Algebra and Semantics

Effect systems were originally conceived as an inference-based program analysis to capture program behaviour—as a set of (representations of) effects. Two orthogonal developments have since happened. First, motivated by static analysis, effects were generalised to values in an algebra, to better model control flow (e.g. for may/must analyses and concurrency). Second, motivated by semantic questions, the syntactic notion of set- (or semilattice-) based effect system was linked to the semantic notion of monads and more recently to graded monads which give a more precise semantic account of effects. We give a lightweight tutorial explanation of the concepts involved in these two threads and then unify them via the notion of an effect-directed semantics for a control-flow algebra of effects. For the case of effectful programming with sequencing, alternation and parallelism—illustrated with music—we identify a form of graded joinads as the appropriate structure for unifying effect analysis and semantics

Relational Algebra by Way of Adjunctions

Bulk types such as sets, bags, and lists are monads, and therefore support a notation for database queries based on comprehensions. This fact is the basis of much work on database query languages. The monadic structure easily explains most of standard relational algebra—specifically, selections and projections—allowing for an elegant mathematical foundation for those aspects of database query language design. Most, but not all: monads do not immediately offer an explanation of relational join or grouping, and hence important foundations for those crucial aspects of relational algebra are missing. The best they can offer is cartesian product followed by selection. Adjunctions come to the rescue: like any monad, bulk types also arise from certain adjunctions; we show that by paying due attention to other important adjunctions, we can elegantly explain the rest of standard relational algebra. In particular, graded monads provide a mathematical foundation for indexing and grouping, which leads directly to an efficient implementation, even of joins

Quantitative program reasoning with graded modal types

In programming, data is often considered to be infinitely copiable, arbitrarily discardable, and universally unconstrained. However this view is naive: some data encapsulates resources that are subject to protocols (e.g., file and device handles, channels); some data should not be arbitrarily copied or communicated (e.g., private data). Linear types provide a partial remedy by delineating data in two camps: "resources" to be used but never copied or discarded, and unconstrained values. However, this binary distinction is too coarse-grained. Instead, we propose the general notion of graded modal types, which in combination with linear and indexed types, provides an expressive type theory for enforcing fine-grained resource-like properties of data. We present a type system drawing together these aspects (linear, graded, and indexed) embodied in a fully-fledged functional language implementation, called Granule. We detail the type system, including its metatheoretic properties, and explore examples in the concrete language. This work advances the wider goal of expanding the reach of type systems to capture and verify a broader set of program properties