Strong Monad Research Articles

Continuous R-valuations

We introduce continuous $R$-valuations on directed-complete posets (dcpos, for short), as a generalization of continuous valuations in domain theory, by extending values of continuous valuations from reals to so-called Abelian d-rags $R$. Like the valuation monad $\mathbf{V}$ introduced by Jones and Plotkin, we show that the construction of continuous $R$-valuations extends to a strong monad $\mathbf{V}^R$ on the category of dcpos and Scott-continuous maps. Additionally, and as in recent work by the two authors and C. Th\'eron, and by the second author, B. Lindenhovius, M. Mislove and V. Zamdzhiev, we show that we can extract a commutative monad $\mathbf{V}^R_m$ out of it, whose elements we call minimal $R$-valuations. We also show that continuous $R$-valuations have close connections to measures when $R$ is taken to be $\mathbf{I}\mathbb{R}^\star_+$, the interval domain of the extended nonnegative reals: (1) On every coherent topological space, every non-zero, bounded $\tau$-smooth measure $\mu$ (defined on the Borel $\sigma$-algebra), canonically determines a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation; and (2) such a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation is the most precise (in a certain sense) continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation that approximates $\mu$, when the support of $\mu$ is a compact Hausdorff subspace of a second-countable stably compact topological space. This in particular applies to Lebesgue measure on the unit interval. As a result, the Lebesgue measure can be identified as a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation. Additionally, we show that the latter is minimal.

What Makes a Strong Monad?

Strong monads are important for several applications, in particular, in the denotational semantics of effectful languages, where strength is needed to sequence computations that have free variables. Strength is non-trivial: it can be difficult to determine whether a monad has any strength at all, and monads can be strong in multiple ways. We therefore review some of the most important known facts about strength and prove some new ones. In particular, we present a number of equivalent characterizations of strong functor and strong monad, and give some conditions that guarantee existence or uniqueness of strengths. We look at strength from three different perspectives: actions of a monoidal category V, enrichment over V, and powering over V. We are primarily motivated by semantics of effects, but the results are also useful in other contexts.

Normalization by Evaluation for the Lambek Calculus

The syntactic calculus of Lambek is a deductive system for the multiplicative fragment of intuitionistic non-commutative linear logic. As a fine-grained calculus of resources, it has many applications, mostly in formal computational investigations of natural languages. This paper introduces a calculus of beta-eta-long normal forms for derivations in the Lambek calculus with multiplicative unit and conjunction among its logical connectives. Reduction to normal form follows the normalization by evaluation (NbE) strategy: (i) evaluate a derivation in a Kripke model of Lambek calculus; (ii) extract normal forms from the obtained semantic values. The implementation of the NbE algorithm requires the presence of a strong monad in the Kripke interpretation of positive formulae, in analogy with the extension of intuitionistic propositional logic with falsity and disjunction. The NbE algorithm can also be instantiated with minor variations to calculi for related substructural logics, such as multiplicative and dual intuitionistic linear logic (MILL and DILL).

A metalanguage for guarded iteration

Notions of guardedness serve to delineate admissible recursive definitions in various settings in a compositional manner. In recent work, we have introduced an axiomatic notion of guardedness in symmetric monoidal categories, which serves as a unifying framework for various examples from program semantics, process algebra, and beyond. In the present paper, we propose a generic metalanguage for guarded iteration based on combining this notion with the fine-grain call-by-value paradigm, which we intend as a unifying programming language for guarded and unguarded iteration in the presence of computational effects. We give a generic (categorical) semantics of this language over a suitable class of strong monads supporting guarded iteration, and show it to be in touch with the standard operational behaviour of iteration by giving a concrete big-step operational semantics for a certain specific instance of the metalanguage and establishing soundness and (computational) adequacy for this case.

Gabriel-Morita Theory for Excisive Model Categories

We develop a Gabriel-Morita theory for strong monads on pointed monoidal model categories. Assuming that the model category is excisive, i.e. the derived suspension functor is conservative, we show that if the monad T preserves cofibre sequences up to homotopy and has a weakly invertible strength, then the category of T-algebras is Quillen equivalent to the category of T(I)-modules where I is the monoidal unit. This recovers Schwede’s theorem on connective stable homotopy over a pointed Lawvere theory as special case.

Commutants for Enriched Algebraic Theories and Monads

We define and study a notion of commutant for $$\mathscr {V}$$ -enriched $${\mathscr {J}}$$ -algebraic theories for a system of arities $${\mathscr {J}}$$ , recovering the usual notion of commutant or centralizer of a subring as a special case alongside Wraith’s notion of commutant for Lawvere theories as well as a notion of commutant for $$\mathscr {V}$$ -monads on a symmetric monoidal closed category $$\mathscr {V}$$ . This entails a thorough study of commutation and Kronecker products of operations in $${\mathscr {J}}$$ -theories. In view of the equivalence between $${\mathscr {J}}$$ -theories and $${\mathscr {J}}$$ -ary monads we reconcile this notion of commutation with Kock’s notion of commutation of cospans of monads and, in particular, the notion of commutative monad. We obtain notions of $${\mathscr {J}}$$ -ary commutant and absolute commutant for $${\mathscr {J}}$$ -ary monads, and we show that for finitary monads on $$\text {Set}$$ the resulting notions of finitary commutant and absolute commutant coincide. We examine the relation of the notion of commutant to both the notion of codensity monad and the notion of algebraic structure in the sense of Lawvere.

THE HERBRAND FUNCTIONAL INTERPRETATION OF THE DOUBLE NEGATION SHIFT

AbstractThis paper considers a generalisation of selection functions over an arbitrary strong monad T, as functionals of type $J_R^T X = (X \to R) \to TX$. It is assumed throughout that R is a T-algebra. We show that $J_R^T$ is also a strong monad, and that it embeds into the continuation monad $K_R X = (X \to R) \to R$. We use this to derive that the explicitly controlled product of T-selection functions is definable from the explicitly controlled product of quantifiers, and hence from Spector’s bar recursion. We then prove several properties of this product in the special case when T is the finite powerset monad ${\cal P}_{\rm{f}} \left( \cdot \right)$. These are used to show that when $TX = {\cal P}_{\rm{f}} \left( X \right)$ the explicitly controlled product of T-selection functions calculates a witness to the Herbrand functional interpretation of the double negation shift.

When are enriched strong monads double exponential monads?

When are enriched strong monads double exponential monads?

Eilenberg–Moore Monoids and Backtracking Monad Transformers

We develop an algebraic underpinning of backtracking monad transformers in the general setting of monoidal categories. As our main technical device, we introduce Eilenberg--Moore monoids, which combine monoids with algebras for strong monads. We show that Eilenberg--Moore monoids coincide with algebras for the list monad transformer ('done right') known from Haskell libraries. From this, we obtain a number of results, including the facts that the list monad transformer is indeed a monad, a transformer, and an instance of the MonadPlus class. Finally, we construct an Eilenberg--Moore monoid of endomorphisms, which, via the codensity monad construction, yields a continuation-based implementation a la Hinze.

Commutativity

We describe a general framework for notions of commutativity based on enriched category theory. We extend Eilenberg and Kelly's tensor product for categories enriched over a symmetric monoidal base to a tensor product for categories enriched over a normal duoidal category; using this, we re-find notions such as the commutativity of a finitary algebraic theory or a strong monad, the commuting tensor product of two theories, and the Boardman–Vogt tensor product of symmetric operads.

Notions of Monad Strength

Over the past two decades the notion of a strong monad has found wide applicability in computing. Arising out of a need to interpret products in computational and semantic settings, different approaches to this concept have arisen. In this paper we introduce and investigate the connections between these approaches and also relate the results to monad composition. We also introduce new methods for checking and using the required laws associated with such compositions, as well as provide examples illustrating problems and issues that arise.

Relating computational effects by ⊤⊤-lifting

We consider the problem of establishing a relationship between two interpretations of base type terms of a λc-calculus extended with algebraic operations. We show that the given relationship holds if it satisfies a set of natural conditions. We apply this result to 1) comparing two monadic semantics related by a strong monad morphism, and 2) comparing two monadic semantics of fresh name creation: Starkʼs new name creation monad and the global counter monad. We also consider the same problem, relating semantics of computational effects, in the presence of recursive functions. We apply this additional by extending the previous monad morphism comparison result to the recursive case.

Cartesian effect categories are Freyd-categories

Most often, in a categorical semantics for a programming language, the substitution of terms is expressed by composition and finite products. However this does not deal with the order of evaluation of arguments, which may have major consequences when there are side-effects. In this paper Cartesian effect categories are introduced for solving this issue, and they are compared with strong monads, Freyd-categories and Haskell’s Arrows. It is proved that a Cartesian effect category is a Freyd-category where the premonoidal structure is provided by a kind of binary product, called the sequential product. The universal property of the sequential product provides Cartesian effect categories with a powerful tool for constructions and proofs. To our knowledge, both effect categories and sequential products are new notions.

Categorifying Computations into Components via Arrows as Profunctors

The notion of arrow by Hughes is an axiomatization of the algebraic structure possessed by structured computations in general. We claim that an arrow also serves as a basic component calculus for composing state-based systems as components—in fact, it is a categorified version of arrow that does so. In this paper, following the second author's previous work with Heunen, Jacobs and Sokolova, we prove that a certain coalgebraic modeling of components—which generalizes Barbosa's—indeed carries such arrow structure. Our coalgebraic modeling of components is parametrized by an arrow A that specifies computational structure exhibited by components; it turns out that it is this arrow structure of A that is lifted and realizes the (categorified) arrow structure on components. The lifting is described using the first author's recent characterization of an arrow as an internal strong monad in Prof, the bicategory of small categories and profunctors.

Selection functions, bar recursion and backward induction

Bar recursion arises in constructive mathematics, logic, proof theory and higher-type computability theory. We explain bar recursion in terms of sequential games, and show how it can be naturally understood as a generalisation of the principle of backward induction that arises in game theory. In summary, bar recursion calculates optimal plays and optimal strategies, which, for particular games of interest, amount to equilibria. We consider finite games and continuous countably infinite games, and relate the two. The above development is followed by a conceptual explanation of how the finite version of the main form of bar recursion considered here arises from a strong monad of selections functions that can be defined in any cartesian closed category. Finite bar recursion turns out to be a well-known morphism available in any strong monad, specialised to the selection monad.

Commutative monads, diagrams and knots

There is certain diverse class of diagram that is found in a variety of branches of mathematics and which all share this property: there is a common scheme for translating all of these diagrams into useful functional code. These diagrams include Bayesian networks, quantum computer circuits [1], trace diagrams for multilinear algebra [2], Feynman diagrams and even knot diagrams [3]. I will show how a common thread lying behind these diagrams is the presence of a commutative monad and I will show how we can use this fact to translate these diagrams directly into Haskell code making use of do-notation for monads. I will also show a number of examples of such translated code at work and use it to solve problems ranging from Bayesian inference to the topological problem of untangling tangled strings. Along the way I hope to give a little insight into the subjects mentioned above and illustrate how a functional programming language can be a valuable tool in mathematical research and experimentation.

The connected Vietoris powerlocale

The connected Vietoris powerlocale is defined as a strong monad V c on the category of locales. V c X is a sublocale of Johnstone's Vietoris powerlocale VX, a localic analogue of the Vietoris hyperspace, and its points correspond to the weakly semifitted sublocales of X that are “strongly connected”. A product map × : V c X × V c Y → V c ( X × Y ) shows that the product of two strongly connected sublocales is strongly connected. If X is locally connected then V c X is overt. For the localic completion Y ¯ of a generalized metric space Y, the points of V c Y ¯ are certain Cauchy filters of formal balls for the finite power set F Y with respect to a Vietoris metric. Application to the point-free real line R gives a choice-free constructive version of the Intermediate Value Theorem and Rolle's Theorem. The work is topos-valid (assuming natural numbers object). V c is a geometric construction.

Comonadic Notions of Computation

We argue that symmetric (semi)monoidal comonads provide a means to structure context-dependent notions of computation such as notions of dataflow computation (computation on streams) and of tree relabelling as in attribute evaluation. We propose a generic semantics for extensions of simply typed lambda calculus with context-dependent operations analogous to the Moggi-style semantics for effectful languages based on strong monads. This continues the work in the early 90s by Brookes, Geva and Van Stone on the use of computational comonads in intensional semantics.

Monad compositions II: Kleisli strength

In this paper we introduce the concept of Kleisli strength for monads in an arbitrary symmetric monoidal category. This generalises the notion of commutative monad and gives us new examples, even in the cartesian-closed category of sets. We exploit the presence of Kleisli strength to derive methods for generating distributive laws. We also introduce linear equations to extend the results to certain quotient monads. Mechanisms are described for finding strengths that produce a large collection of new distributive laws, and consequently monad compositions, including the composition of monadic data types such as lists, trees, exceptions and state.

Combining algebraic effects with continuations

We consider the natural combinations of algebraic computational effects such as side-effects, exceptions, interactive input/output, and nondeterminism with continuations. Continuations are not an algebraic effect, but previously developed combinations of algebraic effects given by sum and tensor extend, with effort, to include commonly used combinations of the various algebraic effects with continuations. Continuations also give rise to a third sort of combination, that given by applying the continuations monad transformer to an algebraic effect. We investigate the extent to which sum and tensor extend from algebraic effects to arbitrary monads, and the extent to which Felleisen et al.’s C operator extends from continuations to its combination with algebraic effects. To do all this, we use Dubuc’s characterisation of strong monads in terms of enriched large Lawvere theories.