Monad Structure Research Articles

On $\ast$-measure monads on the category of ultrametric spaces

The functor of $\ast$-measures of compact support on the category of ultrametric spaces and non-expanding maps is introduced in the previous publication of the authors. In the present note, we prove that this functor determines a monad on this category. The monad structure allows us to define the tensor product of $\ast$-measures. We consider some applications of this notion to equilibrium theory.

Semialgebras and Weak Distributive Laws

Motivated by recent work on weak distributive laws and their applications to coalgebraic semantics, we investigate the algebraic nature of semialgebras for a monad. These are algebras for the underlying functor of the monad subject to the associativity axiom alone-the unit axiom from the definition of an Eilenberg-Moore algebras is dropped. We prove that if the underlying category has coproducts, then semialgebras for a monad M are in fact the Eilenberg-Moore algebras for a suitable monad structure on the functor id + M , which we call the semifree monad M^s. We also provide concrete algebraic presentations for semialgebras for the maybe monad, the semigroup monad and the finite distribution monad. A second contribution is characterizing the weak distributive laws of the form M T => T M as strong distributive laws M^s T => T M^s subject to an additional condition.

Multinomial and Hypergeometric Distributions in Markov Categories

Markov categories, having tensors with copying and discarding, provide a setting for categorical probability. This paper uses finite colimits and what we call uniform states in such Markov categories to define a (fixed size) multiset functor, with basic operations for sums and zips of multisets, and a graded monad structure. Multisets can be used to represent both urns filled with coloured balls and also draws of multiple balls from such urns. The main contribution of this paper is the abstract definition of multinomial and hypergeometric distributions on multisets, as draws. It is shown that these operations interact appropriately with various operations on multisets.

Fuzzy Automata as Coalgebras

The coalgebraic method is of great significance to research in process algebra, modal logic, object-oriented design and component-based software engineering. In recent years, fuzzy control has been widely used in many fields, such as handwriting recognition and the control of robots or air conditioners. It is then an interesting topic to analyze the behavior of fuzzy automata from a coalgebraic point of view. This paper models different types of fuzzy automata as coalgebras with a monad structure capturing fuzzy behavior. Based on the coalgebraic models, we can define a notion of fuzzy language and consider several versions of bisimulation for fuzzy automata. A group of combinators is defined to compose fuzzy automata of two branches: state transition and output function. A case study illustrates the coalgebraic models proposed and their composition.

A Computational Treatment of Anaphora and Its Algorithmic Implementation

In this paper, we propose a framework capable of dealing with anaphora and ellipsis which is both general and algorithmic. This generality is ensured by the compination of two general ideas. First, we use a dynamic semantics which reperent effects using a monad structure. Second we treat scopes flexibly, extending them as needed. We additionally implement this framework as an algorithm which translates abstract syntax to logical formulas. We argue that this framework can provide a unified account of a large number of anaphoric phenomena. Specifically, we show its effectiveness in dealing with pronominal and VP-anaphora, strict and lazy pronouns, lazy identity, bound variable anaphora, e-type pronouns, and cataphora. This means that in particular we can handle complex cases like Bach–Peters sentences, which require an account dealing simultaneously with several phenomena. We use Haskell as a meta-language to present the theory, which also consitutes an implementation of all the phenomena discussed in the paper. To demonstrate coverage, we propose a test suite that can be used to evaluate computational approaches to anaphora.

Interaction trees: representing recursive and impure programs in Coq

Interaction trees (ITrees) are a general-purpose data structure for representing the behaviors of recursive programs that interact with their environments. A coinductive variant of “free monads,” ITrees are built out of uninterpreted events and their continuations. They support compositional construction of interpreters from event handlers , which give meaning to events by defining their semantics as monadic actions. ITrees are expressive enough to represent impure and potentially nonterminating, mutually recursive computations, while admitting a rich equational theory of equivalence up to weak bisimulation. In contrast to other approaches such as relationally specified operational semantics, ITrees are executable via code extraction, making them suitable for debugging, testing, and implementing software artifacts that are amenable to formal verification. We have implemented ITrees and their associated theory as a Coq library, mechanizing classic domain- and category-theoretic results about program semantics, iteration, monadic structures, and equational reasoning. Although the internals of the library rely heavily on coinductive proofs, the interface hides these details so that clients can use and reason about ITrees without explicit use of Coq’s coinduction tactics. To showcase the utility of our theory, we prove the termination-sensitive correctness of a compiler from a simple imperative source language to an assembly-like target whose meanings are given in an ITree-based denotational semantics. Unlike previous results using operational techniques, our bisimulation proof follows straightforwardly by structural induction and elementary rewriting via an equational theory of combinators for control-flow graphs.

On Distinguishing Sets of Structures by First-Order Sentences of Minimal Quantifier Rank

We investigate the distinguishability of sets of relational structures concerning a class of structures in the following sense: for a fixed class of structures, given two sets of structures in this class, find a first-order formula of minimal quantifier rank that distinguishes one set from the other. We consider the following classes of structures: monadic structures, equivalence structures, and disjoint unions of linear orders. We use results of the Ehrenfeucht–Fraïssé game on these classes of structures in order to design an algorithm to find such a sentence. For these classes of structures, the problem of determining if the Duplicator has a winning strategy in an Ehrenfeucht–Fraïssé game is solved in polynomial time. We also introduce the distinguishability sentences which are sentences that distinguish between two given structures. We define the distinguishability sentences based on necessary and sufficient conditions for a winning strategy in an Ehrenfeucht–Fraïssé game. Our algorithm returns a boolean combination of such sentences. We also show that any first-order sentence is equivalent to a boolean combination of distinguishability sentences. Finally, we also show that our algorithm's running time is polynomial in the size of the input.

A functional representation of the capacity multiplication monad

Functional representations of the capacity monad based on the max and min operations were considered in \cite{Ra1} and \cite{Ny1}. Nykyforchyn considered in \cite{Ny2} some alternative monad structure for the possibility capacity functor based on the max and usual multiplication operations. We show that such capacity monad (which we call the capacity multiplication monad) has a functional representation, i.e. the space of capacities on a compactum $X$ can be naturally embedded (with preserving of the monad structure) in some space of functionals on $C(X,I)$. We also describe this space of functionals in terms of properties of functionals.

Constructive Galois Connections

AbstractGalois connections are a foundational tool for structuring abstraction in semantics, and their use lies at the heart of the theory of abstract interpretation. Yet, mechanization of Galois connections using proof assistants remains limited to restricted modes of use, preventing their general application in mechanized metatheory and certified programming. This paper presents constructive Galois connections, a variant of Galois connections that is effective both on paper and in proof assistants; is complete with respect to a large subset of classical Galois connections; and enables more general reasoning principles, including the “calculational” style advocated by Cousot. To design constructive Galois connections, we identify a restricted mode of use of classical ones which is both general and amenable to mechanization in dependently typed functional programming languages. Crucial to our metatheory is the addition of monadic structure to Galois connections to control a “specification effect.” Effectful calculations may reason classically, while pure calculations have extractable computational content. Explicitly moving between the worlds of specification and implementation is enabled by our metatheory. To validate our approach, we provide two case studies in mechanizing existing proofs from the literature: the first uses calculational abstract interpretation to design a static analyzer, and the second forms a semantic basis for gradual typing. Both mechanized proofs closely follow their original paper-and-pencil counterparts, employ reasoning principles not captured by previous mechanization approaches, support the extraction of verified algorithms, and are novel.

Iterated Covariant Powerset is not a Monad

We prove that the double covariant powerset functor PP does not admit any monad structure. The same applies to the n-fold composition of P for any n>1.

Biwreaths: A self-contained system in a 2-category that encodes different known algebraic constructions and gives rise to new ones

We introduce bimonads in a 2-category K and define biwreaths as bimonads in the 2-category bEM(K) of bimonads, in the analogous fashion as Lack and Street defined wreaths. A biwreath is then a system containing a wreath, a cowreath and their mixed versions, but also a 2-cell λ in bEM(K) governing the compatibility of the monad and the comonad structure of the biwreath. We deduce that the monad laws encode 2-(co)cycles and the comonad laws so called 3-(co)cycles, while the 2-cell conditions of the (co)monad structure 2-cells of the biwreath encode (co)actions twisted by these 2- and 3-(co)cycles. The compatibilities of λ deliver concrete expressions of the latter structure 2-cells. We concentrate on the examples of biwreaths in the 2-category induced by a braided monoidal category C and take for the distributive laws in a biwreath the braidings of the different categories of Yetter–Drinfel'd modules in C. We prove that the before-mentioned properties of a biwreath specified to the latter setting recover on the level of C different algebraic constructions known in the category MR of modules over a commutative ring R, such as Radford biproduct, Sweedler's crossed product algebra, comodule algebras over a quasi-bialgebra and the Drinfel'd twist. In this way we obtain that the known examples of (mixed) wreaths coming from MR are not merely examples, rather they are consequences of the structure of a biwreath, and that the form of their structure morphisms originates in the laws inside of a biwreath. Choosing different distributive laws and different 2-cells λ in a biwreath, leads to different and possibly new algebraic constructions.

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.

Applications of the Kleisli and Eilenberg-Moore 2-adjunctions

In 2010, J. Climent Vidal and J. Soliveres Tur developed, among other things, a pair of 2-adjunctions between the 2-category of adjunctions and the 2-category of monads. One is related to the Kleisli adjunction and the other to the Eilenberg-Moore adjunction for a given monad.Since any 2-adjunction induces certain natural isomorphisms of categories, these can be used to classify bijections and isomorphisms for certain structures in monad theory. In particular, one important example of a structure, lying in the 2-category of adjunctions, where this procedure can be applied to is that of a lifting. Therefore, a lifting can be characterized by the associated monad structure,lying in the 2-category of monads, through the respective 2-adjunction. The same can be said for Kleisli extensions.Several authors have been discovered this type of bijections and isomorphisms but these pair of 2-adjunctions can collect them all at once with an extra property, that of naturality.

Simplicitly: foundations and applications of implicit function types

Understanding a program entails understanding its context; dependencies, configurations and even implementations are all forms of contexts. Modern programming languages and theorem provers offer an array of constructs to define contexts, implicitly. Scala offers implicit parameters which are used pervasively, but which cannot be abstracted over. This paper describes a generalization of implicit parameters to implicit function types , a powerful way to abstract over the context in which some piece of code is run. We provide a formalization based on bidirectional type-checking that closely follows the semantics implemented by the Scala compiler. To demonstrate their range of abstraction capabilities, we present several applications that make use of implicit function types. We show how to encode the builder pattern, tagless interpreters, reader and free monads and we assess the performance of the monadic structures presented.

Wreaths, mixed wreaths and twisted coactions

Distributive laws between two monads in a 2-category $\mathscr K$, as defined by Jon Beck in the case $\mathscr{K} = \mathrm{Cat}$, were pointed out by the author to be monads in a 2-category $\mathrm{Mnd}\mathscr{K}$ of monads. Steve Lack and the author defined wreaths to be monads in a 2-category $\mathrm{EM}\mathscr{K}$ of monads with different 2-cells from $\mathrm{Mnd}\mathscr{K}$. Mixed distributive laws were also considered by Jon Beck, Mike Barr and, later, various others; they are comonads in $\mathrm{Mnd}\mathscr{K}$. Actually, as pointed out by John Power and Hiroshi Watanabe, there are a number of dual possibilities for mixed distributive laws. It is natural then to consider mixed wreaths as we do in this article; they are comonads in $\mathrm{EM}\mathscr{K}$. There are also mixed opwreaths: comonads in the Kleisli construction completion $\mathrm{Kl}\mathscr{K}$ of $\mathscr{K}$. The main example studied here arises from a twisted coaction of a bimonoid on a monoid. A wreath determines a monad structure on the composite of the two endomorphisms involved; this monad is called the wreath product. For mixed wreaths, corresponding to this wreath product, is a convolution operation analogous to the convolution monoid structure on the set of morphisms from a comonoid to a monoid. In fact, wreath convolution is composition in a Kleisli-like construction. Walter Moreira’s Heisenberg product of linear endomorphisms on a Hopf algebra, is an example of such convolution, actually involving merely a mixed distributive law. Monoidality of the Kleisli-like construction is also discussed.

Constructive Galois connections: taming the Galois connection framework for mechanized metatheory

Galois connections are a foundational tool for structuring abstraction in semantics and their use lies at the heart of the theory of abstract interpretation. Yet, mechanization of Galois connections remains limited to restricted modes of use, preventing their general application in mechanized metatheory and certified programming. This paper presents constructive Galois connections, a variant of Galois connections that is effective both on paper and in proof assistants; is complete with respect to a large subset of classical Galois connections; and enables more general reasoning principles, including the "calculational" style advocated by Cousot. To design constructive Galois connection we identify a restricted mode of use of classical ones which is both general and amenable to mechanization in dependently-typed functional programming languages. Crucial to our metatheory is the addition of monadic structure to Galois connections to control a "specification effect". Effectful calculations may reason classically, while pure calculations have extractable computational content. Explicitly moving between the worlds of specification and implementation is enabled by our metatheory. To validate our approach, we provide two case studies in mechanizing existing proofs from the literature: one uses calculational abstract interpretation to design a static analyzer, the other forms a semantic basis for gradual typing. Both mechanized proofs closely follow their original paper-and-pencil counterparts, employ reasoning principles not captured by previous mechanization approaches, support the extraction of verified algorithms, and are novel.

Material Geometry , The Periodic Table of Atoms , and Chemistry

: Point in E-Geometry , which is nothing and dimensionless , can be derived from the addition of a Positive (+)and a Negative (-) number while Material point has dimension , the Segment ds=[ ⊕⊝ ] , and originates in the same way . Adding it to numbers i.e. Monads , creates Primary Particles, the Rest-Gravity constituent and the Atoms of Periodic System in Planck`s Space-Level. Monads are Spinning because of the inner Electromagnetic Waves E,P which create External Spin and again the Inner Electromagnetic Waves E , P continuing this eternal Cycle . Now is shown the , Why and How the Spinning of Particles and the Way and the Place conserved. Euclidean-Geometry elucidated the definitions of Geometry-content {Point, Segment, Straight Line, Plane ,Volume Space [S] Anti-space [AS] ,Sub-space [SS] ,Cave, Space-Anti-Space Mechanism of the Six-Triple-Points-Line ,that produces and transfers Points of Spaces , Anti-Spaces and Sub-Spaces in a Common Inertial Sub -Space , the Field [MFMF] and Particles} and the same for Material-Geometry which describes the Space andand Energy beyond Plank´s length-level [ G -Length 3,969.10 62] , to Point = m → 0 . Energy monads presuppose Energy-Space Base which is the cause of Spaces existence and the motion of particles. This Energy-Space Base , MFMF = Medium-Field of Material-Fragments is the Material-Geometry in the way used in Mechanics and Physics . Property for Doubling Energy in the same Space is encountered in Stationary wave where Energy is proportional to angular velocity ,w . Now is shown the How Material-Fragments are built-in E -Geometry and the Why follow its moulds . The Special Problems of E-geometry consist the , Mould Quantization of Euclidean Geometry in it to become → Monad , through mould of Space -Anti-space in itself , which is the material dipole in inner monad Structure as is the Electromagnetic cycloidal field . The Unification of Space and Energy becomes through [STPL] Geometrical Mould Mechanism of the minimum Energy-Quanta to Monads → Particles, Anti-particles, Bosons, Gravity-Force, Gravity-Field, Photons, Dark Matter , and Dark-Energy , consisting the Material Dipoles in inner monad Structures , with light velocity and continually formulating the ,Zero → Discrete ,Particles-Atoms-Elements →Infinite Geometrical Universe. Conservation of Energy, as Torsional momentum, is obtained by Dynamical stability of Nucleus-Wheel-Rims

Material Geometry the Periodic Table and Chemistry

Point in E-Geometry , which is nothing and dimensionless , can be derived from the addition of a Positive (+)and a Negative (-) number while Material point has dimension , the Segment ds=[⊕⊝] , and originates in the same way . Adding it to numbers i.e. Monads , creates Primary Particles, the Rest-Gravity constituent and the Atoms of Periodic System in Planck`s Space-Level. Monads are Spinning because of the inner ElectromagneticWaves E,P which create External Spin and again the Inner Electromagnetic Waves E , P continuing this eternal Cycle . Now is shown the , Why and How the Spinning of Particles and the Way and the Place conserved. Euclidean-Geometry elucidated the definitions of Geometry-content {Point, Segment, Straight Line, Plane , Volume Space [S] Anti-space [AS] ,Sub-space [SS] ,Cave, Space-Anti-Space Mechanism of the Six-TriplePoints -Line ,that produces and transfers Points of Spaces , Anti-Spaces and Sub-Spaces in a Common Inertial Sub -Space , the Field [MFMF] and Particles} and the same for Material-Geometry which describes the Space and and Energy beyond Plank s length -level [ G -Length 3,969.10 62] , to Point L v = ei.(Nπ2)b=10 N= − ∞ m → 0 . Energy monads presuppose Energy-Space Base which is the cause of Spaces existence and the motion of particles. This Energy-Space Base , MFMF = Medium-Field of Material-Fragments is the Material-Geometry in the way used in Mechanics and Physics . Property for Doubling Energy in the same Space is encountered in Stationary wave where Energy is proportional to angular velocity ,w . Now is shown the How MaterialFragments are built-in E -Geometry and the Why follow its moulds . The Special Problems of E-geometry consist the , Mould Quantization of Euclidean Geometry in it to become → Monad , through mould of Space -Anti-space in itself , which is the material dipole in inner monad Structure as is the Electromagnetic cycloidal field . The Unification of Space and Energy becomes through [STPL] Geometrical Mould Mechanism of the minimum Energy-Quanta to Monads → Particles, Anti-particles, Bosons, Gravity-Force, Gravity-Field, Photons, Dark Matter , and Dark-Energy , consisting the Material Dipoles in inner monad Structures , with light velocity and continually formulating the ,Zero → Discrete ,Particles-Atoms-Elements →Infinite Geometrical Universe. Conservation of Energy, as Torsional momentum, is obtained by Dynamical stability of Nucleus-Wheel-Rims.

THE ANCIENT-GREEK SPECIAL PROBLEMS, AS THE QUANTIZATION MOULDS OF SPACES.

The Special Problems of E-geometry consist the , Mould Quantization , of Euclidean Geometry in it , to become → Monad , through mould of Space –Anti-space in itself , which is the material dipole in inner monad Structure as the Electromagnetic cycloidal field → Linearly , through mould of Parallel Theorem [44- 45] , which are the equal distances between points of parallel and line → In Plane , through mould of Squaring the circle [46] , where the two equal and perpendicular monads consist a Plane acquiring the common Plane-meter → and in Space (volume) , through mould of the Duplication of the Cube [46] , where any two Unequal perpendicular monads acquire the common Space-meter to be twice each other , as analytically all methods are proved and explained . [39-41] 
 The Unification of Space and Energy becomes through [STPL] Geometrical Mould Mechanism of Elements , the minimum Energy-Quanta , In monads → Particles, Anti-particles , Bosons , Gravity – Force , Gravity -Field , Photons , Dark Matter , and Dark-Energy ,consisting Material Dipoles in inner monad Structures i.e. the Electromagnetic Cycloidal Field of monads. [39-41]
 Euclid’s elements consist of assuming a small set of intuitively appealing axioms, proving many other propositions. Because nobody until [9] succeeded to prove the parallel postulate by means of pure geometric logic , many self consistentnon - Euclidean geometries have been discovered , based on Definitions , Axioms or Postulates , in order that none of them contradicts any of the other postulates . In [39] the only Space-Energy geometry is Euclidean , agreeing with the Physical reality , on unit AB = Segment which is The Electromagnetic field of the Quantized on AB Energy Space Vector , on the contrary to the General relativity of Space-time which is based on the rays of the non-Euclidean geometries to the limited velocity of light and Planck`s cavity . Euclidean geometry elucidated the definitions of geometry-content ,{ for Point , Segment , Straight Line , Plane , Volume, Space [S] , Antispace [AS] , Sub-space [SS] , Cave, Space-Anti-Space Mechanism of the Six-Triple-Points-Line , that produces and transfers Points of Spaces , Anti-Spaces and Sub-Spaces in a Common Inertial Sub-Space and a cylinder ,Gravity field [MFMF] , Particles } and describes the Space-Energy beyond Plank´s length level [ Gravity Length 3,969.10 ̄ 62 m ] , reaching the Point = m = 0 m , which is nothing and zero space .[43-46] -The Geometrical solution of the Special Problems is now presented.

Azumaya Monads and Comonads

The definition of Azumaya algebras over commutative rings \(R\) requires the tensor product of modules over \(R\) and the twist map for the tensor product of any two \(R\)-modules. Similar constructions are available in braided monoidal categories, and Azumaya algebras were defined in these settings. Here, we introduce Azumaya monads on any category \(\mathbb{A}\) by considering a monad \((F,m,e)\) on \(\mathbb{A}\) endowed with a distributive law \(\lambda: FF\to FF\) satisfying the Yang–Baxter equation (BD%please define -law). This allows to introduce an opposite monad \((F^\lambda,m\cdot \lambda,e)\) and a monad structure on \(FF^\lambda\). The quadruple \((F,m,e,\lambda)\) is called an Azumaya monad, provided that the canonical comparison functor induces an equivalence between the category \(\mathbb{A}\) and the category of \(FF^\lambda\)-modules. Properties and characterizations of these monads are studied, in particular for the case when \(F\) allows for a right adjoint functor. Dual to Azumaya monads, we define Azumaya comonads and investigate the interplay between these notions. In braided categories (V\(,\otimes,I,\tau)\), for any V-algebra \(A\), the braiding induces a BD-law \(\tau_{A,A}:A\otimes A\to A\otimes A\), and \(A\) is called left (right) Azumaya, provided the monad \(A\otimes-\) (resp. \(-\otimes A\)) is Azumaya. If \(\tau\) is a symmetry or if the category V admits equalizers and coequalizers, the notions of left and right Azumaya algebras coincide.