Finite Powerset Research Articles

Quantale-valued convex structures as lax algebras

Based on a unital and commutative quantale (Q,⁎), a Q-valued lax extension of the nonempty finite powerset monad and a Q-valued finitary closure space (also called algebraic Q-valued closure space) are introduced. It is proved that the category of (Pf,Q)-categories with respect to the Q-valued lax extension of the nonempty finite powerset monad Pf is isomorphic to that of Q-valued finitary closure spaces. Considering the Q-valued finitary closure spaces as linkages, it is shown that balanced Q-convex structures can be treated as (Pf,Q)-categories when Q is required to be a frame and Q-fuzzifying convex structures can be treated as (Pf,Q)-categories when Q is required to be a completely distributive De Morgan algebra.

Fractals and Finite Distributions of Power Sets

In this paper, we are committed to investigating the fractal decomposition of power sets. Our main result is that every power set can be decomposed into a sum of a power set and an isomorphic set that does not intersect with it. For the finite power set, this property can be drawn on the ordinal line by constructing the fractal number axis of the ordinal line, and the fractal distribution and fractal graph of the finite power set can be obtained by using the parallel translation drawing method. Moreover, the distributions do not overlap or cross. The results in this paper provide a new perspective for further investigation of the fractal distribution of power sets.

Two Guarded Recursive Powerdomains for Applicative Simulation

Clocked Cubical Type Theory is a new type theory combining the power of guarded recursion with univalence and higher inductive types (HITs). This type theory can be used as a metalanguage for synthetic guarded domain theory in which one can solve guarded recursive type equations, also with negative variable occurrences, and use these to construct models for reasoning about programming languages. Combining this with HITs allows for the use of type constructors familiar from set-theory based approaches to semantics, such as quotients and finite powersets in these models. In this paper we show how to reason about the combination of finite non-determinism and recursion in this type theory. Unlike traditional domain theory which takes an ordering of programs as primitive, synthetic guarded domain theory takes the notion of computation step as primitive in the form of a modal operator. We use this extra intensional information to define two guarded recursive (finite) powerdomain constructions differing in the way non-determinism interacts with the computation steps. As an example application of these we show how to prove applicative similarity a congruence in the cases of may- and must-convergence for the untyped lambda calculus with finite non-determinism. Such results are usually proved using operational reasoning and Howe's method. Here we use an adaptation of a denotational method developed by Pitts in the context of domain theory.

Quantifiers on languages and codensity monads

AbstractThis paper contributes to the techniques of topo-algebraic recognition for languages beyond the regular setting as they relate to logic on words. In particular, we provide a general construction on recognisers corresponding to adding one layer of various kinds of quantifiers and prove a corresponding Reutenauer-type theorem. Our main tools are codensity monads and duality theory. Our construction hinges on a measure-theoretic characterisation of the profinite monad of the free S-semimodule monad for finite and commutative semirings S, which generalises our earlier insight that the Vietoris monad on Boolean spaces is the codensity monad of the finite powerset functor.

Lattices do not distribute over powerset

We show that there is no distributive law of the free lattice monad over the powerset monad. The proof presented here also works for other classes of lattices such as (bounded) distributive/modular lattices and also for some variants of the powerset monad such as the (nonempty) finite powerset monad.

Properties of the atoms in finitely supported structures

The goal of this paper is to present a collection of properties of the set of atoms and the set of finite injective tuples of atoms, as well as of the (finite and cofinite) powersets of atoms in the framework of finitely supported structures. Some properties of atoms are obtained by translating classical Zermelo–Fraenkel results into the new framework, but several important properties are specific to finitely supported structures (i.e. they do not have related classical Zermelo–Fraenkel and related non-atomic correspondents).

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.

A characterization of operators on functionals of discrete-time normal martingales

ABSTRACTIn this article, we aim at characterizing operators acting on functionals of discrete-time normal martingales. Let be a discrete-time normal martingale that has the chaotic representation property. We first introduce a transform, called 2D-Fock transform, for operators from the testing functional space to the generalized functional space of M. Then we characterize continuous linear operators from to via their 2D-Fock transforms. Our characterization theorems show that there exists a one-to-one correspondence between continuous linear operators from to and functions on Γ × Γ that only satisfy some type of growth condition, where Γ designates the finite power set of . Finally, we give some applications of our characterization theorems.

Exploring the Boundaries of Monad Tensorability on Set

We study a composition operation on monads, equivalently presented as large equational theories. Specifically, we discuss the existence of tensors, which are combinations of theories that impose mutual commutation of the operations from the component theories. As such, they extend the sum of two theories, which is just their unrestrained combination. Tensors of theories arise in several contexts; in particular, in the semantics of programming languages, the monad transformer for global state is given by a tensor. We present two main results: we show that the tensor of two monads need not in general exist by presenting two counterexamples, one of them involving finite powerset (i.e. the theory of join semilattices); this solves a somewhat long-standing open problem, and contrasts with recent results that had ruled out previously expected counterexamples. On the other hand, we show that tensors with bounded powerset monads do exist from countable powerset upwards.

Generalised powerlocales via relation lifting

This paper introduces an endofunctor VT on the category of frames that is parametrised by an endofunctor T on the category Set that satisfies certain constraints. This generalises Johnstone's construction of the Vietoris powerlocale in the sense that his construction is obtained by taking for T the finite covariant power set functor. Our construction of the T-powerlocale VT out of a frame is based on ideas from coalgebraic logic and makes explicit the connection between the Vietoris construction and Moss's coalgebraic cover modality.We show how to extend certain natural transformations between set functors to natural transformations between T-powerlocale functors. Finally, we prove that the operation VT preserves some properties of frames, such as regularity, zero-dimensionality and the combination of zero-dimensionality and compactness.

An alternative approach to Privault's discrete-time chaotic calculus

In this paper, we present an alternative approach to Privault's discrete-time chaotic calculus. Let Z be an appropriate stochastic process indexed by N (the set of nonnegative integers) and l 2 ( Γ ) the space of square summable functions defined on Γ (the finite power set of N ). First we introduce a stochastic integral operator J with respect to Z, which, unlike discrete multiple Wiener integral operators, acts on l 2 ( Γ ) . And then we show how to define the gradient and divergence by means of the operator J and the combinatorial properties of l 2 ( Γ ) . We also prove in our setting the main results of the discrete-time chaotic calculus like the Clark formula, the integration by parts formula, etc. Finally we show an application of the gradient and divergence operators to quantum probability.

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.

Localic completion of generalized metric spaces II: Powerlocales

The work investigates the powerlocales (lower, upper, Vietoris) of localic completions of generalized metric spaces. The main result is that all three are localic completions of generalized metric powerspaces, on the Kuratowski finite powerset. This is a constructive, localic version of spatial results of Bonsangue et al. and of Edalat and Heckmann. As applications, a localic completion is always overt, and is compact iff its gener- alized metric space is totally bounded. The representation is used to discuss closed intervals of the reals, with the localic Heine-Borel Theorem as a consequence. The work is constructive in the topos-valid sense. 2000 Mathematics Subject Classification 54B20 (primary); 06D22, 03G30,54E50,03F60 (secondary)

Widening operators for powerset domains

The finite powerset construction upgrades an abstract domain by allowing for the representation of finite disjunctions of its elements. While most of the operations on the finite powerset abstract domain are easily obtained by “lifting” the corresponding operations on the base-level domain, the problem of endowing finite powersets with a provably correct widening operator is still open. In this paper we define three generic widening methodologies for the finite powerset abstract domain. The widenings are obtained by lifting any widening operator defined on the base-level abstract domain and are parametric with respect to the specification of a few additional operators that allow all the flexibility required to tune the complexity/precision trade-off. As far as we know, this is the first time that the problem of deriving non-trivial, provably correct widening operators in a domain refinement is tackled successfully. We illustrate the proposed techniques by instantiating our widening methodologies on powersets of convex polyhedra, a domain for which no non-trivial widening operator was previously known.

A Category Theoretic Formulation for Engeler-style Models of the Untyped λ-Calculus

We give a category-theoretic formulation of Engeler-style models for the untyped λ-calculus. In order to do so, we exhibit an equivalence between distributive laws and extensions of one monad to the Kleisli category of another and explore the example of an arbitrary commutative monad together with the monad for commutative monoids. On Set as base category, the latter is the finite multiset monad. We exploit the self-duality of the category Rel, i.e., the Kleisli category for the powerset monad, and the category theoretic structures on it that allow us to build models of the untyped λ-calculus, yielding a variant of the Engeler model. We replace the monad for commutative monoids by that for idempotent commutative monoids, which, on Set, is the finite powerset monad. This does not quite yield a distributive law, so requires a little more subtlety, but, subject to that subtlety, it yields exactly the original Engeler construction.

Entailment systems for stably locally compact locales

The category SCFr U of stably continuous frames and preframe homomorphisms (preserving finite meets and directed joins) is dual to the Karoubi envelope of a category Ent whose objects are sets and whose morphisms X→ Y are upper closed relations between the finite powersets FX and FY . Composition of these morphisms is the “cut composition” of Jung et al. that interfaces disjunction in the codomains with conjunctions in the domains, and thereby relates to their multi-lingual sequent calculus. Thus stably locally compact locales are represented by “entailment systems” ( X,⊢) in which ⊢, a generalization of entailment relations, is idempotent for cut composition. Some constructions on stably locally compact locales are represented in terms of entailment systems: products, duality and powerlocales. Relational converse provides Ent with an involution, and this gives a simple treatment of the duality of stably locally compact locales. If A and B are stably continuous frames, then the internal preframe hom A⋔B is isomorphic to A ̃ ⊗B where A ̃ is the Hofmann–Lawson dual. For a stably locally compact locale X, the lower powerlocale of X is shown to be the dual of the upper powerlocale of the dual of X.

On complementary and independent mappings on databases

We define the notion of independent views to indicate whether the range values of the two views may be achieved independently The concept of complementary views indicates when the domain element can be uniquely determined by the range values of the two complementary views We consider the relationship between independent and complementary views In unrestricted domains, a view (but not the identity or empty view) can have more than one complementary, independent view Databases, however, are more restricted domains They are finite power sets A view is monotonic if it preserves inclusion However, in finite power sets when all views are monotonic, if a given view has another view which is independent and complementary, then this view is unique.