Related papers: Set-reconstructibility of Post classes

Let $X$ be a Borel subset of the Cantor set \textbf{C} of additive or multiplicative class ${\alpha},$ and $f: X \to Y$ be a continuous function with compact preimages of points onto $Y \subset \textbf{C}.$ If the image $f(U)$ of every…

Motivated by reconstruction results by Rubin, we introduce a new reconstruction notion for permutation groups, transformation monoids and clones, called automatic action compatibility, which entails automatic homeomorphicity. We further…

We employ the notions of `sequential function' and `interrogation' (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley's preorder-enriched category of…

In a recent paper we proposed the study of aggregation functions on lattices via clone theory approach. Observing that aggregation functions on lattices just correspond to $0,1$-monotone clones, we have shown that all aggregation functions…

For each clone C on a set A there is an associated equivalence relation, called C-equivalence, on the set of all operations on A, which relates two operations iff each one is a substitution instance of the other using operations from C. In…

This paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted Boolean constraint satisfaction problem. The problem is parameterised by a finite set F of non-negative functions that…

Canonical functions are a powerful concept with numerous applications in the study of groups, monoids, and clones on countable structures with Ramsey-type properties. In this short note, we present a proof of the existence of canonical…

Extending Sparks's theorem, we determine the cardinality of the lattice of $(C_1,C_2)$-clonoids of Boolean functions for certain pairs $(C_1,C_2)$ of clones of essentially unary, linear, or $0$- or $1$-separating functions or semilattice…

A full subcategory of a Grothendieck category is called deconstructible if it consists of all transfinite extensions of some set of objects. This concept provides a handy framework for structure theory and construction of approximations for…

We generalise classical reconstruction results in algebra, using the language of monads, monoidal categories, module categories, as well as various notions of duality, such as closedness, Grothendieck--Verdier duality (also known as…

Reversible logic represents the basis for many emerging technologies and has recently been intensively studied. However, most of the Boolean functions of practical interest are irreversible and must be embedded into a reversible function…

In this note we give a precise statement and a detailed proof for reconstruction problem of weak bialgebra maps. As an application we characterize indecomposability of weak algebras in categorical setting.

Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in…

The clone of term operations of an algebraic structure consists of all operations that can be expressed by a term in the language of the structure. We consider bounds for the length and the height of the terms expressing these functions,…

The process of replacing an arbitrary Boolean function by a bijective one, a fundamental tool in reversible computing and in cryptography, is interpreted algebraically as a particular instance of a certain group homomorphism from the X-fold…

Given a level set $E$ of an arbitrary multiplicative function $f$, we establish, by building on the fundamental work of Frantzikinakis and Host [13,14], a structure theorem which gives a decomposition of $\mathbb{1}_E$ into an almost…

We propose learning flexible but interpretable functions that aggregate a variable-length set of permutation-invariant feature vectors to predict a label. We use a deep lattice network model so we can architect the model structure to…

Splines can be constructed by convolving the indicator function of a cell whose shifts tessellate $\R^k$. This paper presents simple, non-algebraic criteria that imply that, for regular shift-invariant tessellations, only a small subset of…

In this paper, a technique on constructing nonlinear resilient Boolean functions is described. By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number…

Scaling-invariant functions preserve the order of points when the points are scaled by the same positive scalar (with respect to a unique reference point). Composites of strictly monotonic functions with positively homogeneous functions are…