Related papers: Override and restricted union for partial function…
We present axiomatisations for a number of partial function signatures that include domain restriction, modelled as a right normal band operation. Other operations considered are override and update, difference, minus, intersection,…
Disjoint union is a partial binary operation returning the union of two sets if they are disjoint and undefined otherwise. A disjoint-union partial algebra of sets is a collection of sets closed under disjoint unions, whenever they are…
We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, {\em if-then-else} and…
Domain operations on semirings have been axiomatised in two different ways: by a map from an additively idempotent semiring into a boolean subalgebra of the semiring bounded by the additive and multiplicative unit of the semiring, or by an…
We give complete, finite quasiequational axiomatisations for algebras of unary partial functions under the operations of composition, domain, antidomain, range and intersection. This completes the extensive programme of classifying algebras…
We investigate notions of complete representation by partial functions, where the operations in the signature include antidomain restriction and may include composition, intersection, update, preferential union, domain, antidomain, and set…
Unification and generalization are operations on two terms computing respectively their greatest lower bound and least upper bound when the terms are quasi-ordered by subsumption up to variable renaming (i.e., $t_1\preceq t_2$ iff $t_1 =…
We investigate the representation and complete representation classes for algebras of partial functions with the signature of relative complement and domain restriction. We provide and prove the correctness of a finite equational…
In this paper, a new axiomatization for unbounded functional calculi is proposed and the associated theory is elaborated comprising, among others, uniqueness and compatibility results and extension theorems of algebraic and topological…
Override and update are natural constructions for combining partial functions, which arise in various program specification contexts. We use an unexpected connection with combinatorial geometry to provide a complete finite system of…
We present a formally verified global optimization framework. Given a semialgebraic or transcendental function $f$ and a compact semialgebraic domain $K$, we use the nonlinear maxplus template approximation algorithm to provide a certified…
We define antidomain operations for algebras of multiplace partial functions. For all signatures containing composition, the antidomain operations and any subset of intersection, preferential union and fixset, we give finite equational or…
Various problems on integers lead to the class of congruence preserving functions on rings, i.e. functions verifying $a-b$ divides $f(a)-f(b)$ for all $a,b$. We characterized these classes of functions in terms of sums of rational…
A semigroup of binary relations (under composition) on a set $X$ is \emph{complemented} if it is closed under the taking of complements within $X\times X$. We resolve a 1991 problem of Boris Schein by showing that the class of finite unary…
We prove that the finite representation property holds for representation by partial functions for the signature consisting of composition, intersection, domain and range and for any expansion of this signature by the antidomain, fixset,…
We introduce a new class of functions that can be minimized in polynomial time in the value oracle model. These are functions $f$ satisfying $f(x)+f(y)\ge f(x \sqcap y)+f(x \sqcup y)$ where the domain of each variable $x_i$ corresponds to…
Overlap functions were introduced as class of bivariate aggregation functions on [0, 1] to be applied in image processing. This paper has as main objective to present appropriates definitions of overlap functions considering the scope of…
We use the Aubry-Perret bound for singular curves, a generalization of the Hasse-Weil bound, to prove the following curious result about rational functions over finite fields: Let $f(X),g(X)\in\Bbb F_q(X)\setminus\{0\}$ be such that $q$ is…
The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The…
In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of equations of continuous functions in bounded domains, and for which all functions are computable in the sense that it is possible to…