Idempotence

About: Idempotence is a research topic. Over the lifetime, 1860 publications have been published within this topic receiving 19976 citations. The topic is also known as: idempotent.

Rule formats for determinism and idempotence

Abstract: Determinism is a semantic property of (a fragment of) a language that specifies that a program cannot evolve operationally in several different ways. Idempotence is a property of binary composition operators requiring that the composition of two identical specifications or programs will result in a piece of specification or program that is equivalent to the original components. In this paper, we propose two (related) meta-theorems for guaranteeing determinism and idempotence of binary operators. These meta-theorems are formulated in terms of syntactic templates for operational semantics, called rule formats. We show the applicability of our formats by applying them to various operational semantics from the literature.

Algorithmic Complexity of a Problem of Idempotent Convex Geometry

Abstract: Properties of the idempotently convex hull of a two-point set in a free semimodule over the idempotent semiring Rmax min and in a free semimodule over a linearly ordered idempotent semifield are studied. Construction algorithms for this hull are proposed. Some asymptotic physical problems (such as quasiclassical approximation in quantum mechan- ics (1)), as well as many problems of optimization theory, mathematical economics, etc., admit a natural and simple formulation in terms of algebraic structures involving the operations of mini- mization or maximization (2, 3). Such algebraic structures are the object of the actively developing field of idempotent mathematics (3-5). Interesting, important and useful constructions and re- sults of traditional mathematics over number fields and similar structures have counterparts over idempotent semifields and semirings formulated in the spirit of Bohr's correspondence principle in quantum theory (6, 7). This correspondence can be far from obvious, though. In this paper, we consider the simplest problem of idempotent convex geometry (developed, in particular, in (8) and (7)), the construction of the convex hull of a two-point set in an idempotent semimodule, and prove that the algorithmic complexity of its solution increases with the growth of the semimodule's dimension. We consider the number line with the operations ⊕ = max and � = + and the additional element −∞, which plays the part of 0, i.e., is assumed to have the properties −∞ ⊕ a = a, −∞ � a = −∞. The ⊕ operation is commutative, associative, and idempotent ( a ⊕ a = a), the � operation is commutative and distributive with respect to ⊕. In idempotent analysis, the above- mentioned properties are considered to be the axioms of an idempotent semiring. The structure defined above has also the property of invertibility of theoperation and, for that reason, is called the idempotent semifield Rmax +. The idempotent semiring Rmax min is an important example of a semiring which is not a semi- field. It includes the entire number line with the additional elements −∞ and +∞ and has two operations ⊕ = max and � = min. The elements −∞ and +∞ are considered to be the elements 0 and 1 of the semiring, i.e., are assumed to have the properties −∞ ⊕ a = a, a � (+∞ )= a. In any idempotent semiring, the ⊕ operation induces a partial order: ab if and only if a ⊕ b = b; a ≺ b if and only if ab and ab. In both semirings considered above, this order is linear, because for any elements a and b ,w e haveab or ba; therefore, for any elements a and b of these semirings the operation a ∧ b of taking the lower bound is also defined. In the paper, we consider semimodules S n of column vectors of the form (a 1 , . .. ,a n ), a i ∈ S, with coordinatewise operations of generalized addition and multiplication by a scalar from the

Conservative algebras of 2-dimensional algebras, II

Abstract: In 1990 Kantor defined the conservative algebra W(n) of all algebras (i.e. bilinear maps) on the n-dimensional vector space. If n>1, then the algebra W(n) does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). We describe automorphisms, one-sided ideals, and idempotents of W(2). Also similar problems are solved for the algebra W2 of all commutative algebras on the 2-dimensional vector space and for the algebra S2 of all commutative algebras with trace zero multiplication on the 2-dimensional vector space.

Multivalued linear projections

Abstract: A multivalued linear projection operator P defined on linear space X is a multivalued linear operator which is idempotent and has invariant domain. We show that a multivalued projection can be characterized in terms of a pair of subspaces and then establish that the class of multivalued linear projections is closed under taking adjoints and closures. We apply the characterizations of the adjoint and completion of projection together with the closed graph and closed range theorems to give criteria for the continuity of a projection defined ona normed linear space. A new proof of the theorem on closed sums of closed subspaces in a Banach space (cf. Mennicken and Sagraloof [9, 10]) follows as a simple corollary. We then show that the topological decomposition of a space may be expressed in terms of multivalued projections. The paper is concluded with an application to multivalued semi-Fredholm relations with generalized inverses. Mathematics Subject Classification (2000): 47A06, 47A53 Quaestiones Mathematicae 25 (2002), 503-512

Idempotent Expansions for Continuous-Time Stochastic Control

Abstract: Max-plus methods have previously been used to solve deterministic control problems. The methods are based on max-plus (or min-plus) expansions and can yield curse-of-dimensionality-free numerical methods. In this paper, we explore min-plus methods for continuous-time stochastic control on a finite-time horizon. We first approximate the original value function via time-discretization. By generalizing the min-plus distributive property to continuum spaces, we obtain an algorithm for recursive computation of the time-discretized values, which we refer to as the idempotent distributed dynamic programming principle (IDDPP). Under the IDDPP, the value function at each step can be represented as an infimum of functions in a certain class. This is a min-plus expansion for the value function. For the specific class of problems considered here, we see that the class can be taken as that consisting of the quadratic functions. A means for reducing the numbers of constituent quadratic functions is discussed.