Rational Closure Research Articles

Nonmonotonic reasoning, conditional objects and possibility theory

This short paper relates the conditional object-based and possibility theory-based approaches for reasoning with conditional statements pervaded with exceptions, to other methods in nonmonotonic reasoning which have been independently proposed: namely, Lehmann's preferential and rational closure entailments which obey normative postulates, the infinitesimal probability approach, and the conditional (modal) logics-based approach. All these methods are shown to be equivalent with respect to their capabilities for reasoning with conditional knowledge although they are based on different modeling frameworks. It thus provides a unified understanding of nonmonotonic consequence relations. More particularly, conditional objects, a purely qualitative counterpart to conditional probabilities, offer a very simple semantics, based on a 3-valued calculus, for the preferential entailment, while in the purely ordinal setting of possibility theory both the preferential and the rational closure entailments can be represented.

Cyclides in Geometric Modeling: Computational Tools for an Algorithmic Infrastructure

The Dupin cyclide is a quartic surface with useful properties such as circular lines of curvature, rational parametric representations and closure under offsetting. All natural quadrics (cone, cylinder, sphere) and the torus are special cases of the cyclide. Applications of cyclides include variable radius blending, piping design and design of tubular geometry, such as mold gates and wire harnesses. While the mathematical treatment of cyclides is well developed, the tools required to incorporate this surface into conventional modeling applications have not received sufficient attention. In this paper, we detail various methods for manipulating and computing with the cyclide. Issues such as auxiliary representations, inverse parameter mapping, point classification, normal projection of a point onto the surface, distance computation, bounding volumes and surface intersection detection are discussed in detail. We also provide implemented examples of design applications that utilize these computational tools.

Another perspective on default reasoning

The lexicographic closure of any given finite setD of normal defaults is defined. A conditional assertiona ❘∼b is in this lexicographic closure if, given the defaultsD and the facta, one would concludeb. The lexicographic closure is essentially a rational extension ofD, and of its rational closure, defined in a previous paper. It provides a logic of normal defaults that is different from the one proposed by R. Reiter and that is rich enough not to require the consideration of non-normal defaults. A large number of examples are provided to show that the lexicographic closure corresponds to the basic intuitions behind Reiter's logic of defaults.

What does a conditional knowledge base entail?

This paper presents a logical approach to nonmonotonic reasoning based on the notion of a nonmonotonic consequence relation. A conditional knowledge base, consisting of a set of conditional assertions of the type if … then …, represents the explicit defeasible knowledge an agent has about the way the world generally behaves. We look for a plausible definition of the set of all conditional assertions entailed by a conditional knowledge base. In a previous paper, Kraus and the authors defined and studied preferential consequence relations. They noticed that not all preferential relations could be considered as reasonable inference procedures. This paper studies a more restricted class of consequence relations, rational relations. It is argued that any reasonable nonmonotonic inference procedure should define a rational relation. It is shown that the rational relations are exactly those that may be represented by a ranked preferential model, or by a (nonstandard) probabilistic model. The rational closure of a conditional knowledge base is defined and shown to provide an attractive answer to the question of the title. Global properties of this closure operation are proved: it is a cumulative operation. It is also computationally tractable. This paper assumes the underlying language is propositional.

Rational closure of the group algebra of a linearly ordered group in the division ring of Mal'tsev series

Rational closure of the group algebra of a linearly ordered group in the division ring of Mal'tsev series

Dérivées directionnelles et développements de Taylor combinatoires

Let U be the set of all isomorphism classes of atomic species, let K be a binomial half-ring and K ̄ its rational closure. The differential half-ring K [[ U ]] of all K -species in the sense of Yeh is a combinatorial and algebraic extension of the half-ring K [[ X]] of all formal power series in one indeterminate X. Using the operation of substitution in K ̄ [[ U ]] and the Q -species X̂ of “pseudo-singletons” we study two new notions: the combinatorial directional derivative of a K -species in the direction of another K -species and Taylor expansions in K [[ U ]]. The use of K -species is essential here. We show, along the way, certain similarities and differences between these two notions and their classical analogues in K [[ X]]. Tables are given for small cardinalities.

On the generalized iterates of Yeh's combinatorial [formula omitted]-species

Let f = f( x) = x + a 2 x 2 + … ∈ K[[ x]] be a “normalized” power series over a (commutative) field K of characteristic zero. The operator Δ f : K[[ x]] → K[[ x]], defined by Δ f g = g ∘ f − g, has been used in ( G. Labelle, European J. Combin. 1 (1980) , 113–138) to obtain formulas for the inverse f 〈−1〉 and the generalized iterates f 〈 t〉 , t ∈ K, of the series f. A. Joyal (in Lect. Notes in Math. Vol. 1234, pp. 126–159, Springer-Verlag, New York/Berlin, 1986) was the first to realize that Δ f can be lifted to the combinatorial level. He made use of this fact to obtain a formula for a virtual species F 〈−1〉 which is the inverse (under substitution) of any given normalized species F = X + …. Using the same operator, we show that the concept of K -species in the sense of Y.-N. Yeh (ibid.) (where K is now only a binomial half-ring) is a good context for the definition of the generalized iterates F 〈 t〉 , t ∈ K, of any normalized species (or K -species). We present a new approach to Yeh's extension of substitution to K -species. We also introduce the notions of “infinitesimal generator,” “directional derivatives,” and “Lie bracket” of K -species, which turn out to be K -species, where K denotes the “rational closure” of K . These concepts give, in return, a better insight into substitution itself. For example, G ∘ F can be written in the form G ∘ F = (exp D Φ ) G for a suitably chosen derivation D Φ . More generally, G ∘ F 〈 t〉 = (exp tD Φ ) G. Two normalized K -species commute under substitution if and only if the Lie bracket of their infinitesimal generators is zero. Explicitly computed examples are also given.

Density functional theory of nonuniform polyatomic systems. II. Rational closures for integral equations

With the density functional theory outlined in paper I, we address and formally solve the nonlinear inversion problem associated with identifying the entropy density functional for systems with bonding constraints. With this development, we derive a nonlinear integral equation for the average site density fields of a polyatomic system. When external potential fields are set to zero, the integral equation represents a mean field theory for symmetry breaking and thus phase transformations of polyatomic systems. In the united atom limit where the intramolecular interaction sites become coincident, the mean field theory becomes identical to that developed for simple atomic systems by Ramakrishnan, Yussouff, and others. When the external potential fields are particle producing fields (in the sense introduced long ago by Percus), the integral equation represents a theory for the solvation of a simple spherical solute by a polyatomic solvent. In the united atom limit for the solvent, the theory reduces to the hypernetted chain (HNC) integral equation. This reduction is not found with the so-called ‘‘extended’’ RISM equation; indeed, the extended RISM equation—the theory in which the HNC closure of simple systems is inserted directly into the Chandler–Andersen (i.e., RISM or SSOZ) equation—behaves poorly in the united atom limit. The integral equation derived herein with the density functional approach however suggests a rational closure of the RISM equation which does pass over to the HNC theory in the united atom limit. The new integral equation for pair correlation functions arising from this suggested closure is presented and discussed.

Testing a closure for velocity/pressure-gradient correlations in nonuniform turbulent flow

Certain velocity/pressure-gradient correlations arise in the Reynolds stress equation through the interaction of the turbulent velocity field with the mean strain rate field. An existing, rational closure representation for these correlations is extended here and then tested. The closure permits good agreement to be obtained between observed and computed Reynolds stress components as they evolve under the influence of uniform shear and plane strain.

On the Derived Quotient Module

With every R-module M associate the direct limit of ${\operatorname {Hom}_R}(D,M)$ over the dense right ideals of R, the derived quotient module $\mathcal {D}(M)$ of M. $\mathcal {D}(M)$ is a module over the complete ring of right quotients of R. Relationships between $\mathcal {D}(M)$ and the torsion theory of Gentile-Jans are explored and functorial properties of $\mathcal {D}$ are discussed. When M is torsion free, results are given concerning rational closure, rational completion, and injectivity.

On the derived quotient module

With every R-module M associate the direct limit of HomR (D, M) over the dense right ideals of R, the derived quotient module 9(M) of M. 9(M) is a module over the complete ring of right quotients of R. Relationships between 9(M) and the torsion theory of Gentile-Jans are explored and functorial properties of 9 are discussed. When M is torsion free, results are given concerning rational closure, rational completion, and injectivity.