In this paper we investigate a formalism for solving planning problems based on ordered task decomposition using Answer Set Programming (ASP). Our planning methodology is an adaptation of Hierarchical Task Network (HTN) planning, an approach that has led to some very efficient planners. The ASP paradigm evolved out of the stable semantics for logic programs in recent years and is strongly relat...

We define the total weak discrepancy of a poset P as the minimum nonnegative integer k for which there exists a function f : V → Z satisfying (i) if a ≺ b then f(a) + 1 ≤ f(b) and (ii) ∑ |f(a)− f(b)| ≤ k, where the sum is taken over all unordered pairs {a, b} of incomparable elements. If we allow k and f to take real values, we call the minimum k the fractional total weak discrepancy of P . The...

Every model of ZFC contains a product of two linear orders, each of size א1, with the property that every subset or complement thereof contains a maximal chain. Mathematics Subject Classification (2000): 06A10.

We present a finite set theory implementation for ACL2. Our library represents sets as fully ordered lists, and provides efficient implementations of the typical set theory operations such as insertion, deletion, union, intersection, difference, cardinality, and sorting lists to create sets. It also includes facilities for quantifying predicates over sets, filtering sets by some criteria, and t...

This paper considers the problem of computing an interpolating skin of a ordered set of discrete 2D balls. By construction, the skin is constrained to be C continuous, and for each ball, it touches the ball at a point and is tangent to the ball at the point of contact. Using an energy formulation, we derive differential equations that are designed to minimize the skin’s arc length, curvature, o...

R. Redheffer described an n×n matrix of 0’s and 1’s the size of whose determinant is connected to the Riemann Hypothesis. We describe the permutations that contribute to its determinant and its permanent in terms of integer factorizations. We generalize the Redheffer matrix to finite posets that have a 0 element and find the analogous results in the more general situation.

In this paper we introduce the notion of the fractional weak discrepancy of a poset, building on previous work on weak discrepancy in [5, 8, 9]. The fractional weak discrepancy wdF (P ) of a poset P = (V,≺) is the minimum nonnegative k for which there exists a function f : V → R satisfying (1) if a ≺ b then f(a) + 1 ≤ f(b) and (2) if a ‖ b then |f(a)− f(b)| ≤ k. We formulate the fractional weak...

We study statistics on ordered set partitions whose generating functions are related to p, q-Stirling numbers of the second kind. The main purpose of this paper is to provide bijective proofs of all the conjectures of Steingŕımsson(Arxiv:math.CO/0605670). Our basic idea is to encode ordered partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structu...

Ordered Choice Logic Programming (OCLP) allows for dynamic preference-based decision-making with multiple alternatives without the need for any form of negation. This complete absence of negation does not weaken the language as both forms (classical and as-failure) can be intuitively simulated in the language and eliminated using a simple pre-processor, making it also an easy language for users...

Through the development of set-valued information systems from models of single-valued information systems, we see there are important applications in the pattern recognition and intelligent decision-making. In this paper, we first introduce concept of dominance rough entropy in set-valued ordered information systems. When the average uncertainty quantity of rough set is measured in set-valued ...