George Markowsky

Chain-complete posets and directed sets with applications

Let a poset P be called chain-complete when every chain, including the empty chain, has a sup in P. Many authors have investigated properties of posets satisfying some sort of chain-completeness condition (see [,11, [-31, [6], I-71, [17], [,181, [191, [,211, [,221), and used them in a variety of applications. In this paper we study the notion of chain-completeness and demonstrate its usefulness for various applications. Chain-complete posets behave in many respects like complete lattices; in fact, a chaincomplete lattice is a complete lattice. But in many cases it is the existence of sups of chains, and not the existence of arbitrary sups, that is crucial. More generally, let P be called chain s-complete when every chain of cardinality not greater than ~ has a sup. We first show that if a poset P is chain s-complete, then every directed subset of P with cardinality not exceeding ct has a sup in P. This sharpens the known result ([,8], [,181) that in any chain-complete poset, every directed set has a sup. Often a property holds for every directed set i f and only if it holds for every chain. We show that direct (inverse) limits exist in a category if and only if chain colimits (chain limits) exist. Since every chain has a well-ordered cofinal subset [11, p. 681, one need only work with well-ordered collections of objects in a category to establish or disprove the existence of direct and inverse limits. Similarly, a topological space is compact if and only if every chain of points has a cluster point. A chain of points is a generalization of a sequence. Chain-complete posers, like complete lattices, arise from closure operators in a fairly direct manner. Using closure operators we show how to form the chaincompletion P of any poset P. The chain-completion/~ of a poset P is a chain-complete poset with the property that any chain-continuous map from a poser P into a chain-complete poset Q extends uniquely to a chain-continuous map from the completion/~ into Q, where by a chaincontinuous map we mean one that preserves sups of chains. If P is already chaincomplete, then/~ is naturally isomorphic to P. This completion is not the MacNeille

Exact and approximate membership testers

In this paper we consider the question of how much space is needed to represent a set. Given a finite universe U and some subset V (called the vocabulary), an exact membership tester is a procedure that for each element s in U determines if s is in V. An approximate membership tester is allowed to make mistakes: we require that the membership tester correctly accepts every element of V, but we allow it to also accept a small fraction of the elements of U - V.

On the number of prime implicants

It is shown that any Boolean expression in disjunctive normal form having k conjuncts, can have at most 2k prime implicants. However, there exist such expressions that have 2k2 prime implicants. It is also shown that any Boolean expression on n distinct propositional variables can have at most O(3nn) prime implicants, and that there exist expressions with ?(3nn prime implicants.

Learning probabilistic prediction functions

The question of how to learn rules, when those rules make probabilistic statements about the future, is considered. Issues are discussed that arise when attempting to determine what a good prediction function is, when those prediction functions make probabilistic assumptions. Learning has at least two purposes: to enable the learner to make predictions in the future and to satisfy intellectual curiosity as to the underlying cause of a process. Two results related to these distinct goals are given. In both cases, the inputs are a countable collection of functions which make probabilistic statements about a sequence of events. One of the results shows how to find one of the functions, which generated the sequence, the other result allows to do as well in terms of predicting events as the best of the collection. In both cases the results are obtained by evaluating a function based on a tradeoff between its simplicity and the accuracy of its predictions.<<ETX>>

Multidimensional bin packing algorithms

A comparative study is made of algorithms for a general multidimensional problem involving the packing of k-part objects in k compartments in a large supply of bins. The goal is to pack the objects using a minimum number of bins. The properties and limitations of the algorithms are discussed, including k-dimensional analogs of some popular one-dimensional algorithms. An application of the algorithms is the design of computer networks.

A mathematical analysis of human leukocyte antigen serology

Abstract This paper presents and explores a comprehensive mathematical model for human leukocyte antigen serology, based on a mathematical formalization of the concept of specificity. This model is general enough to take into account such factors as absorption, elution, cross-reactivity, and incomplete immunization. The paper includes a presentation of the relevant immunological background and a short discussion of the underlying computational difficulty of the basic problems. Upper and lower bounds are derived for the minimal number of specificities required to explain a given set of HLA reactions, and it is shown that the numbers of antibodies and antigens involved must be no less then this minimal number of specificities. Other techniques and theorems are also presented to aid in reducing and analyzing HLA reaction matrices.

On sets of Boolean n-vectors with all k-projections surjective

SummaryGiven a set, S, of Boolean n-vectors, one can choose k of the n coordinate positions and consider the set of k-vectors which results by keeping only the designated k positions of each vector, i.e., from k-projecting S. In this paper, we study the question of finding sets S as small as possible such that every k-projection of S yields all the 2k possible k-vectors. We solve this problem constructively and almost optimally for k=2 and all n. For k≧3, the constructive solutions we describe are much larger than an O(k 2k log n) nonconstructive upper bound which we derive. The nonconstructive approach allows us to generate fairly small sets S which have a very high probability of having the surjective k-projection property.

Categories of chain-complete posets

Abstract We investigate the existence of various limits and colimits in three categories: CPI (chain-complete posets and isotone maps); CPC (chain-complete posets and chain-continuous maps); CPC ∗ (chain-complete posets and chain- ∗ continuous maps). Among other things we show CPC ∗ to be complete and cocomplete. By way of contrast we show that LC ∗ (complete lattices and chain- ∗ continuous maps) is neither complete nor cocomplete. We also introduce a construction which yields the chain-completion of a poset and other “completions” as special cases.

An Evaluation of Local Path ID Swapping in Computer Networks

This paper analyzes a method for identifying end-to-end connections in computer networks which is designed to provide reductions in the sizes of the packet headers and routing tables stored in the nodes. The method, known as Local Path ID Swapping, uses a shortened connection identifier, called the LPID, in the message headers and routing tables. In general, the LPID field is swapped in the message header from node to node along the path of the route. Some analytical results are presented for evaluating the important tradeoffs involved in LPID swapping. Most notable is the tradeoff between the size of the LPID field and the number of connections which can be defined in the network.

Best Huffman trees

SummaryGiven a sequence of positive weights, W=w1≧...≧wn>0, there is a Huffman tree, T↑ (“T-up”) which minimizes the following functions: max{d(wi)}; Σd(wi); Σf(d(wi)) wi(here d(wi) represents the distance of a leaf of weight wi to the root and f is a function defined for nonnegative integers having the property that g(x) = f(x + 1) − f(x) is monotone increasing) over the set of all trees for W having minimal expected length. Minimizing the first two functions was first done by Schwartz [5]. In the case of codes where W is a sequence of probabilities, this implies that the codes based on T↑ have all their absolute central moments minimal. In particular, they are the least variance codes which were also described by Kou [3]. Furthermore, there exists a Huffman tree T↓, (“T-down”) which maximizes the functions considered above.However, if g(x) is monotone decreasing, T↑ and T↓, respectively maximize and minimize Σf(d(wi) wi) over the set of all trees for W having minimal expected length. In addition, we derive a number of interesting results about the distribution of labels within Huffman trees. By suitable modifications of the usual Huffman tree construction, (see [1]) T↑ and T↓ can also be constructed in time O(n log n).