Set Disjointness Research Articles

A probabilistic distributed algorithm for set intersection and its analysis

A probabilistic algorithm for checking set disjointness and performing set intersection of two sets stored in different machines is presented. The algorithm is intended to minimize the amount of communication between the machines. If n is the total number of elements and k is the number of bits required to represent each of the element, then it is shown that the expected running time of the set disjointness algorithm is O(log log n) rounds, each round consisting of exchanging one message with O( n + k) bits and performing O( n) steps of local computation (all the constants are small). The analysis of the algorithm involves approximating Markov chains by deterministic models.

Set containment inference and syllogisms

Type hierarchies and type inclusion (isa) inference are now standard in many knowledge representation schemes. In this paper, we show how to determine consistency and inference for collections of statements of the form mammal isa vertebrate. These containment statements relate the contents of two sets (or types). The work here is new in permitting statements with negative information: disjointness of sets, or non-inclusion of sets. For example, we permit the following statements also: mammal isanon(reptile), non(vertebrate) isanon(mammal), not(reptile isa amphibian). Binary containment inference is the problem of determining the consequences of positive constraints P and negative constraints not(P) on sets, where positive constraints have the form P:X⊆Y. Negations of these constraints therefore have the form not(P):X∩non(Y)≠ > 0slash;, so two sets have a nonempty intersection.We show binary containment inference is solved by rules essentially equivalent to Aristotle's syllogisms. Necessary and sufficient conditions for consistency, as well as sound and complete sets of inference rules, are presented for binary containment. The sets of inference rules are compact, and lead to polynomial-time inference algorithms, so permitting negative constraints does not result in intractability for this problem.

Existence of Drèze equilibrium under set-up costs

A fixed number of firms operate under technologies which are described as the union of the origin and a compact convex set disjoint from the origin: increasing returns thus prevail over some range. It is shown that a zero trade parametric price, quantity-constrained (Drèze) equilibrium exists at all prices and non-trivial equilibrium will exist at some prices if (in particular) the consumer side of the economy is replicated sufficiently. Following recent work by Weddepohl the objective is to provide a logically consistent and useful tool for the analysis of increasing return economies.

A probabilistic lower bound for checking disjointness of sets

On prouve, dans cet article, que pour un probleme particulier, determiner si deux ensembles (pas necessairement egaux en taille) sont disjoints montre que l'acceleration est limitee a une constante. Une borne inferieure probabiliste Ω(m log n) est prouvee, ou m et n sont les cardinalites d'ensembles et m≥n

Some Area-Time Tradeoffs for VLSI

Area-time bounds on VLSI circuits for context-free language recognition, for the evaluation of propositional calculus formulae and for set equality and disjointness questions, are considered. In all cases, a lower bound $AT^{2\alpha } = \Omega (n^{1 + \alpha } )$ is proved, where A is the chip area, T the execution time, and $0 \leqq \alpha \leqq 1$. Similar results were known for computations with $\Omega (n)$-bit outputs, but the computations considered here have only 1-bit outputs. Upper bounds are also discussed.

The IF THEN ELSE statement and interval-valued fuzzy sets of higher type

A new fuzzy relation which represents an IF THEN ELSE (abbreviated to “ITE”) statement is constructed. It is shown that a relation which (a) always gives correct inference and (b) does not contain false information which is not present in the ITE statement, must be of a higher degree of fuzziness than the antecedents and consequents of the ITE statement. Three different ways of increasing the fuzziness of the relation are used here: (1) the fuzzy relation is of higher type; (2) it is interval-valued; (3) contains a BLANK or “don't know” component. These three types of fuzziness come about naturally because the relation is a restriction of an initial relation which represents the second approximation to the state of complete ignorance. There exist successive approximations to the state of complete ignorance, each of them being an intervalvalued fuzzy set BLANK of one type higher than the previous one. Similar representations of the zeroth and first approximation to the state of ignorance have been used in the theory of probability, though in a rather heuristic fashion. The assignment of a value to a variable is represented as a complete restriction of the BLANK state of type N; the “value” being any pure (non-interval-valued) fuzzy set of type N. With the new relation, the inferred set is a superposition of the consequents of the ITE statement and of the BLANK state, each of these components being multiplied by an interval-valued coefficient. In the case of modus ponens inference, the component with the highest coefficient (determined from a specially defined ordering relation for interval-values) is always the correct consequent, provided that the original ITE statement is logically consistent. A mathematical test for logical consistency of the (possibly fuzzy) ITE statement is given. Disjointness of fuzzy sets is defined and connected up with logical consistency. The paradoxes of the implication of mathematical logic disappear in the fuzzy set treatment of the ITE statement. Subnormal fuzzy singletons find their natural interpretation. When used as an antecedent in an ITE statement, such a singleton does not have enough strength to induce the consequent with complete certainty. Instead it induces a superposition of an interval-valued fuzzy set and the BLANK state. New definitions for union, intersection and complementation of fuzzy sets of higher type are suggested. A new interpretation of an interval-valued fuzzy set of type N as a collection of fuzzy sets of type N, not as a type N+1 set, is given. There exist two types of union, intersection and complementation for intervalvalued fuzzy sets. They are called the fuzzy and the crisp operations, respectively. It is suggested that the negation be represented by an interval-valued fuzzy set. We conclude that increased fuzziness in a description means increased ability to handle inexact information in a logically correct manner.

Disjointness and Weak Mixing of Minimal Sets

Disjointness and Weak Mixing of Minimal Sets

Disjointness and weak mixing of minimal sets

In [3, Problem G, p. 34] the following problem is proposed: characterize the class o' of all flows which are disjoint from every distal flow. We show here that ai consists precisely of the weakly mixing minimal flows. In arriving at this conclusion we make use of a result (Corollary to Theorem 1) which has been arrived at independently by Keynes and Robertson [4, Theorem 3.4, p. 366]. In the following all transformation groups will be assumed to have compact Hausdorff phase spaces. For any unexplained notation or terminology the reader is referred to [1 ] and [3 ]. This research is part of the author's doctoral dissertation prepared at Yale University under the guidance of Professor S. Kakutani, whose assistance the author gratefully acknowledges. We say that a transformation group (X, T) is weakly mixing if given nonempty open subsets A, B, C, D of X there is tET such that AtnCC0 and BtnDS 0.

Approximation by polynomials with restricted zeros

Let D be a simply connected domain in the complex plane and let I’ be an arbitrary point set. A r-polynomial is a polynomial whose zeros belong to r. r is called a polynomial approximation set (PA-set) relative to D if every zero-free holomorphic function in D can be uniformly approximated on compacts of D by r-polynomials. Special cases of P-d-sets were first considered by G. R. MacLane [I I] and later by M. D. Thompson [12]. Each proved that Jordan curves of special types are PA-sets for their interiors. More generally, J. Korevaar [8] has proved that if D is a bounded simply connected domain and r is a bounded set disjoint from D, then I’is a Pz4-set for D if and only if clos I’separates the plane and D belongs to a bounded component of its complement. In particular, the boundary of every bounded simply connected domain is a PA-set for its interior. When D and r are unbounded, the situation is not quite so simple. In fact, T. Ganelius [2] has shown that if D contains a half-plane, any sequence of De, the complement of D, polynomials which converges uniformly on a disc in D to f f 0, converges uniformly on every compact of the plane to an entire function. Thus any zero-free holomorphic function in D which is approximable by DC-polynomials would be the restriction of an entire function so that DC is not a P/l-set for D. On the other hand, Thompson [12] has shown that if D is contained in a half-plane and the boundary of D is a special type of Jordan curve, then the boundary is a Pi4-set for D. The principal result of this paper is:

Disjointness of minimal sets.

Disjointness of minimal sets.

A lattice homomorphism of a lattice ordered group

The homomorphism we have in mind has been introduced by Jaffard, [1],2 for lattice ordered groups, and has been discussed by Pierce, [2], for any distributive lattice L with a minimal element. It may be defined as the homomorphism a of L for which a(x) =a(y) if and only if the set of elements in L disjoint with x is the same as the set disjoint with y. Pierce has shown that this is the maximal lattice homomorphism of L whose kernel is [0], the set whose only element is 0, and that it is the only lattice homomorphism of L, with kernel [0], whose image is disjunctive. Jaffard called the elements of the image lattice (i.e., the equivalence sets of the homomorphism) filets in L, but we prefer the term carriers in L because of their role in function spaces. Our purpose is to obtain a characterization of a, which involves the group operation, for the case where L is the positive cone of an archimedean lattice ordered group. Jaffard has shown, in this case, that a has the following properties: (a) a: O implies a(a) >a(O). (b) a is a lattice homomorphism; i.e., a(aUb) =a(a)Ua(b), a(anb) a (xa) rxab) . (c) a(a+b) =a(aUb). We show that a has the additional property: (d) If x=sup [xi|iEI], then sup [a(xi)IiEI] exists and a(x) = sup [a(xifl|iEII]. PROOF. First, a(x) _a(xi) for all iEI. Suppose there is z>O such that a(z) >a(x,), for all iEI, but a(z) 0 such that a(w) ? a(x) and a(w)ra(z) =a(0). Then a(w)Oa(xi) =a(0), for all iEI. Let v=xCxw. Then a(v) =a(x) a(w) =a(w), and v>O. It follows that vqxi=O, for all iEI. Now, v<x, xs<x, and vflxi=O implies that v+xi=vUxi<x for all iEI. If we let y=x-v, then y<x and y2xi for all iEI. But this contradicts the definition of x.