Kenneth Kunen

Negation in logic programming

Abstract We define a semantics for negation as failure in logic programming. Our semantics may be viewed as a cross between the approaches of Clark [5] and Fitting [7]. As does [7], our semantics corresponds well with real PROLOG in the standard examples used in the literature to illustrate problems with [5]. Also, PROLOG and the common variants of it are sound but not complete for our semantics. Unlike [7], our semantics is constructive, in that the set of supported queries is recursively enumerable. Thus, a complete interpreter exists in theory, although we point out that there are serious difficulties in building one that works well in practice.

Signed data dependencies in logic programs

Abstract Logic programming with negation has been given a declarative semantics by Clarks completed database (CDB), and one can consider the consequences of the CDB in either two-valued or three-valued logic. Logic programming also has a proof theory given by SLDNF derivations. Assuming the data-dependency condition of strictness , we prove that the two-valued and three-valued semantics are equivalent. Assuming allowedness (a condition on occurrences of variables), we prove that SLDNF is complete for the three-valued semantics. Putting these two results together, we have completeness of SLDNF deductions for strict and allowed databases and queries under the standard two-valued semantics. This improves a theorem of Cavedon and Lloyd, who obtained the same result under the additional assumption of stratifiability .

Some applications of iterated ultrapowers in set theory

Let 91 be a ~-complete ultrafilter on the measurable cardinal ~. Scott [ 1 3 ] proved V ¢ L by using 91 to take the ultrapower of V. Gaifman [ 2] considered iterated ultrapowers of V by cg to conclude even stronger results; for example, that L n ~(6o) is count-able. In this paper we discuss some new applications of iterated ultra-powers. In § § 1-4, we develop a straightforward generalization of Gaif-mans method which is needed fo~ some of the restqts in § § 6-1 1. Namely, we consider iterated uhrapowers of a sub-model, M, of the universe by an ultrafilter which need not be in M. § 5 discusses some known results within our present framework. In § 6, we investigate the universe constructed from a normal ultrafilter on the measurable cardinal ~:, and show that in this universe tb~ normal ultrafilter is unique. In § 7, we obtain a character-~zation of arbitrary ~-complete free ultrafilters in this universe, and in § 8, we show that this universe has some pathological model-theoretic properties. § 9 uses methods of § 6 to discuss the problem of GCH at a measurable cardinal. We show that in the theory

Ultrafilters and independent sets

Independent families of sets and of functions are used to prove some theorems about ultrafilters. All of our results are well known to be provable from some form of the generalized continuum hypothesis, but had remained open without such an assumption. Independent sets are used to show that the Rudin-Keisler ordering on ultrafilters is nonlinear. Independent functions are used to prove the existence of good ultrafilters.

Elementary embeddings and infinitary combinatorics

One of the standard ways of postulating large cardinal axioms is to consider elementary embeddings, j , from the universe, V , into some transitive submodel, M . See Reinhardt–Solovay [7] for more details. If j is not the identity, and κ is the first ordinal moved by j , then κ is a measurable cardinal. Conversely, Scott [8] showed that whenever κ is measurable, there is such j and M . If we had assumed, in addition, that , then κ would be the κ th measurable cardinal; in general, the wider we assume M to be, the larger κ must be.

Random and Cohen Reals

Publisher Summary This chapter focuses on facts in common between the random and Cohen extensions. Continuum Hypothesis can be violated by adding Cohen generic reals and by adding random reals. These generic extensions are similar in many respects but differ greatly in their effects on measure and category. The chapter presents a unified treatment of both extensions simultaneously. Their properties can actually be derived from some abstract properties shared by the ideal of meagre sets and the ideal of null set. The chapter presents results that also apply to the extensions arising from any other ideals that share these properties.

Weak P-Points in N*

N denotes the space of natural numbers (i.e., ω) with the discrete topology. sN is the Cech compactification of N, and N* = sN\N. We often identify the points of sN with the ultrafilters on a?, in which case the points of N* are the non-principal ultrafilters.

A compact L-space under CH

Abstract The continuum Hypothesis implies that there is a compact Hausdorff space which is hereditarily Lindelof but not separable. The space is the support of a Borel probability measure for which the measure-0 subsets, the first-category subsets, and the separable subsets all coincide.

The structure of conjugacy closed loops

We study structure theorems for the conjugacy closed (CC-) loops, a specific variety of G-loops (loops isomorphic to all their loop isotopes). These theorems give a description all such loops of small order. For example, if p and q are primes, p 2, there is exactly one non-group CC-loop in order 2q, and there are exactly three in order q2. We also derive a number of equations valid in all CC-loops. By contrast, every equation valid in all G-loops is valid in all loops.

Diassociativity in Conjugacy Closed Loops

Abstract Let Q be a conjugacy closed loop, and N(Q) its nucleus. Then Z(N(Q)) contains all associators of elements of Q. If in addition Q is diassociative (i.e., an extra loop), then all these associators have order 2. If Q is power-associative and |Q| is finite and relatively prime to 6, then Q is a group. If Q is a finite non-associative extra loop, then 16 ∣ |Q|.