Kinds Of Counterexamples Research Articles

No simple dual to the causal holographic information?

In AdS/CFT, the fine grained entropy of a boundary region is dual to the area of an extremal surface X in the bulk. It has been proposed that the area of a certain ‘causal surface’ C — i.e. the ‘causal holographic information’ (CHI) — corresponds to some coarse-grained entropy in the boundary theory. We construct two kinds of counterexamples that rule out various possible duals, using (1) vacuum rigidity and (2) thermal quenches. This includes the ‘one-point entropy’ proposed by Kelly and Wall, and a large class of related procedures. Also, any coarse-graining that fixes the geometry of the bulk ‘causal wedge’ bounded by C, fails to reproduce CHI. This is in sharp contrast to the holographic entanglement entropy, where the area of the extremal surface X measures the same information that is found in the ‘entanglement wedge’ bounded by X.

Are There Good Mistakes? A Theoretical Analysis of CEGIS

Counterexample-guided inductive synthesis CEGIS is used to synthesize programs from a candidate space of programs. The technique is guaranteed to terminate and synthesize the correct program if the space of candidate programs is finite. But the technique may or may not terminate with the correct program if the candidate space of programs is infinite. In this paper, we perform a theoretical analysis of counterexample-guided inductive synthesis technique. We investigate whether the set of candidate spaces for which the correct program can be synthesized using CEGIS depends on the counterexamples used in inductive synthesis, that is, whether there are good mistakes which would increase the synthesis power. We investigate whether the use of minimal counterexamples instead of arbitrary counterexamples expands the set of candidate spaces of programs for which inductive synthesis can successfully synthesize a correct program. We consider two kinds of counterexamples: minimal counterexamples and history bounded counterexamples. The history bounded counterexample used in any iteration of CEGIS is bounded by the examples used in previous iterations of inductive synthesis. We examine the relative change in power of inductive synthesis in both cases. We show that the synthesis technique using minimal counterexamples MinCEGIS has the same synthesis power as CEGIS but the synthesis technique using history bounded counterexamples HCEGIS has different power than that of CEGIS, but none dominates the other.

A STRUCTURE ON COEFFICIENTS OF NILPOTENT POLYNOMIALS

We observe a structure on the products of coecients of nilpotent polynomials, introducing the concept of n-semi-Armendariz that is a generalization of Armendariz rings. We first obtain a classi- fication of reduced rings, proving that a ring R is reduced if and only if the n by n upper triangular matrix ring over R is n-semi-Armendariz. It is shown that n-semi-Armendariz rings need not be (n+1)-semi-Armendariz and vice versa. We prove that a ring R is n-semi-Armendariz if and only if so is the polynomial ring over R. We next study interesting proper- ties and useful examples of n-semi-Armendariz rings, constructing various kinds of counterexamples in the process.

Promises beyond assurance

Breaking a promise is generally taken to involve committing a certain kind of moral wrong, but what (if anything) explains this wrong? According to one influential theory that has been championed most recently by Scanlon, the wrong involved in breaking a promise is a matter of violating an obligation that one incurs to a promisee in virtue of giving her assurance that one will perform or refrain from performing certain acts. In this paper, we argue that the “Assurance View”, as we call it, is susceptible to two kinds of counterexamples. The first show that giving assurance is not sufficient for incurring the kind of obligation of fulfillment that one violates in breaking a promise. The second show that giving assurance is not necessary. Having shown that the Assurance View fails in these ways, we then very briefly sketch the outline of what we take to be a better view—a view that we claim is not only attractive in its own right and that avoids the earlier counterexamples, but that also affords us a deeper explanation of why the Assurance View seems initially plausible, yet nonetheless turns out to be ultimately inadequate.

Some counterexamples to a generalized Saari’s conjecture

For the Newtonian n-body problem, Saari's conjecture states that the only solutions with a constant moment of inertia are relative equilibria, solutions rigidly rotating about their center of mass. We consider the same conjecture applied to Hamiltonian systems with power-law potential functions. A family of counterexamples is given in the five-body problem (including the Newtonian case) where one of the masses is taken to be negative. The conjecture is also shown to be false in the case of the inverse square potential and two kinds of counterexamples are presented. One type includes solutions with collisions, derived analytically, while the other consists of periodic solutions shown to exist using standard variational methods.

Three Potential Problems for Powers' One-Fallacy Theory

Lawrence Powers advocates a one-fallacy theory in which the only real fallacies are fallacies of ambiguity. He defines a fallacy, in general, as a bad argument that appears good. He claims that the only legitimate way that an argument can appear valid, while being invalid, is when the invalid inference is covered by an ambiguity. Several different kinds of counterexamples have been offered from begging the question, to various forms of ad hominem fallacies. In this paper, I outline three potential counterexamples to Powers' theory, including one that has been addressed already by Powers, and two which are well known problems, but until now have never been applied as counterexamples to Powers' theory. I argue that there is a simpler explanation of these 'hard' cases than positing ambiguities that are not obviously there.

On the Structure of Counterexamples to Symmetric Orderings for BDD's

Binary Decision Diagrams (BDDs) are used to represent boolean functions in a variety of applications. The size of a reduced ordered BDD depends on the ordering of variables. Several researchers have suggested grouping symmetric variables as a promising heuristic for finding good orderings. In this paper we study the conjecture which states that symmetric variables gather in at least one of the optimum variable orders. First, we prove some useful properties of partially symmetric functions. Next, we develop a faster procedure for finding counterexamples to this conjecture that exploits the partitioning of boolean functions into nn-equivalence classes. Third, we study the structure of counterexamples and devise a new and simple method to generate new counterexamples from given counterexamples. Finally, we present different kinds of counterexamples, which show that boolean functions are very diverse with respect to where symmetric orders can fall in the range from optimal orders to worst-case orders.

Resolutions of subsets of finite sets of points in projective space

Given a finite setX, of points in projective space for which the Hilbert function is known, a standard result says that there exists a subset of this finite set whose Hilbert function is"as big as possible"inside X. Given a finite set of points in projective space for which the minimal free resolution of its homogeneous ideal is known, what can be said about possible resolutions of ideals of subsets of this finite set? We first give a maximal rank type description of the most generic possible resolution of a subset. Then we show, via two very different kinds of counterexamples, that this generic resolution is not always achieved. However, we show that it is achieved for sets of points in projective two space: given any finite set of points in projective two space for which the minimal free resolution is known, there must exist a subset having the predicted resolution.

Comment on Peter of Spain, Jim Mackenzie, and begging the question

Jim Mackenzie, in (1984a), offers an account of question-begging arguments that is an elaboration of a conjecture made by Peter of Spain: that begging the question is necessary and sufficient for an argument which is inferentially valid to fail to be probative. Mackenzie's analysis is set in the context of systems of formal dialogue. While sympathetic to this approach to the fallacy of begging the question, I will show that Mackenzie's account is susceptibile to counterexamples. In the first part of this note I will briefly describe Mackenzie's account of question-begging; in the second part I will present several different kinds of counterexamples which show that the condition he proposes as necessary and sufficient for the fallacy is not necessary.

Reasoning with cases and hypotheticals in HYPO

HYPO is a case-based reasoning system that evaluates problems by comparing and contrasting them with cases from its Case Knowledge Base (CKB). It generates legal arguments citing the past cases as justifications for legal conclusions about who should win in problem disputes involving trade secret law. HYPO's arguments present competing adversarial views of the problem and it poses hypotheticals to alter the balance of the evaluation. HYPO uses Dimensions as a generalization scheme for accessing and evaluating cases. HYPO's reasoning process and various computational definitions are described and illustrated, including its definitions for computing relevant similarities and differences, the most on point and best cases to cite, four kinds of counter-examples, targets for hypotheticals and the aspects of a case that are salient in various argument roles. These definitions enable HYPO to make contextually sensitive assessments of relevance and salience without relying on either a strong domain theory or a priori weighting schemes.

Nelson Goodman's Entrenchment Theory

One of the fundamental problems in the fields of inductive logic and the philosophy of science is the one concerning inferences or projections containing so-called “grue-like” or “pathological” predicates. This problem was first put into sharp focus by Nelson Goodman, who called it the “new riddle of induction.”Goodman has shown that the few attempts by others to solve this problem (which appear in the literature) are not adequate. However, very little has been written concerning Goodman's own attempt to solve the problem, namely his theory of entrenchment. The purpose of this article is to show that Goodman's entrenchment theory also is inadequate as a solution to the new riddle of induction. I shall try to do this by presenting two kinds of counterexamples to the entrenchment theory: one kind illustrating a general objection to the theory as a whole ; the other kind, specific objections to particular (vital) parts of the theory.