Counting Equivalence Classes For Co-κ-Souslin Equivalence Relations

The experimenter asked if documented equivalence class membership would influence the development of shared discriminative stimulus function established through simple successive discrimination training. In Experiment 1, equivalence classes were established with two sets of 9 stimuli. Common stimulus functions were then trained within or across the equivalence classes. Greater acquisition rates of the simple discriminations with stimuli drawn from within the equivalence classes were observed. In Experiment 2, a third stimulus set was added with which no equivalence relations were explicitly trained. The findings of Experiment 1 were replicated, but the Set 3 results were inconsistent across subjects. The outcomes of the two experiments demonstrate that equivalence classes have an effect on other behavioral relations which requires further investigation.

Two experiments investigated possible interactions between existing equivalence relations among stimuli and the acquisition of simple discriminations involving the same stimuli. During the first phase of Experiment 1, two adult humans learned two sets of six conditional relations among arbitrary visual stimuli designed to produce two independent sets of three, three-member equivalence classes (Set 1 and Set 2). Subjects whose performance met accuracy criteria on tests for equivalence relations proceeded to a second phase in which subjects were taught, via reinforcement, to make the same response to a cluster of three stimuli (e.g., response 1 in the presence of A1, B1, and C1). The clusters of three stimuli were initially consistent with documented equivalence relations for both Set 1 and Set 2 stimuli. In the second half of this phase, the contingencies were realigned such that the simple discriminations being established were consistent with documented equivalence relations for Set 1 stimuli but inconsistent for Set 2 stimuli. Results showed that the simple discriminations were acquired faster for Set 1 stimuli than for Set 2 stimuli. Experiment 2 introduced a third set of stimuli as a control set and systematically replicated Experiment 1 with four subjects. This study found that three of four subjects acquired simple discriminations faster when the cluster of stimuli were drawn from within equivalence classes rather than across equivalence classes. The development of equivalence relations with Set 3 stimuli constrained the utility of the control condition in this experiment. The results are best interpreted as suggesting that existing equivalence relations among stimuli can interact with the development of analytic units such as simple discriminations to either facilitate or retard their development.

Recent research has demonstrated the transfer of induced mood functions through equivalence relations by means of a musical mood-induction procedure. The research described in this article replicated and extended such work, primarily with the inclusion of a baseline and two types of reversal procedures. First, 16 adult participants were trained and tested for the formation of two three-member equivalence classes (A1-B1-C1 and A2-B2-C2). A musical mood-induction procedure was used to induce happy and sad mood states in the presence of the B stimuli, and a transfer of mood functions test was administered to determine whether the corresponding mood functions had transferred via equivalence to the C stimuli. Participants were then exposed to either Reversal 1, in which only the functions attached to the B stimuli were reversed, or to Reversal 2, in which the structure of the equivalence classes was altered. At all stages throughout the experiment, participants were asked to rate their current mood state by using mood and incentive rating scales. The findings of the current study replicated the results of previous research in demonstrating a clear transfer of mood functions via equivalence relations. This outcome is compared with the existing literature about mood induction and the transfer of mood functions.

The purpose of this study was to examine the transfer of consequential (reinforcement and punishment) functions through equivalence relations. In Experiment 1, 9 subjects acquired three three-member equivalence classes through matching-to-sample training using arbitrary visual forms. Comparison stimuli were then given conditioned reinforcement or punishment functions by pairing them with verbal feedback during a sorting task. For 8 of the 9 subjects, trained consequential functions transferred through their respective equivalence classes without additional training. In Experiment 2, transfer of function was initially tested before equivalence testing per se. Three of 4 subjects showed the transfer without a formal equivalence test. In Experiment 3, 3 subjects were given training that gave rise to six new three-member conditional equivalence classes. For 2 of the subjects, the same stimulus could have either a reinforcement or punishment function on the basis of contextual cues that defined its class membership. Experiment 4 assessed whether equivalence training had established general or specific consequential functions primarily by adding novel stimuli in the transfer test. Subjects treated even novel feedback stimuli in the transfer test as consequences, but the direction of consequential effects depended upon the transfer of specific consequential functions through equivalence relations.

Given two (positive) equivalence relations ~1, ~2 on the set ω of natural numbers, we say that ~1 is m-reducible to ~2 if there exists a total recursive function h such that for every x, y ∈ ω, we have x ~1y iff hx ~2hy. We prove that the equivalence relation induced in ω by a positive precomplete numeration is complete with respect to this reducibility (and, moreover, a “uniformity property” holds). This result allows us to state a classification theorem for positive equivalence relations (Theorem 2). We show that there exist nonisomorphic positive equivalence relations which are complete with respect to the above reducibility; in particular, we discuss the provable equivalence of a strong enough theory: this relation is complete with respect to reducibility but it does not correspond to a precomplete numeration.From this fact we deduce that an equivalence relation on ω can be strongly represented by a formula (see Definition 8) iff it is positive. At last, we interpret the situation from a topological point of view. Among other things, we generalize a result of Visser by showing that the topological space corresponding to a partition in e.i. sets is irreducible and we prove that the set of equivalence classes of true sentences is dense in the Lindenbaum algebra of the theory.

The paper presents a novel, rough set-based clustering method that enables the evaluation of classification knowledge simplicity during the clustering procedure. The method iteratively refines equivalence relations so that they become a more simple set of relations that give adequate coarse classification to the objects. At each step of the iteration, the importance of the equivalence relation is evaluated on the basis of the newly introduced measure, indiscernibility degree. An indiscernibility degree is defined as a ratio of equivalence relations that classify the two objects into the same equivalence class. If an equivalence relation has the ability to discern two objects that have a high indiscernibility degree, a very fine classification is performed and then modified to regard them as indiscernible objects. The refinement is repeated, decreasing the threshold level of indiscernibility degree, and finally simple clusters can be obtained. Experimental results on the artificial data shows that iterative refinement of equivalence relation leads to successful generation of coarse clusters that can be represented by simple knowledge.

For p prime and for a field F containing a root of unity of order p, we generalize Marshall's equivalence relation on orderings to arbitrary subgroups of F× of index p. The equivalence classes then correspond to free pro-p factors of the maximal pro-p Galois group of F. We generalize to this setting results of Jacob on the maximal pro-2 Galois group of a Pythagorean field.

Two experiments used post-class formation within-class relational assessment test performances to evaluate whether participants demonstrated preference for certain members of an equivalence class based on the type of relation that existed between class members. In Experiment 1, two 5-node 7-member equivalence classes, consisting entirely of nonsense syllables, were established using the simultaneous protocol. Only 1 of the 6 participants in Experiment 1 formed classes. After class formation, the effects of the different relations between stimuli were evaluated using within-class relational assessment tests, and the 1 participant showed an absolute preference for transitive over equivalence relations, and for baseline over symmetrical relations. Experiment 2 was identical to Experiment 1, except that one of the nonsense syllable stimuli in each class was replaced by a pictorial stimulus. Under these conditions, classes were formed by 5 of 13 participants. During the relational assessment tests, the 5 participants who formed classes demonstrated almost exclusive preferences for transitive relations over equivalence relations and for trained baseline relations over symmetrical relations. Thus, this research demonstrates that the members of equivalence classes are differentially related to each other based on relational type.

In a family of nonlinear optimal control problems, equivalence classes of problems that yield identical optimal closed-loop systems are delineated. The equivalence relations are established as one-to-one correspondences between the solutions of the Hamilton-Jacobi equations associated with the problems. Two such equivalence relations are presented. These allow an arbitrary problem in the family to be reduced to an equivalent basic standard form of problem. Also, a new formal procedure for solving this basic problem is given that allows the determination of coefficients in the series expansion of the optimal closed-loop system. The set of algebraic equations to be solved is considerably simpler than that derived by other methods.

Let be a non-empty set and an equivalence relation on . Then, is called an approximation space. The equivalence relation on forms disjoint equivalence classes. If , then we can form a lower approximation and an upper approximation of . If X⊆U, then we can form a lower approximation and an upper approximation of X. In this research, rough group and rough subgroups are constructed in the approximation space for commutative and non-commutative binary operations.

Equivalence relations induced by the U-partial order

Similarity is an important way of categorizing mathematical objects. In this paper we consider how similarity between undirected graphs, or networks, can be defined. We do this by reducing any undirected graph, G = (V,E) to its irreducible spine, I. First we show that the reduction process, ω, is a unique transformation and thus is a well-defined function. Each irreducible spine, I, thus defines an equivalence class consisting of all graphs {G_i} such that G_i.ω = I. We then show that reduction by ω preserves many key properties. Specifically it preserves shortest path lengths in I, and (under rather general conditions) preserves the location of distance and betweeness centers. In essence, every member graph of the equivalence class will have much the same path structure. Next, an algorithm is presented which, given any irreducible spine I, will generate (randomly) a graph Gk in the equivalence class defined by I, that is, we must have G_k.ω = I. Since I.e.ω = I, e = ω^-1. It is of some importance in network analysis to be able to quantify the notion of network similarity; to be able to say that a network G1 is closer to the network G2 than to G3. A popular method of comparing similarity between two networks G1 and G2 is to calculate the Spearman coefficient based on corresponding node degrees. We do this for 5 random graphs in the equivalence class of our running example. Finally, we show that any irreducible spine consists of a system of cycles. We then note that every spine, I, is topologically equivalent to a cubic graph. Thus individual cubic graphs generate a second, and coarser, equivalence relation on the family of all undirected graphs, which we call its genus.

In a recent paper by H. KARZEL and R. STANIK, Abh. Math. Sem. Univ. Hamburg, a characterization of euclidean metric planes (the metric is defined there by a separable quadratic field extension L ⊃ K of the base field K) by means of a congruence relation on segments is given. Here we develop a theory of congruence of angles for these planes to get an axiomatic foundation, which also works when L ⊃ K is inseparable. The main result is of the following type: Let be an affine plane together with an equivalence relation $$\sphericalangle $$ onG x G, whereG denotes the set of lines — any equivalence class is then what we call an angle — such that some simple properties (W1) to (W3) of angles and one further condition (V) on equality of certain angles in quadrangles hold, then is pappian and $$\sphericalangle $$ results from a quadratic field extension L ⊃ K.

Hepaxpert-I is a medical expert system for interpretive analysis of hepatitis A and B serology findings. The knowledge base of HEPAXPERT-I contains 13 rules for hepatitis A and 106 rules for hepatitis B serology. After entering an input pattern of test results, a pattern matching algorithm based on indexing is employed as efficient look-up procedure for the interpretive text. Formally, a partition of the sets of possible serological finding patterns—64 for hepatitis A and 4,096 for hepatitis B serology—induced by an equivalence relation, divides the elements of the sets into disjoint subsets, the equivalence classes. Each equivalence class together with its associated interpretive text is represented as one rule that assigns to every member of the equivalence class one interpretive text. Since September 1, 1989, Hepaxpert-I has been routinely used at the hepatitis serology laboratory of the 2nd Department for Gastroenterology and Hepatology of the University of Vienna Medical School (Vienna General Hospital). By February 28, 1990, 3,408 interpretive reports had been automatically generated and sent out. Beforehand, the system had been validated with 24,242 serological test results. The acceptance of the generated reports by the physicians requiring the findings is extremely high.

An examination of the translation invariance of V0 under dyadic rationals is presented, generating a new equivalence relation on the collection of wavelets. The equivalence classes under this relation are completely characterized in terms of the support of the Fourier transform of the wavelet. Using operator interpolation, it is shown that several equivalence classes are non-empty.