Interactions Between Equivalence Relations and the Development of Analytic Units

The present study investigated whether simple discrimination procedures produce emergent relations that are consistent with functional and equivalence classes. In Experiment 1, 4 normally capable adults were exposed to simple successive discrimination reversal training. Responses to the S+ (A1, B1, and C1) but not S− (A2, B2, and C2) were reinforced. The participants were then exposed to repeated-reversal training, followed by tests of emergent equivalence relations. In Experiment 2, 4 additional adults were exposed to a simple successive discrimination training procedure using differential responses. Response 1 (R1) was reinforced only when emitted in the presence of A1, B1, or C1 (A1-R1, B1-R1, or C1-R1). Response 2 (R2) was reinforced only in the presence of A2, B2, or C2 (A2-R2, B2-R2, or C2-R2). A new response was then trained in the presence of 1 member of each class (A1-R3, A2-R4). A transfer-of-function test was then conducted to verify whether the new responses would be emitted in the presence of B1, B2, C1, and C2. In both experiments, emergent conditional relations were tested using a go/no-go procedure with compound stimuli. In Experiment 1, all of the participants showed functional classes, and 3 showed patterns that are consistent with the formation of equivalence classes. In Experiment 2, all 4 participants showed patterns that are consistent with functional and equivalence class formation. In these experiments, simple discrimination training generated a range of emergent stimulus relations that mirrored those that are indicative of equivalence class formation.

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.

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.

9 New Procedures for establishing emergent matching performances in children and adults: Implications for stimulus equivalence

The current study aimed to develop a behavior-analytic model of analogical reasoning. In Experiments 1 and 2 subjects (adults and children) were trained and tested for the formation of four, three-member equivalence relations using a delayed matching-to-sample procedure. All subjects (Experiments 1 and 2) were exposed to tests that examined relations between equivalence and non-equivalence relations. For example, on an equivalence-equivalence relation test, the complex sample B1/C1 and the two complex comparisons B3/C3 and B3/C4 were used, and on a nonequivalence-nonequivalence relation test the complex sample B1/C2 was presented with the same two comparisons. All subjects consistently related equivalence relations to equivalence relations and nonequivalence relations to nonequivalence relations (e.g., picked B3/C3 in the presence of B1/C1 and picked B3/C4 in the presence of B1/C2). In Experiment 3, the equivalence responding, the equivalence-equivalence responding, and the nonequivalence-nonequivalence responding was successfully brought under contextual control. Finally, it was shown that the contextual cues could function successfully as comparisons, and the complex samples and comparisons could function successfully as contextual cues and samples, respectively. These data extend the equivalence paradigm and contribute to a behaviour-analytic interpretation of analogical reasoning and complex human functioning, in general.

The occurrence of "blocking" was investigated in jungle fowl chicks (Gallus gallus spadiceus B.) in an imprinting situation. In Experiment 1, chicks were simultaneously exposed to two stationary coloured cylinders, either two red cylinders (Group RR), a yellow and a red cylinder (YR), or two yellow cylinders (YY). After six days of exposure, the cylinders were removed from the cages and replaced by a yellow and a blue cylinder (i.e. YB) for each chick. This second phase of the experiment lasted for seven days. When the blue cylinder was presented alone during tests at different stages in Phase 2, the RR birds spent significantly more time with this cylinder and emitted fewer shrill calls than the chicks in the YR and YY groups. In Experiment 2, RR and YY birds were reared as in Experiment 1, except that in the second phase of the experiment they were exposed to a blue cylinder only. In this experiment the development of an attachment to the novel blue cylinder proceeded similarly in the RR and YY birds. In Experiment 3, it was found that chicks that were reared with a yellow and a red cylinder preferred the latter stimulus. Thus, although in the first phase of Experiment 1 the RR birds had been exposed to a more attractive stimulus, in tests during the second phase they spent more time with a novel stimulus (B) than the YY birds. These results are consistent with the suggestion that imprinting to a novel stimulus is "blocked" to some extent when that stimulus is presented in compound with another stimulus to which the animal has previously been exposed.

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.

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.

Children and adolescents from four age groups (4, 8, 11, and 14 years of age) learned the baseline conditional relations for the emergence of two three-member equivalence classes. Then, they were exposed to exclusion trials that included baseline stimuli as defined stimuli to be rejected, while another, undefined stimulus had to be selected in the presence of an undefined sample. Emergent performance in symmetry and transitivity relations with stimuli related by exclusion was tested to probe for evidence of the emergence of a new stimulus class (class 3). Then, participants were exposed to new exclusion trials which employed class 3 stimuli as defined stimulus, and symmetry and transitivity conditional relations were again tested with the new stimuli related by exclusion to identify the emergence of a new class (class 4). Participants of the four age groups showed emergence and expansion of equivalence classes by exclusion, and use of the exclusion class stimuli as a basis for the emergence of a new class by exclusion. The likelihood of equivalence emergence with stimuli related by exclusion decreased in 5-year-old children when symmetry and transitivity were tested for exclusion classes, but did not for baseline classes. Some sources of control for accurate responding to the tests are discussed.

The effects of response topography on stimulus class formation were studied in two experiments. In Experiment 1, 32 college students were assigned to 2 response topographies and 2 stimulus sets, in a 2 × 2 design. Students selected stimuli by either moving a mouse to place an arrow-shaped cursor on the stimulus or pressing a key corresponding to stimulus location. After they learned conditional discriminations EF, DE, CD, BC, and AB and a simple simultaneous discrimination, tests for class formation were conducted. The number of students showing class formation was larger for the group using the mouse. Stimulus set had no effect. Experiment 2 included probes for controlling relations in the baseline and showed that the response topography using the mouse increases the probability of sample-S+ relations, thus increasing the probability of class formation.

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.