On Categorical Equivalences of Commutative BCK-algebras

A commutative BCK-algebra with the relative cancellation property is a commutative BCK-algebra (X;*,0) which satisfies the condition: if a ≤ x, a ≤ y and x * a = y * a, then x = y. Such BCK-algebras form a variety, and the category of these BCK-algebras is categorically equivalent to the category of Abelian ℓ-groups whose objects are pairs (G, G ₀), where G is an Abelian ℓ-group, G ₀ is a subset of the positive cone generating G ⁺ such that if u, v ∈ G ₀, then 0 ∨ (u - v) ∈ G ₀, and morphisms are ℓ-group homomorphisms h: (G, G ₀) → (G′,G′₀) with f(G ₀) ⫅ G′₀. Our methods in particular cases give known categorical equivalences of Cornish for conical BCK-algebras and of Mundici for bounded commutative BCK-algebras (= MV-algebras). This revised version was published online in June 2006 with corrections to the Cover Date.     

Generalization of deferred imitation during the first year of life

Infants first generalize across contexts and cues at 3 months of age in operant tasks but not until 12 months of age in imitation tasks. Three experiments using an imitation task examined whether infants younger than 12 months of age might generalize imitation if conditions were more like those in operant studies. Infants sat on a distinctive mat in a room in their home (the context) while an adult modeled actions on a hand puppet (the cue). When they were tested 24 h later, 6-month-olds generalized imitation when either the mat or the room (but not both) differed, whereas 9-month-olds generalized when both the mat and the room differed. In addition, 9-month-olds who imitated immediately also generalized to a novel test cue, whereas 6-month-olds did not. These results parallel results from operant studies and reveal that the similarity between the conditions of encoding and retrieval-not the type of task-determines whether infants generalize. The findings offer further evidence that memory development during infancy is a continuous function.     

Equivalent MIRID models

It is shown that in the context of the Model with Internal Restrictions on the Item Difficulties (MIRID), different componential theories about an item set may lead to equivalent models. Furthermore, we provide conditions for the identifiability of the MIRID model parameters, and it will be shown how the MIRID model relates to the Linear Logistic Test Model (LLTM). While it is known that the LLTM is a special case of the MIRID, we show that it is possible to construct an LLTM that encompasses the MIRID. The MIRID model places a bilinear restriction on the item parameters of the Rasch model. It is explained how this fact is used to simplify the results of Bechger, Verhelst, and Verstralen (2001) and Bechger, Verstralen, and Verhelst (2002), and extend their scope to a wider class of models.     

On aims and methods in the neuroimaging of derived relations

Ingenious and seemingly powerful technologies have been developed recently that enable the visualization in some detail of events in the brain concomitant upon the ongoing behavioral performance of a human participant. Measurement of such brain events offers at the very least a new set of dependent variables in relation to which the independent variables familiarly manipulated in the operant laboratory may be explored. Two related paradigms in which a start has been made in such research concern the derivation of novel or emergent relations from a baseline set of trained relations, and include the phenomenon of transitive inference (TI), observed in studies of stimulus equivalence (SE) and serial learning (SL) or seriation. This paper reviews some published and forthcoming neuroimaging studies of these and related phenomena, and considers how this line of research both demands and represents a welcome synthesis between types of question and levels of explanation in behavioral science that often have been seen as antithetical.     

A functional analytic approach to computer-interactive mathematics

Following a pretest, 11 participants who were naive with regard to various algebraic and trigonometric transformations received an introductory lecture regarding the fundamentals of the rectangular coordinate system. Following the lecture, they took part in a computer-interactive matching-to-sample procedure in which they received training on particular formula-to-formula and formula-to-graph relations as these formulas pertain to reflections and vertical and horizontal shifts. In training A-B, standard formulas served as samples and factored formulas served as comparisons. In training B-C, factored formulas served as samples and graphs served as comparisons. Subsequently, the program assessed for mutually entailed B-A and C-B relations as well as combinatorially entailed C-A and A-C relations. After all participants demonstrated mutual entailment and combinatorial entailment, we employed a test of novel relations to assess 40 different and complex variations of the original training formulas and their respective graphs. Six of 10 participants who completed training demonstrated perfect or near-perfect performance in identifying novel formula-to-graph relations. Three of the 4 participants who made more than three incorrect responses during the assessment of novel relations showed some commonality among their error patterns. Derived transfer of stimulus control using mathematical relations is discussed.     

Recombinative generalization of within-syllable units in nonreading adults with mental retardation

Two adults with mental retardation demonstrated the recombination of within-syllable units (onsets and rimes) using a spoken-to-printed-word matching-to-sample (MTS) procedure. Further testing with 1 participant showed comprehension of the printed words. Printed-word naming was minimal before, but greater after, comprehension tests. The findings suggest that these procedures hold promise for further basic and applied analyses of word-attack skills.     

Classification of vowels and consonants by individuals with moderate mental retardation: development of arbitrary relations via match-to-sample training with compound stimuli.

This study explored whether an identity-matching-based stimulus equivalence procedure could be used to teach vowel and consonant stimulus classes to 2 adolescent females with moderate mental retardation. Delayed match-to-sample trials presented a compound sample stimulus consisting of printed letters and a spoken word (“vowel” or “consonant”). The correct comparison stimulus matched only one of the letters in the compound sample. Subsequently, test trials assessed whether arbitrary relations had formed among the individual stimuli from each compound sample and whether stimuli from different compound samples had merged into larger stimulus classes. Both participants acquired five-member classes of vowel and consonant stimuli, which subsequently generalized to vocal classification and to identification in the context of four-letter words. Follow-up tests showed that the generalized performances remained intact after 6 weeks. These procedures suggest an economical approach to stimulus class development.     

Generalization of cross-modal stimulus equivalence classes: operant processes as components in human category formation.

This study employed a stimulus-class rating procedure to explore whether stimulus equivalence and stimulus generalization can combine to promote the formation of open-ended categories incorporating cross-modal stimuli. A pretest of simple auditory discrimination indicated that subjects (college students) could discriminate among a range of tones used in the main study. Before beginning the main study, 10 subjects learned to use a rating procedure for categorizing sets of stimuli as class consistent or class inconsistent. After completing conditional discrimination training with new stimuli (shapes and tones), the subjects demonstrated the formation of cross-modal equivalence classes. Subsequently, the class-inclusion rating procedure was reinstituted, this time with cross-modal sets of stimuli drawn from the equivalence classes. On some occasions, the tones of the equivalence classes were replaced by novel tones. The probability that these novel sets would be rated as class consistent was generally a function of the auditory distance between the novel tone and the tone that was explicitly included in the equivalence class. These data extend prior work on generalization of equivalence classes, and support the role of operant processes in human category formation.     

Basic research needed for stimulating the development of behavioral technologies

The costs of disconnection between the basic and applied sectors of behavior analysis are reviewed, and some solutions to these problems are proposed. Central to these solutions are collaborations between basic and applied behavioral scientists in programmatic research that addresses the behavioral basis and solution of human behavior problems. This kind of collaboration parallels the deliberate interactions between basic and applied researchers that have proven to be so profitable in other scientific fields, such as medicine. Basic research questions of particular relevance to the development of behavioral technologies are posed in the following areas: response allocation, resistance to change, countercontrol, formation and differentiation/discrimination of stimulus and response classes, analysis of low-rate behavior, and rule-governed behavior. Three interrelated strategies to build connections between the basic and applied analysis of behavior are identified: (a) the development of nonhuman animal models of human behavior problems using operations that parallel plausible human circumstances, (b) replication of the modeled relations with human subjects in the operant laboratory, and (c) tests of the generality of the model with actual human problems in natural settings.     

AUDITORY SUCCESSIVE CONDITIONAL DISCRIMINATION AND AUDITORY STIMULUS EQUIVALENCE CLASSES

This paper describes an experimental demonstration of stimulus equivalence classes consisting entirely of auditory stimuli. Stimuli were digitized arbitrary syllables (e.g., “cug,” “vek”) presented via microcomputer. Training and testing were conducted with a two-choice auditory successive conditional discrimination procedure. On each trial, auditory samples and comparisons were presented successively. As each comparison was presented, a response location (a rectangle) appeared on the computer screen. After all stimuli for a trial were presented, subjects selected one of the response locations. Six subjects acquired the conditional discrimination baseline, 4 subjects demonstrated the formation of three-member auditory equivalence classes resulting from sample-S+ relations, and 1 subject demonstrated equivalence classes resulting from sample-S— relations. Four subjects received additional training and subsequently demonstrated expansion of the three-member classes to four members each.