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.     

Nonsimultaneous stimulus presentations and their role in listener naming

Previous studies on naming have presented the object and its name simultaneously during both training and testing, and thus the training component may establish a transformation of function directly between the object and the name. Successful tests for listener naming may thus not require the emergence of a novel (entailed) transformation of function. The current study aimed to control for this possibility by presenting the object and the name sequentially and nonsimultaneously. Eight typically developing toddlers participated in the current study. During name training, objects and names were presented nonsimultaneously, and all participants failed to emit listener-naming responses during the first test session. Subsequently, 4 participants received multiple exemplar training, which led to improvements in listener naming for all 4; and speaker naming for only 1 participant. As a control condition, the remaining 4 participants were tested repeatedly, without multiple exemplar training, and did not show any consistent improvements in their listener or speaker performances. Multiple exemplar training thus appeared to be effective in establishing generalized listener responses, which involved generating entailed transformation of functions. The strategy of using nonsimultaneous stimulus presentations could allow for greater precision in identifying the behavioral processes involved in listener-naming.     

Transformations of mathematical and stimulus functions

Following a pretest, 8 participants who were unfamiliar with algebraic and trigonometric functions received a brief presentation on the rectangular coordinate system. Next, they participated in a computer-interactive matching-to-sample procedure that trained formula-to-formula and formula-to-graph relations. Then, they were exposed to 40 novel formula-to-graph tests and 10 novel graph-to-formula tests. Seven of the 8 participants showed substantial improvement in identifying formula-to-graph relations; however, in the test of novel graph-to-formula relations, participants tended to select equations in their factored form. Next, we manipulated contextual cues in the form of rules regarding mathematical preferences. First, we informed participants that standard forms of equations were preferred over factored forms. In a subsequent test of 10 additional novel graph-to-formula relations, participants shifted their selections to favor equations in their standard form. This preference reversed during 10 more tests when financial reward was made contingent on correct identification of formulas in factored form. Formula preferences and transformation of novel mathematical and stimulus functions are discussed.     

Derived relational responding as generalized operant behavior

The major aim of the present study was to demonstrate that derived relational responding may be viewed as a form of generalized operant behavior. In Experiment 1, 4 subjects were divided into two conditions (2 in each condition). Using a two-comparison matching-to-sample procedure, all subjects were trained and tested for the formation of two combinatorially entailed relations. Subjects were trained and tested across multiple stimulus sets. Each set was composed of novel stimuli. Both Conditions 1 and 2 involved explicit performance-contingent feedback presented at the end of each block of test trials (i.e., delayed feedback). In Condition 1, feedback was accurate (consistent with the experimenter-designated relations) following exposure to the initial stimulus sets. When subjects' responding reached a predefined mastery criterion, the feedback then switched to inaccurate (not consistent with the experimenter-designated relations) until responding once again reached a predefined criterion. Condition 2 was similar to Condition 1, except that exposure to the initial stimulus sets was followed by inaccurate feedback and once the criterion was reached feedback switched to accurate. Once relational responding emerged and stabilized, response patterns on novel stimulus sets were controlled by the feedback delivered for previous stimulus sets. Experiment 2 replicated Experiment 1, except that during Conditions 3 and 4 four comparison stimuli were employed during training and testing. Experiment 3 was similar to Condition 1 of Experiment 1, except that after the mastery criterion was reached for class-consistent responding, feedback alternated from accurate to inaccurate across each successive stimulus set. Experiment 4 involved two types of feedback, one type following tests for mutual entailment and the other type following tests for combinatorial entailment. Results from this experiment demonstrated that mutual and combinatorial entailment may be controlled independently by accurate and inaccurate feedback. Overall, the data support the suggestion, made by relational frame theory, that derived relational responding is a form of generalized operant behavior.     

Arbitrarily applicable spatial relational responding

Spatial reasoning, where novel spatial relationships are inferred based on trained relationships, can be conceptualized as arbitrarily applicable spatial relational responding. Here, we conducted two experiments to develop and validate, for the first time, a laboratory procedure to establish arbitrarily applicable spatial relational responding in adult humans. In Experiment 1, participants were trained on nonarbitrary spatial relational tasks designed to establish contextual cues for left of, right of, above, and below. Contextual cues were then used to train a series of arbitrary spatial relations involving four abstract shapes. Following training in a subset of arbitrary relations (A is left of B, B is above C, C is right of D), subsequent testing examined the emergence of untrained spatial relations (B is right of A, C is below B, D is left of C, D is below A and A is above D). When absent in initial tests, spatial relational responding was facilitated by a remedial training procedure incorporating nonarbitrary relational guidance. Participants showed patterns of spatial relational responding consistent with test relations. In Experiment 2, a variant reversal design yielded predictable, reversed spatial relational responses. Overall, the present procedures represent the first empirical demonstration of arbitrarily applicable spatial relational responding and thus, arguably, the first functional analytic model of spatial reasoning.     

Constructing Relational Sentences: Establishing Arbitrarily AppliCable Comparative Relations with the Relational Completion Procedure

The present study examined the effectiveness of the Relational Completion Procedure (RCP) in establishing arbitrarily applicable comparative relations after training with either linear (Experiment 1) or non-linear (Experiment 2) arbitrary relations. Linear relations were those presented in a sequential order (i.e., A-B, B-C, C-D, etc.), while nonlinear relations were presented in a nonsequential order (i.e., C-D, B-C, A-B, etc.). After nonarbitrary relational training and testing to establish the contextual functions of MORE THAN and LESS THAN, adults underwent constructed-response nonarbitrary relational training and testing with physically dissimilar stimuli in which they were required to “complete” relational sentences in the presence of the cues. Next, during arbitrary relational training with abstract stimuli, participants were randomly assigned to 1 of 3 training groups: All-More (E > D, D > C., C > B, and B > A), All-Less (A < B, B < C., C < D, and D < E), and Less-More (A < B, B < C., D > C., and E > D). Finally, during arbitrary relational testing, probes for mutually entailed relations were followed by probes for 1-and 2-node combinatorially entailed relations. The yield, or percentage of participants passing, was 83% in Experiment 1 and 91.6% in Experiment 2. Our findings demonstrate that the RCP was successful in establishing arbitrarily applicable comparative relations with adults regardless of the order in which the arbitrary relational training trials were presented.

     

The transfer of Crel contextual control (same,opposite, less than,more than) through equivalence relations

According to Relational Frame Theory (RFT) C_rel denotes a contextual stimulus that controls a particular type of relational response (sameness, opposition, comparative, temporal, hierarchical etc.) in a given situation. Previous studies suggest that contextual functions may be indirectly acquired via transfer of function. The present study investigated the transfer of C_rel contextual control through equivalence relations. Experiment 1 evaluated the transfer of C_rel contextual functions for relational responses based on sameness and opposition. Experiment 2 extended these findings by evaluating transfer of function using comparative C_rel stimuli. Both experiments followed a similar sequence of phases. First, abstract forms were established as C_rel stimuli via multiple exemplar training (Phase 1). The contextual cues were then applied to establish arbitrary relations among nonsense words and to test derived relations (Phase 2). After that, equivalence relations involving the original C_rel stimuli and other abstract forms were trained and tested (Phase 3). Transfer of function was evaluated by replacing the directly established C_rel stimuli with their equivalent stimuli in the former experimental tasks (Phases 1 and 2). Results from both experiments suggest that C_rel contextual control may be extended via equivalence relations, allowing other arbitrarily related stimuli to indirectly acquire C_rel functions and regulate behavior by evoking appropriate relational responses in the presence of both previously known and novel stimuli.     

DERIVED MORE‐LESS RELATIONAL MANDS IN CHILDREN DIAGNOSED WITH AUTISM

In Experiment 1, more and less relations were trained for arbitrary Stimuli A1 and A2 with 3 children with autism. The following conditional discriminations were then trained: A1‐B1, A2‐B2, B1‐C1, B2‐C2. In subsequent tests, participants showed derived more‐less mands (mand with C1 for more and mand with C2 for less). A training procedure reversed the B‐C conditional discriminations, and participants then showed derived reversed more—less mands (mand with C1 for less, C2 for more). Baseline B‐C relations were retrained, and participants subsequently demonstrated a return to the original derived manding. A second experiment with 1 prior participant and 1 naive participant removed a possible confounding effect. Establishing derived manding may be an advantageous component when teaching a mand repertoire in applied settings.     

Producing and Recognizing Analogical Relations

Analogical reasoning is an important component of intelligent behavior, and a key test of any approach to human language and cognition. Only a limited amount of empirical work has been conducted from a behavior analytic point of view, most of that within Relational Frame Theory (RFT), which views analogy as a matter of deriving relations among relations. The present series of four studies expands previous work by exploring the applicability of this model of analogy to topography-based rather than merely selection-based responses and by extending the work into additional relations, including nonsymmetrical ones. In each of the four studies participants pretrained in contextual control over nonarbitrary stimulus relations of sameness and opposition, or of sameness, smaller than, and larger than, learned arbitrary stimulus relations in the presence of these relational cues and derived analogies involving directly trained relations and derived relations of mutual and combinatorial entailment, measured using a variety of productive and selection-based measures. In Experiment 1 participants successfully recognized analogies among stimulus networks containing same and opposite relations; in Experiment 2 analogy was successfully used to extend derived relations to pairs of novel stimuli; in Experiment 3 the procedure used in Experiment 1 was extended to nonsymmetrical comparative relations; in Experiment 4 the procedure used in Experiment 2 was extended to nonsymmetrical comparative relations. Although not every participant showed the effects predicted, overall the procedures occasioned relational responses consistent with an RFT account that have not yet been demonstrated in a behavior-analytic laboratory setting, including productive responding on the basis of analogies.     

Teaching foundational perspective‐taking skills to children with autism using the PEAK‐T curriculum: single‐reversal “I–You” deictic frames

We taught basic perspective‐taking tasks to 3 children with autism and evaluated their ability to derive mutually entailed single‐reversal deictic relations of those newly established perspective‐taking skills. Furthermore, we examined the possibility of transfers of perspective‐taking function to novel untrained stimuli. The methods were taken from the PEAK‐T training curriculum, and results yielded positive gains for all 3 children to learn basic perspective taking as well as for 2 of the 3 to derive untrained single‐reversal I relations following direct training of single‐reversal You relations. All participants demonstrated a transfer of stimulus function to untrained stimuli after the single‐reversal deictic relations had been mastered.     

Identity concept learning in matching-to-sample tasks by tufted capuchin monkeys (Cebus apella)

The abstract concept of equivalence is considered one of the bases of higher-order cognition, and it has been the subject of considerable research in comparative cognition. This study examined the conditions under which tufted capuchin monkeys (Cebus apella) are able to acquire an identity concept. Six capuchin monkeys were trained to solve a visual matching-to-sample (MTS) task on the basis of perceptual identity. The acquisition of the identity rule was inferred from the subject’s ability to solve transfer tests with novel stimuli. We evaluated the ability of the capuchin monkeys to match the shape of novel stimuli after training with both several small stimulus sets (Experiment 1) and a large stimulus set (Experiment 2). Moreover, we examined the ability of capuchins to transfer the concept to novel visual dimensions, such as colour and size and to transfer to novel spatial arrangements of the stimuli (Experiment 2). We demonstrated that the ability of capuchins to match novel stimuli was improved by increasing the number of stimuli used during training (Experiments 1 and 2) and that after a widely applicable identity concept based on the stimulus shape was acquired, the capuchins were able to match stimuli according to an identity rule based on both the colour and size of the stimuli and when the spatial arrangement of the stimuli was varied (Experiment 2). This study is the first to demonstrate that the size of the training set affects the acquisition of an abstract identity concept in an MTS task in non-human primates.     

Context and Perceptual Salience Influence the Formation of Novel Stereotypes via Cumulative Cultural Evolution

We use a transmission chain method to establish how context and category salience influence the formation of novel stereotypes through cumulative cultural evolution. We created novel alien targets by combining features from three category dimensions—color, movement, and shape—thereby creating social targets that were individually unique but that also shared category membership with other aliens (e.g., two aliens might be the same color and shape but move differently). At the start of the transmission chains each alien was randomly assigned attributes that described it (e.g., arrogant, caring, confident). Participants were given training on the alien‐attribute assignments and were then tested on their memory for these. The alien‐attribute assignments participants produced during test were used as the training materials for the next participant in the transmission chain. As information was repeatedly transmitted an increasingly simplified, learnable stereotype‐like structure emerged for targets who shared the same color, such that by the end of the chains targets who shared the same color were more likely to share the same attributes (a reanalysis of data from Martin et al., 2014 which we term Experiment 1). The apparent bias toward the formation of novel stereotypes around the color category dimension was also found for objects (Experiment 2). However, when the category dimension of color was made less salient, it no longer dominated the formation of novel stereotypes (Experiment 3). The current findings suggest that context and category salience influence category dimension salience, which in turn influences the cumulative cultural evolution of information.     

Children's identity matching and oddity: assessing control by specific and general sample-comparison relations.

After children in Experiments 1 and 2 learned identity matching or oddity, control by sample-comparison relations was assessed. Tests for generalized control displayed novel samples and two comparison stimuli, one identical to the sample. Specific relations were tested with identical or nonidentical sample-comparison stimuli from one set of stimuli and substitute comparisons from either the other training set or from a novel set. When tests displayed identical stimuli, patterns of comparison selection suggested control by generalized identity and oddity. However, selection patterns varied when stimuli were nonidentical and familiar or novel substitute comparisons were used. Therefore, control by specific relations is not a precondition for generalized identity and oddity. One set of training stimuli was used in Experiment 3, and generalized performances occurred again. Moreover, control by specific relations was shown by the oddity subjects and 2 of 6 identity subjects. Generalized and specific control may therefore exist simultaneously. In Experiment 4, selections were irregular on tests displaying substitute comparisons and samples and familiar comparison stimuli; this finding supported the relational account of specific sample-comparison control found in Experiment 3.     

The transfer of specific and general consequential functions through simple and conditional equivalence relations

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.     

Circle graphs and monadic second-order logic

This article is part of a project consisting in expressing, whenever possible, graph properties and graph transformations in monadic second-order logic or in its extensions using modulo p cardinality set predicates or auxiliary linear orders. A circle graph is the intersection graph of a set of chords of a circle. Such a set is called a chord diagram. It can also be described by a word with two occurrences of each letter, called a double occurrence word. If a circle graph is prime for the split (or join) decomposition defined by Cunnigham, it has a unique representation by a chord diagram, and this diagram can be defined by monadic second-order formulas with the even cardinality set predicate. By using the (canonical) split decomposition of a circle graph, we define in monadic second-order logic with auxiliary linear orders all its chord diagrams. This construction uses the fact that the canonical split decomposition of a graph can be constructed in monadic second-order logic with help of an arbitrary linear order. We prove that the order of first occurrences of the letters in a double occurrence word w that represents a connected circle graph determines this word in a unique way. The word w can be defined by a monadic second-order formula from the word of first occurrences of letters. We also prove that a set of circle graphs has bounded clique-width if and only if all the associated chord diagrams have bounded tree-width.     

Parallels Between Action‐Object Mapping and Word‐Object Mapping in Young Children

Across a series of four experiments with 3‐ to 4‐year‐olds we demonstrate how cognitive mechanisms supporting noun learning extend to the mapping of actions to objects. In Experiment 1 (n = 61) the demonstration of a novel action led children to select a novel, rather than a familiar object. In Experiment 2 (n = 78) children exhibited long‐term retention of novel action‐object mappings and extended these actions to other category members. In Experiment 3 (n = 60) we showed that children formed an accurate sensorimotor record of the novel action. In Experiment 4 (n = 54) we demonstrate limits on the types of actions mapped to novel objects. Overall these data suggest that certain aspects of noun mapping share common processing with action mapping and support a domain‐general account of word learning.     

Preschoolers trust particular informants when learning new names and new morphological forms

Across three studies, we investigated whether 4‐year‐olds would trust a previously reliable informant when learning novel morphological forms. In Experiment 1, children (N= 16) were presented with two informants: one who correctly named familiar objects and another who named them incorrectly. Children were invited to turn to these informants when learning novel labels and morphological forms. The majority of children chose the previously correct labeller when learning novel label and morphology. In Experiment 2, children (N= 16) were presented with an informant who used familiar plurals correctly and one who used them incorrectly. Children chose the previously correct morphologist when learning novel labels and past tense forms. Thus, children track both semantic and morphological accuracy. In Experiment 3, some children (N= 16) were presented with two informants who differed in naming accuracy, whereas others (N= 16) were presented with two informants who differed in morphological accuracy. To forestall any risk of experimenter cuing, one experimenter blind to the training children had received, tested children with novel labels and morphology. The results replicated those of Experiments 1 and 2. Implications for how children's trust in an informant might play a role in their acquisition of morphological forms are discussed.     

A transformation of self-discrimination response functions in accordance with the arbitrarily applicable relations of sameness, more than, and less than

In Experiment 1, 2 experimental subjects were given pretraining of nonarbitrary relations that brought their responses under the control of four contextual stimuli; same, opposite, more than, and less than. One control subject was not exposed to this pretraining. The 2 pretrained subjects and the 3rd nonpretrained subject then received training in six arbitrary relations, the following four relations being the most critical: same/A1-B1, same/A1-C1, less than/A1-B2, more than/A1-C2. All 3 subjects were then tested for seven derived relations, the following three relations being the most important: same/B1-C1, more than/B1-C2, less than/B1-B2. The 2 pretrained subjects, but not the nonpretrained subject, showed the derived relations. One of the stimuli (B1) from the relational network and two novel stimuli (X1 and X2) were then used to train three different self-discrimination responses on three complex schedules of reinforcement. That is, all 3 subjects were trained to pick X1 if they had not emitted a response, to pick B1 if they had emitted one response only, and to pick X2 if they had emitted two responses only. The 2 pretrained subjects, but not the nonpretrained subject, showed the predicted transformation of self-discrimination response functions in accordance with the relations of sameness, more than, and less than (i.e., no response, pick B2; one response only, pick C1; and two responses only, pick C2). In Experiment 2, 2 new subjects were employed, and the arbitrary relational training and testing phases were modified to control for a procedural artifact that may have contributed to the results of the first experiment. Experiment 2 replicated the findings of Experiment 1. The pattern of results support the utility of a relational frames approach to understanding derived stimulus relations.     

Categorical flexibility in preschoolers: contributions of conceptual knowledge and executive control

The current study evaluated the relative roles of conceptual knowledge and executive control on the development of categorical flexibility, the ability to switch between simultaneously available but conflicting categorical representations of an object. Experiment 1 assessed conceptual knowledge and executive control together; Experiment 2 differentiated conceptual knowledge from costly executive processes. In Experiment 1, 3‐ to 5‐year‐olds were given a three‐choice (taxonomic, thematic, and nonassociate) match‐to‐sample task and asked to match two associates. In Experiment 2, same‐aged children were assessed on another match‐to‐sample task that reduced executive costs by presenting thematic and taxonomic associates on separate trials. By comparing performance across tasks, age‐related changes resulting from conceptual knowledge and executive control indicated that conceptual knowledge of superordinate relations showed gains between 3 and 4 years, whereas gains in executive control were seen between 4 and 5 years, suggesting a décalage in the development of conceptual and executive processes underlying categorical flexibility.     

Using the stimulus equivalence paradigm to teach course material in an undergraduate rehabilitation course

In 2 experiments, we examined whether the stimulus equivalence instructional paradigm could be used to teach relations among names, definitions, causes, and common treatments for disabilities using a selection-based intraverbal training format. Participants were pre- and posttested on vocal intraverbal relations and were trained using multiple-choice worksheets in which selection-based intraverbal relations were taught and feedback was delivered until mastery. Most participants in Experiment 1 showed the emergence of vocal intraverbal relations, but responding did not generalize to final written intraverbal tests. Participants in Experiment 2 showed the emergence of not only vocal intraverbal relations but also written intraverbal relations on final tests. Results suggest that the stimulus equivalence paradigm can be effectively implemented using a selection-based intraverbal training format, the protocol may be an effective means of emphasizing important concepts in a college course, and emergent skills may generalize to novel response topographies.