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.     

Derived relational responding: a comparison of match-to-sample and the relational completion procedure

Previous research suggests that the Relational Completion Procedure may be an effective alternative procedure for studying derived relational responding. However, the parameters that make it effective, relative to traditional match-to-sample, remain to be determined. The present experiment compared the Relational Completion Procedure and match-to-sample protocols for training and testing Same and Opposite derived stimulus relations. Trials to criterion and overall pass rate (i.e., yield) in both procedures were compared across three variables: presence versus absence of a confirmatory response requirement, three versus five comparison stimuli, and top-to-bottom versus left-to-right presentation format. Findings demonstrated a facilitative effect of the confirmatory response requirement in both procedures. Training trials to criterion were nominally but not significantly lower during the nonarbitrary training phase in the Relational Completion Procedure compared to match-to-sample, and the overall yield on the arbitrary relational test was greater in the former procedure compared to the latter. The present findings support the further development of the Relational Completion Procedure as an efficient alternative procedure for establishing Same and Opposite relations with adult humans, and with potential applicability to other types of derived relations.     

CONSTRUCTING AND DERIVING RECIPROCAL TRIGONOMETRIC RELATIONS: A FUNCTIONAL ANALYTIC APPROACH

Participants were pretrained and tested on mutually entailed trigonometric relations and combinatorially entailed relations as they pertained to positive and negative forms of sine, cosine, secant, and cosecant. Experiment 1 focused on training and testing transformations of these mathematical functions in terms of amplitude and frequency followed by tests of novel relations. Experiment 2 addressed training in accordance with frames of coordination (same as) and frames of opposition (reciprocal of) followed by more tests of novel relations. All assessments of derived and novel formula‐to‐graph relations, including reciprocal functions with diversified amplitude and frequency transformations, indicated that all 4 participants demonstrated substantial improvement in their ability to identify increasingly complex trigonometric formula‐to‐graph relations pertaining to same as and reciprocal of to establish mathematically complex repertoires.     

Stimulus equivalence and arbitrarily applicable relational responding.

Subjects' responses to nonarbitrary stimulus relations of sameness, oppositeness, or difference were brought under contextual control. In the presence of the SAME context, selecting the same comparison as the sample was reinforced. In the presence of the OPPOSITE context, selecting a comparison as far from the sample as possible on the physical dimension defined by the set of comparisons was reinforced. Given the DIFFERENT context, selecting any comparison other than the sample was reinforced. Subjects were then exposed to arbitrary matching-to-sample training in the presence of these same contextual cues. Some subjects received training using the SAME and OPPOSITE contexts, others received SAME and DIFFERENT, and others received SAME, OPPOSITE, and DIFFERENT. The stimulus networks established allowed testing for a wide variety of derived relations. In two experiments it was shown that derived performances were consistent with relational responding brought to bear by the contextual cues. In contexts relevant to the relation of sameness, stimulus equivalence emerged. Other kinds of relational networks emerged in the other contexts. Arbitrarily applicable relational responding may give rise to a very wide variety of derived stimulus relations. The kinds of performances seen in stimulus equivalence do not appear to be unique.     

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.     

Cross-domain analogies as relating derived relations among two separate relational networks

Contemporary behavior analytic research is making headway in analyzing analogy as the establishment of a relation of coordination among common types of trained or derived relations. Previous studies have been focused on within-domain analogy. The current study expands previous research by analyzing cross-domain analogy as relating relations among separate relational networks and by correlating participants' performance with a standard measure of analogical reasoning. In two experiments, adult participants first completed general intelligence and analogical reasoning tests. Subsequently, they were exposed to a computerized conditional discrimination training procedure designed to create two relational networks, each consisting of two 3-member equivalence classes. The critical test was a two-part analogical test in which participants had to relate combinatorial relations of coordination and distinction between the two relational networks. In Experiment 1, combinatorial relations for each network were individually tested prior to analogical testing, but in Experiment 2 they were not. Across both experiments, 65% of participants passed the analogical test on the first attempt. Moreover, results from the training procedure were strongly correlated with the standard measure of analogical reasoning.     

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.     

Analysis of apparent demonstrations of responding in accordance with relational frames of sameness and opposition

We evaluated whether contextual control over equivalence and nonequivalence and responding by exclusion can explain the outcomes of relational frame theory (RFT) studies on sameness and opposition relations. We trained nine college students to maintain and reverse conditional discriminations with X1 and X2 as contextual stimuli. In Experiment 1, X1 and X2 controlled derived stimulus relations (DSR) analogous to those controlled by Same and Opposite in RFT studies. These results can be explained by at least two hypotheses: X1 and X2 were cues for equivalence and nonequivalence and responding by exclusion, or for sameness and opposition. In Experiment 2, X1 and X2 controlled DSR predicted by the hypothesis that they were cues for equivalence and nonequivalence and responding by exclusion, and not predicted by the hypothesis that they were cues for sameness and opposition. The results of Experiment 2 and the functional equivalence of X1 and X2 with Same and Opposite in Experiment 1 suggest that Same and Opposite were cues for equivalence and nonequivalence and responding by exclusion in RFT studies.     

A precursor to the relational evaluation procedure: searching for the contextual cues that control equivalence responding.

The precursor to the relational evaluation procedure (pREP) is a go/no-go successive discrimination procedure for examining stimulus equivalence. Previous research has shown that it does not readily produce equivalence responding unless some matching-to-sample (MTS) procedures are incorporated into the experimental sequence. Two experiments attempted to identify contextual cues that would generate equivalence responding on the pREP. Experiment 1 examined the effects of using abstract symbols or various verbal labels as response options on the pREP. Only the words same and different, when used as response options, reliably produced equivalence responding across 4 subjects. Experiment 2 examined different pretraining preparations designed to attach the functions of the words same and different to abstract symbols that could then be used as response options on the pREP. The most effective pretraining procedure involved multiple-exemplar training during which subjects were trained to respond to abstract symbols in the presence of pairs of stimuli that were either formally the same or different. The abstract symbols were subsequently used as response options with the pREP, and all subjects reliably demonstrated equivalence responding. The findings suggest that the relations of same and different may be fundamental to equivalence responding. These findings are discussed in terms of what they suggest about the nature of the equivalence phenomenon specifically and derived relational responding more generally.     

Self‐control as generalized operant behavior by adults with autism spectrum disorder

The present investigation attempted to teach self‐control to 3 adults who had been diagnosed with autism spectrum disorder. Using a self‐control training procedure that coupled a gradual fading of delay to access a large delayed reinforcer with a concurrent work requirement, each participant was exposed to conditions in which baseline levels of choices for large delayed reinforcers and task engagement increased 10‐fold. Furthermore, generalization effects were partially demonstrated in a novel context.     

A functional-analytic model of analogy: a relational frame analysis

The aim of this study was to explore a behavior-analytic model of analogical reasoning, defined as the discrimination of formal similarity via equivalence-equivalence responding. In Experiment 1, adult humans were trained and tested for the formation of four three-member equivalence relations: A1-B1-C1, A2-B2-C2, A3-B3-C3, and A4-B4-C4. The B and C stimuli were three-letter nonsense syllables, and the A stimulus was a colored shape. Subjects were then successfully tested for equivalence-equivalence responding (e.g., matching B1/C1 to B2/C2 rather than B3/C4). These tasks were designed such that equivalence-equivalence responding might allow subjects to discriminate a physical similarity between the relations involved. Some participants (color subjects) received only equivalence-equivalence tasks in which they might discriminate a color relation, whereas others (shape subjects) were given tasks in which they might discriminate a shape relation. A control group received both types of task. In a subsequent test for the discrimination of formal similarity, color subjects matched according to color, shape subjects matched according to shape, and the control group showed no consistent matching pattern. In Experiment 2, adult humans showed a transformation of the functions of a block-sorting task via this basic model of analogy. Empirical and conceptual issues related to these results are discussed.     

Toward a theory of verbal behavior

This paper is a reply to an accompanying set of six commentaries by Sidman; Hayes and Barnes; Schusterman, Kastak, and Reichmuth; Tonneau and Sokolowski; Lowenkron; and Moerk. Those commentaries were prompted by our article “On the Origins of Naming and Other Symbolic Behavior” (1996), which was, in turn, followed by 26 commentaries and a reply. In the course of the present reply, we further develop the naming account to embrace more complex verbal relations such as same, different, more, and less. We also examine what we see as the lack of conceptual coherence in equivalence theories, including relational frame theory, and the disparities between these accounts and the findings from empirical research.     

Arbitrarily applicable comparative relations: experimental evidence for a relational operant

Arbitrarily applicable derived relational responding has been argued by relational frame theorists to be a form of operant behavior. The present study examined this idea with 4 female participants, ages 4 to 5 years old, who could not perform a series of problem-solving tasks involving arbitrary more than and less than relations. In a combined multiple baseline (across responses and participants) and multiple probe design (with trained and untrained stimuli), it was shown that reinforced multiple-exemplar training facilitated the development of arbitrary comparative relations, and that these skills generalized not just across stimuli but also across trial types. The sequence of training identified potential prerequisites in the development of comparative relations (e.g., nonarbitrary comparative relations). Taken as a whole, the present data, along with previous work by others in this area, suggest that relating arbitrary events comparatively is an operant. The implications of this conclusion for the analysis of complex behavior are discussed.     

Training class inclusion responding in typically developing children and individuals with autism

In a class inclusion task, a child must respond to stimuli as being involved in two different though hierarchically related categories. This study used a Relational Frame Theory (RFT) paradigm to assess and train this ability in three typically developing preschoolers and three individuals with autism spectrum disorder, all of whom had failed class inclusion tests. For all subjects, relational training successfully established the target repertoire and subsequent testing demonstrated both maintenance and generalization. Limitations and future research directions are discussed.     

Testing response-stimulus equivalence relations using differential responses as a sample

This study tested the notion that an equivalence relation may include a response when differential responses are paired with stimuli presented during training. Eight normal adults learned three kinds of computer mouse movements as differential response topographies (R1, R2, and R3). Next, in matching-to-sample training, one of the response topographies was used to select a comparison stimulus B (B1, B2, or B3) conditionally upon presentation of sample stimulus A (A1, A2, or A3), and to select stimulus D (D1, D2, or D3) conditionally upon presentation of stimulus C (C1, C2, or C3). After two sample-comparison-response relations (ABR and CDR) were established, 18 sample-comparison relations were tested (BA, DC, RA, RB, RC, RD, AC, CA, AD, DA, BC, CB, BD, DB, AA, BB, CC, and DD). In the RA, RB, RC, and RD tests, the differential responses (R1, R2, and R3) were used as sample stimuli. All subjects made class-consistent comparison selections in the tests. This study provides evidence that responses may become members of an equivalence class.     

Relating different perspectives on how outcomes of behavior influence behavior

Many researchers have tackled the question of how behavior is influenced by its outcomes. Some have adopted a nonmechanistic (functional) perspective that attempts to describe the influence of outcomes on behavior. Others have adopted a mechanistic (cognitive) perspective that attempts to explain the influence of outcomes on behavior. Orthogonal to this distinction, some have focused on the influence of outcomes that a behavior had in the past, whereas others also consider the influence of outcomes that a behavior might have in the future. In this article, we relate these different perspectives with the goal of reducing misunderstandings and fostering collaborations between researchers who adopt different perspectives on the common question of how behavior is influenced by its outcomes.     

Fast acquisition of cooperation and trust: A two-stage view of trusting behavior

Trustful behavior was defined in terms of the consecutive numbers of matching-to-sample problems worth money that each subject worked during sessions that ended in an equitable distribution. Two stages of acquisition are inherent in this definition; the first stage requires acquisition of an equitable method of distributing reinforcers (cooperation) to show that the within-session deviations (trust) from equity that develop during the second stage are temporary and are not part of an inequitable method of distributing reinforcers. Previous research has indicated that a contingency to trust is necessary to override the aversiveness of the inequity inherent in trusting and to produce consistent and maximal trust (half of the problems worked consecutively by each subject). The present experiment examined such a contingency. The trust contingency was an increased requirement for changing the direction of problem allocation. Only the subject who had been allocated a problem could change that allocation, by pulling a lever 45 or more times. On the other hand, no separate responses were required to allow the person who worked the last problem to also work the next one (passive trust). Hence, giving a problem was the only way to increase the distribution of problems to the other person and hence prevent oneself from receiving all of the reinforcers. All eight pairs of subjects cooperated from the outset. Trusting behavior developed for all four pairs exposed to the contingency to trust and expanded to maximal levels by the second session for three of the four pairs.     

ALL STIMULI ARE EQUAL,BUT SOME ARE MORE EQUAL THAN OTHERS: MEASURING RELATIONAL PREFERENCES WITHIN AN EQUIVALENCE CLASS

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.