This document outlines how the formal mechanism of cross-cluster function transformation, as described within Relational Frame Theory (RFT), can be applied to teach and empirically manage the emergence of anaphoricresponding and its derivation in naïve learners. Anaphora—where a pronoun or reference term depends on a prior noun or event—requires the learner to integrate across stimulus networks using derived relational responding. The procedures presented below translate the formula FB(v) = Tr(FA(u)) into a structure curriculum for developing this repertoire.
Introduction
Within the analytic architecture of Relational Frame Theory (RFT), the construct of cross-cluster function transformation provides a generative mechanism for explaining how psychological functions shift across distinct stimulus networks. Rather than being confined to directly taught relations, this model reveals the dynamic capacity of derived relational responding to establish novel functions based on prior relational history. This brief explication clarifies the mechanism using formal notation and applied exemplars, grounded in the core concepts of RFT.
Formal Expression of Transformation
The process can be formally expressed as:
FB(v) = Tr(FA(u)), for all u ∈ VA, v ∈ VB
(“∈” means, is an element of)
The above function is read "The meaning or psychological function of a new stimulus (v) is derived from the meaning of a known stimulus (u) using a relational frame, and this applies to all such pairs in each set," or more formally, "The function of v in set VB is the result of transforming the function of u in set VA, using a relational frame—applied for every u in VA and every v in VB,"
This formula explains how and why a learner can assign meaning or emotional function to a new stimulus (v in VB) based on how it is relationally connected to a familiar meaning or function (u in VA), and captures the functional transformation whereby the psychological function of stimulus v (in set VB) is systematically derived from the function of a related stimulus u (in VA), via a relational transformation Tr. Importantly, this transformation is not contingent upon direct reinforcement, but emerges from the contextual control provided by the relational frame.
VA – The Stimulus Hub: Source of Relational Coherence
VA designates a relational hub—a coherent set of stimuli interconnected through multiple relational frames such as Same, Opposite, and Category. These hubs serve as organizational nodes within an individual's relational network, from which functions may propagate outward.
Example:
VA = {“dog”, “puppy”, “hound”}
These stimuli are functionally interrelated through:
• Same: dog ↔ puppy
• Opposite: dog ↔ cat (context-dependent)
• Category: dog ∈ pet ∈ mammal (“∈” means, is an element of)
The reinforcement history embedded in this hub enables the emergence of shared functional properties such as affection, loyalty, and protection.
VB – A Distal Stimulus Set: Relationally Linked, Not Categorically Anchored
VB comprises stimuli that do not reside within the original hub but are relationally linked via analogical, coordinative, or comparative frames. These stimuli are often novel, metaphorically extended, or cross-contextual, yet they acquire function through derived relation to VA.
Example:
VB = {“robot dog”, “loyal soldier”, “AI assistant”}
Although these entities do not fall under the biological category “animal,” they become functionally similar to “dog” via analogical and symbolic mapping, allowing attributes such as loyalty or guardianship to transfer.
Instructional Applications: Examples from Siri Ming
Siri Ming’s empirical work in relational training provides a practical blueprint for implementing the function transformation model using structured stimulus sets and relational networks. Her instructional design methodology demonstrates how practitioners can engineer learning environments to evoke derived relational responses that match targeted psychological functions.
To apply the formula FB(v) = Tr(FA(u)) in empirical teaching contexts, one begins by constructing a relational hub (VA) composed of stimuli with known and reinforced functions. For instance, in Ming’s work with early learners, a hub might include stimuli like “dog,” “puppy,” and “pet,” which reliably evoke responses associated with affection, caregiving, and familiarity (FA(u)). These stimuli are related through frames of Same, Category, and Comparison.
VB stimuli are then selected from conceptually distinct domains—for example, “robot dog,” “guardian,” or “home assistant.” These novel stimuli are presented in analogical or metaphoric relational frames with elements of VA, thereby cueing a derived function (FB(v)) without requiring direct reinforcement.
Ming often tracks the success of such transformations through a derived relational protocol involving match-to-sample, yes/no frames, and intraverbal responses. For example, the learner may be shown a 'robot dog' and asked, “Is it more like a vacuum or a dog?”—a forced-choice frame that sets the occasion for demonstrating functional generalization.
Another application involves establishing equivalence networks to stabilize the transfer. After forming mutual and combinatorial relations between VA and VB items, Ming evaluates transformation of function by presenting VB stimuli in emotionally or socially loaded contexts and observing whether learner responses mirror those previously associated with VA. This allows for empirical verification of the transformation: e.g., whether ‘robot dog’ now evokes comforting behavior in a story-based social interaction probe.
Overall, these methods show how to engineer the relational conditions necessary to produce FB(v) = Tr(FA(u)) outcomes systematically. They also offer metrics to track the strength and generalizability of derived functions—a core concern in both instructional design and cognitive-developmental intervention.
Instructional Progression for Derived Anaphora Responding
To establish derived anaphora responding in a naïve learner, one must first define the core relational transformations involved. Anaphora refers to the use of a linguistic expression (e.g., a pronoun like 'he', 'it', or 'this') whose meaning depends on another expression previously established in the discourse. In RFT terms, this requires the learner to construct functional equivalence or coordination between anaphoric terms (VB) and their antecedents (VA), and to sustain these derived relations across changing contextual frames.
Phase 1: Build Relational Hubs with Referents (VA)
Design a stimulus set VA composed of proper nouns and associated features (e.g., 'Sam', 'Sam is running', 'Sam has red shoes'). Use frames of Coordination, Same, and Perspective-Taking to reinforce multiple exemplar relations among these stimuli. For instance, learners may engage in intraverbal and match-to-sample tasks where 'Sam' is repeatedly matched to images or phrases ('He has red shoes', 'He is fast').
Phase 2: Introduce Derived Pronouns and Reference Terms (VB)
Introduce pronouns such as 'he', 'she', 'it', and demonstratives like 'this' and 'that'. These form the VB stimulus set. Instruct the learner to relate VB items to VA items through derived relational frames of Coordination and Temporal Order. For example, after reading 'Sam is running', the learner is shown 'Who is fast?' or 'He is running. Who is he?' This trains the learner to derive function from VA to VB: FB(‘he’) = Tr(FA(‘Sam’)).
Phase 3: Generalize Anaphoric Derivation via Narrative Integration
To promote derived anaphora responding beyond tightly controlled contexts, embed VA and VB stimuli within short narratives or video-based lessons. For instance, present a story: 'Maria picked up the puppy. She smiled.' Follow with comprehension prompts: 'Who smiled?' Use relational frames of Temporal, Causal, and Deictic perspective to support the inference that 'she' refers to Maria, even when not adjacent in text.
Phase 4: Empirical Verification of Derived Anaphoric Function
To empirically verify transformation of function, present VB stimuli (pronouns) in emotionally salient or behaviorally evocative contexts and track whether the learner responds in accordance with the functions previously conditioned to VA items. For example, if 'Sam' was associated with reinforcement history for humor, a sentence like 'He told a joke. Everyone laughed.' should evoke smiling or verbal laughter from the learner.
Instructional Design Notes for Sustaining Anaphoric Behavior
The progression from within-set relation training (VA) to cross-set derived responding (VB) should follow a precision-teaching model: fluency-building within hubs, followed by recombinative generalization and transformation probes. Siri Ming’s sequencing strategies, especially the use of alternating match-to-sample and narrative intraverbal trials, are recommended to sustain functional flexibility and ensure reliable derived anaphora use.