# Testing Stimulus Equivalence in Transformer-Based Agents
## 1 Introduction
### Stimulus Equivalence  
SE is defined as responding in accordance with the features of reflexivity, symmetry, and transitivity [1]. Stimulus classes can be formed from any unrelated stimulus, arbitrarily assigned, and functionally related regardless of its physical properties [7]. If a stimulus controls one member of the class, it affects all members of an equivalent stimulus class [2]. In SE, functional properties can be transferred to other stimuli without explicit training among members of an equivalence class. A stimulus trained as a conditional stimulus can form new analytical units that have not been previously trained, in which it can either control other three-term units or in which its role changes to a discriminative stimulus under the control of other stimuli [1]. SE provides a framework in the experimental analysis of behaviour to study language, symbolic behaviour, and cognition.  
The most common framework for training and evaluating SE is arbitrary matching-to-sample (MTS). The goal of this procedure is training conditional discriminations presenting a sample stimulus and two or more comparison stimuli. After training, individuals areassessed for responding in accordance with SE by conducting reflexivity, symmetry, and transitivity tests [1; 2; 5; 7; 8].  
An important line of research on SE formation is the conditions that could influence the emergence of equivalence classes [7; 8]. One methodological aspect of SE research is the training structure (TS). TS determines which pairs of stimuli are used for baseline training and which pairs are used to test the properties of reflexivity, symmetry, and transitivity. Three basic patterns of TS are typically used: linear series (LS), many-to-one (MTO), and one-to-many (OTM) [5; 8; 9]. Figure 1 shows a graphical representation of train structures in a four-members (A, B, C, and D) equivalence class with their respective pairs of reflexivity, symmetry, and transitivity. In LS, stimuli are presented sequentially, forming a chain of conditional discriminations. The stimuli are arranged in a specific order, such as A-B-C-D, establishing a clear and sequential relations between each stimulus and its adjacent counterparts. In OTM, a single sample stimulus is associated with multiple comparison stimuli with the pairs A-B, A-C, and A-D. In MTO, multiple sample stimuli are associated with a single comparison stimulus, B-A, C-A, D-A, wherein stimuli B, C, and D serve as samples and are related to a common stimulus A.  
Another methodological consideration is the type of relation that is formed in MTS. In a typical MTS experiment, there is one sample stimulus and three comparison stimuli. During a baseline training trial, one of the comparison stimuli is reinforced, and the other are either punished or non-contingent. Hence, two types of relations between stimuli are formed: select type relations and reject type relations. Select type relations are established when the organism consistently responds to the correct comparison stimulus when presented with the sample stimulus, indicating which comparison stimuli are appropriate matches for the sample stimulus. Reject type relations are established when the organism consistently avoids choosing incorrect comparison stimuli when presented with the sample stimulus, indicating which comparison stimuli are not suitable matches. There is no clear evidence regarding whether SE requires select-only or also requires both select and reject relations. [PERSON] and [PERSON] [10] suggest that select-only relations are essential, though different experiments yield varied results [11; 12; 13; 14].