Equivalence Classes

1. Equivalence Classes Ps625 Concept Formation Dr. Ken Reeve Caldwell College Grad ABA Programs

2. Definition • a finite group of physically disparate stimuli (no perceptual similarity) • stimuli become related as a function of training (Fields, Adams, Buffington, Yang, & Verhave, 1996; Fields & Verhave, 1987; Sidman & Tailby, 1982; Sidman, 1990) • An equivalence class must contain at least three stimuli • To establish a class of N stimuli, N-1 stimulus-stimulus relations are trained such that each stimulus in the potential class is used in at least one relation

3. Example DOG • Written word DOG • The spoken word “DOG” • A picture of a dog DOG

4. Another Example CAT • Written word CAT • The spoken word “CAT” • A picture of a cat CAT

5. Training & Testing Procedures • Usually with conditional discrimination or match-to-sample (but other methods will also be discussed) • Symbolic notation is often used to outline training/testing procedures: • A, B, C, …N represent each of the disparate stimuli that will make up the class • 1, 2, 3, etc. notate the number of classes to be established • Thus, A1 = first stimulus in class 1; B3 = third stimulus in class 2; etc.

6. Training & Testing Procedures • To establish equivalence classes, at least two potential classes must be trained concurrently • training establishes both substitutability of all stimuli within a particular equivalence class in addition to discrimination between classes

7. Training & Testing Procedures • To establish equivalence classes with three members, at least two relations must be trained for each potential class (remember the N-1 rule) • Let’s consider our DOG and CAT potential equivalence classes • First train the AB relation (given stimulus A select stimulus B) • the word DOG (A1) is presented as a sample • The positive comparison (Co+) would be the spoken word dog (B1) (Selection would result in positive feedback or reinforcement) • the negative comparison (Co-) would be the spoken word cat (B2) (Selection would result in corrective feedback or extinction)

8. Training AB relation DOG A1 CAT DOG B1 B2

9. Training & Testing Procedures • In addition, to train the AB relation • the word CAT (A2) is presented as a sample • The positive comparison (Co+) would be the spoken word cat (B2) (Selection would result in positive feedback or reinforcement) • the negative comparison (Co-) would be the spoken word dog (B2) (Selection would result in corrective feedback or extinction)

10. Training AB relation CAT A2 CAT DOG B1 B2

11. Training & Testing Procedures • Once responding is 100% correct, we can conclude that the learner has formed the AB conditional relation • also demonstrates that learner can discriminate between the two different “A” stimuli, the two different “B” stimuli, and the A stimuli from the B stimuli • At this point we can either continue training more conditional discriminations or we can do our first test for an EMERGENT (DERIVED) RELATION (a conditional discrimination that emerges with no direct training history) • If the learner “knows” that A goes with B, can they demonstrate the reverse? (B goes with A) • This emergent relation shows SYMMETRY

12. TESTING BA symmetry relation CAT B2 CAT DOG A1 A2

13. TESTING BA symmetry relation DOG B1 CAT DOG A1 A2

14. Training & Testing Procedures • If BA TESTING is 100% correct, we can conclude that the learner has formed the BA conditional symmetry relation with no direct training • It is called “symmetry” because the relation is a mirror image or reversal of the one directly trained • At this point, we can continue training more conditional discriminations • Let’s train the BC conditional relation

15. Training BC relation DOG B1 C2 C1

16. Training BC relation CAT B2 C2 C1

17. Training & Testing Procedures • Once responding is 100% correct, we can conclude that the learner has formed the BC conditional relation • Now we can either continue training more conditional discriminations or we can do our 2nd test for an EMERGENT (DERIVED) RELATION • If the learner “knows” that B goes with C, can they demonstrate the reverse? (C goes with B) • This emergent relation would show a SECOND SYMMETRY relation

18. TESTING CB symmetry relation C1 DOG CAT B1 B2

19. TESTING CB symmetry relation C2 DOG CAT B1 B2

20. Training & Testing Procedures • If TEST responding is 100% correct, we can conclude that the learner has formed the CB symmetry conditional relation • At this point, we have trained our N-1 relations (3 members subtract 1 equals 2!) • Now we can continue testing for another EMERGENT (DERIVED) RELATION • If the learner “knows” that A goes with B, and B goes with C, can they demonstrate that A goes with C? • This emergent relation would show a TRANSITIVE relation

21. TESTING AC transitive relation DOG A1 C2 C1

22. TESTING AC transitive relation CAT A2 C2 C1

23. Training & Testing Procedures • If TEST responding is 100% correct, we can conclude that the learner has formed the AC transitive conditional relation • We have one last test for another EMERGENT (DERIVED) RELATION • If the learner “knows” that A goes with B, and B goes with C, can they demonstrate that C goes with A? • This emergent relation would shows a combination of symmetry and transitivity; it is called an EQUIVALENCE relation

24. TESTING CA equivalence relation C1 CAT DOG A2 A1

25. TESTING CA equivalence relation C2 CAT DOG A2 A1

26. Training & Testing Summary • If TEST responding is 100% correct, we can conclude that the learner has formed two equivalence classes (one for dogs and one for cats) • We TRAINED 2 relations: • A  B • B  C • We TESTED 4 EMERGENT (DERIVED) relations: • B  A symmetry • C  B symmetry • A  C transitivity • C  A equivalence • Another set of tests for REFLEXIVITY (IDENTITY) is usually omitted unless the learner is less skilled: • A  A • B  B • C  C

27. Training & Testing Summary • Now each member of each class occasions the selection of all the other members of the class • This occurs for all combinations even though we only directly trained a subset of all possible relations

28. Equivalence Class Expansions • Once we have 3-member equivalence classes, we can continue to expand them in a number of ways • One way is by training additional conditional relations and testing for additional emergent conditional relations • For example, train the relation CD in which stimulus D is the Spanish word PERRO for the dog class and GATO for the cat class

29. Training CD relation C2 GATO PERRO D2 D1

30. Training CD relation C1 GATO PERRO D2 D1

31. Training & Testing Summary • Then we would test for all possible new emergent relations • D  C symmetry • A  D transitivity • B  D transitivity • D  A equivalence • D  B equivalence • If test results are 100%, then we can now conclude that we have two 4-member equivalence classes • And so on…

32. More Equivalence Class Expansions • Once we have equivalence classes, we can also expand them by using them as “transfer networks” • This means that we train an operant response (different from the previous selection response) in the presence of only 1 member of each equivalence class • For example, train a child to say WOOF in the presence of A1 and MEOW in the presence of A2

33. Training response transfer DOG A1 WOOF!

34. Training response transfer CAT A2 MEOW!

35. More Equivalence Class Expansions • If we now present stimuli B, C, or D, the child should also say WOOF (in the presence of the dog class members) and MEOW (in the presence of the cat class members) • This should occur without direct training • As such, it demonstrates another type of emergent behavior

36. TESTING response transfer CAT GATO MEOW!

37. TESTING response transfer DOG PERRO WOOF!

38. And…um, victory is mine!

39. More Equivalence Class Expansions • Once we have equivalence classes, we can also expand them by testing to see whether physical variations (“variants”) of each equivalence class member will still occasion selection of all other members • For example, what if we present emergent relations in which the dog or cat picture used in training (stimulus C1 and C2) is substituted with DIFFERENT pictures? (notated as C’1 or C’2) • Will these “variants” still occasion class-consistent responding?

40. TESTING C’A “generalized” equivalence relation C’2 CAT DOG A2 A1

41. TESTING C’A “generalized” equivalence relation C’2 CAT DOG A2 A1

42. TESTING C’A “generalized” equivalence relation C’2 CAT DOG A2 A1

43. TESTING C’A “generalized” equivalence relation C’2 CAT DOG A2 A1

44. Generalized Equivalence Class • Note that many cat pictures occasion selection of the other members of the equivalence class • Thus, the class-consistent selections made by the learner have generalized to variants of one member of the equivalence class • So we call this a “generalized equivalence class” • It is actually a merger of an equivalence class with a perceptual class…

45. One equivalence class + one linked perceptual class = CAT CAT GATO

46. Generalized equivalence class GATO CAT CAT

47. Generalized Equivalence Class • We can also present variants of EACH equivalence class member to see if they occasion selection of the other members of the equivalence class • This would produce an even larger generalized equivalence class • It would be a merger of an equivalence class with perceptual classes linked to each equivalence class member…

48. Generalized equivalence class (expanded further) GATO CAT GATO CAT GATO CAT CAT CAT CAT

49. Training Structures for Equivalence Class • Refers to how the training relations and derived relations are “situated” relative to one another • Can affect likelihood of class formation • See next

50. Linear Training Structure A  B B  C C  D