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) 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) 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 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 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 Written word DOG Written word DOG The spoken word “DOG” The spoken word “DOG” A picture of a dog A picture of a dog DOG

4. Another Example Written word CAT Written word CAT The spoken word “CAT” The spoken word “CAT” A picture of a 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) 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: 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 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 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. 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 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 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) 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 Let’s consider our DOG and CAT potential equivalence classes First train the AB relation (given stimulus A select stimulus B) First train the AB relation (given stimulus A select stimulus B) the word DOG (A1) is presented as a sample 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 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) the negative comparison (Co-) would be the spoken word cat (B2) (Selection would result in corrective feedback or extinction)

8. DOG CAT Training AB relation A1 B2 B1

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

10. Training AB relation CAT DOGCAT A2 B2 B1

11. Training & Testing Procedures Once responding is 100% correct, we can conclude that the learner has formed the AB conditional relation 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 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) 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) If the learner “knows” that A goes with B, can they demonstrate the reverse? (B goes with A) This emergent relation shows SYMMETRY This emergent relation shows SYMMETRY

12. TESTING BA symmetry relation CAT B2 A2 A1 DOG

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

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 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 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 At this point, we can continue training more conditional discriminations Let’s train the BC conditional relation Let’s train the BC conditional relation

15. Training BC relation DOG B1 C2C1

16. Training BC relation CAT B2 C2C1

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

18. TESTING CB symmetry relation DOG C1 B1B2 CAT

19. TESTING CB symmetry relation DOG C2 B1B2 CAT

20. Training & Testing Procedures If TEST responding is 100% correct, we can conclude that the learner has formed the CB symmetry conditional relation 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!) 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 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? 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 This emergent relation would show a TRANSITIVE relation

21. TESTING AC transitive relation A1 C2C1 DOG

22. TESTING AC transitive relation A2 C2C1 CAT

23. Training & Testing Procedures If TEST responding is 100% correct, we can conclude that the learner has formed the AC transitive conditional relation 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 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? 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 This emergent relation would shows a combination of symmetry and transitivity; it is called an EQUIVALENCE relation

24. TESTING CA equivalence relation A1 A2 C1 CAT DOG

25. TESTING CA equivalence relation A1 A2 C2 CAT DOG

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) 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: We TRAINED 2 relations: A  B A  B B  C B  C We TESTED 4 EMERGENT (DERIVED) relations: We TESTED 4 EMERGENT (DERIVED) relations: B  A symmetry B  A symmetry C  B symmetry C  B symmetry A  C transitivity A  C transitivity C  A equivalence C  A equivalence Another set of tests for REFLEXIVITY (IDENTITY) is usually omitted unless the learner is less skilled: Another set of tests for REFLEXIVITY (IDENTITY) is usually omitted unless the learner is less skilled: A  A A  A B  B B  B C  C C  C

27. Training & Testing Summary Now each member of each class occasions the selection of all the other members of the class 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 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 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 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 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 D1 D2 C2 GATO PERRO

30. Training CD relation D1 D2 C1 GATO PERRO

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

32. More Equivalence Class Expansions Once we have equivalence classes, we can also expand them by using them as “transfer networks” 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 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 For example, train a child to say WOOF in the presence of A1 and MEOW in the presence of A2

33. DOG WOOF! Training response transfer A1

34. MEOW! Training response transfer A2 CAT

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) 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 This should occur without direct training As such, it demonstrates another type of emergent behavior As such, it demonstrates another type of emergent behavior

36. MEOW! TESTING response transfer GATO CAT

37. WOOF! TESTING response transfer PERRO DOG

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 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) 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? Will these “variants” still occasion class- consistent responding?

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

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

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

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

44. Generalized Equivalence Class Note that many cat pictures occasion selection of the other members of the 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 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” So we call this a “generalized equivalence class” It is actually a merger of an equivalence class with a perceptual class… It is actually a merger of an equivalence class with a perceptual class…

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

46. CAT Generalized equivalence class CAT GATO

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 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 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… It would be a merger of an equivalence class with perceptual classes linked to each equivalence class member…

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

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

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

51. One-to-Many Training Structure A  B  C  D

52. Many-to-One Training Structure B  A C  D 

53. Training/Testing Variations This refers to when you train and test for various conditional relations This refers to when you train and test for various conditional relations See next… See next…

54. Simultaneous Procedure First, train ALL training relations in one trial block until you reach criterion First, train ALL training relations in one trial block until you reach criterion Then, test ALL derived (emergent) relations in another trial block Then, test ALL derived (emergent) relations in another trial block This procedure tends to produce poor performance This procedure tends to produce poor performance

55. Simple-to-Complex Procedure First, train only 1 relation to criterion in a trial block (e.g., A  B) First, train only 1 relation to criterion in a trial block (e.g., A  B) Then, test the “simplest” derived (emergent) relation in another trial block (e.g., B  A) Then, test the “simplest” derived (emergent) relation in another trial block (e.g., B  A) Then, train the next 1 relation to criterion in a trial block (e.g., B  C) Then, train the next 1 relation to criterion in a trial block (e.g., B  C) Then, test the next “simplest” derived (emergent) relation in another trial block (e.g., C  B) Then, test the next “simplest” derived (emergent) relation in another trial block (e.g., C  B) Followed by A  C testing Followed by A  C testing Followed by C  A testing Followed by C  A testing