The effects of equivalence class structure on test performances.
An equivalence class is a finite group of stimuli that become substitutable for each other as measured in a series of conditional discrimination tests. The stimuli in such a class do not resemble one another (Fields & Verhave, 1987; Sidman, 1990). An example would be the five written stimuli DOS, TWO, ZWEI, 2, and 0010. To establish these stimuli as a class of equivalent terms, four stimulus-stimulus relations must be trained; for example DOS-TWO, TWO-ZWEI, ZWEI-2, and 2-0010 can be used. To demonstrate the existence of the class, the 21 untrained stimulus pairs that can be derived from the five stimuli are presented as tests (Fields, Adams, Newman, & Verhave, 1992; Fields & Verhave, 1987; Fields, Verhave, & Fath, 1984; Sidman, 1971; Sidman & Cresson, 1973; Sidman & Tailby, 1982). The stimuli in a test pair are presented as sample and positive comparison along with a comparison that is not a class member in a conditional discrimination test. If the positive or class-consistent comparisons are selected for all test pairs, the stimuli are said to function as an equivalence class (Fields & Verhave, 1987). These trained relations and test pairs are illustrated symbolically on the left side of Table 1.
In general, a class can be formed from a set of N stimuli as long as TABULAR DATA OMITTED (N-1) stimulus-stimulus relations are trained. Each term must be used in at least one trained relation, and no trained relation can contain the same two stimuli. Within these constraints, however, there are many sets of relations that can be chosen for training (Fields & Verhave, 1987; Fields et al. 1984; O'Mara, 1991; Saunders & Green, 1992). Two sets of trained relations and their corresponding test pairs for a 5-member class are illustrated in Table 1. Each stimulus is represented by a single letter. The letter pairs represent stimulus-stimulus relations. In the matrix on the left, AB, BC, CD, and DE are training pairs; the remaining 21 stimulus pairs would be presented as tests to assess the formation of a class. This matrix is a symbolic representation of the above mentioned class of TWOs. In the matrix on the right, AB, AC, AD, and AE are training pairs. Thus a different set of 21 stimulus pairs would be presented as tests to assess the formation of the class.
The Structure of Equivalence Classes
The set of training relations used to establish an equivalence class defines its structure. All structures can be described by specification of the values of only three parameters: number of nodes, distribution of single stimuli, and directionality of training (Fields & Verhave, 1987).
A node is a stimulus that is linked by training to at least two other stimuli. In a class of N stimuli, 1 to (N-2) stimuli can serve as nodes. The spider diagrams in the upper portion of Figure 1 illustrate the nodal structures formed by three possible training sets that can be used to establish a 9-member class. Each differs in terms of the number of stimuli that function as nodes.
Directionality of training describes the sample and comparison functions served by a stimulus in a trained relation (Fields et al.,1992; Fields & Verhave, 1987). The spider diagrams in the lower portion of Figure 1 illustrate some possible patterns of directionality of training for a 3-node, 5-member class.
The purpose of this article is to show how these two structural parameters influence the likelihood of class formation and the performances occasioned by various tests presented during and after class formation. The data will then be used to reconsider the view that all stimuli in an equivalence class are substitutable for each other. Finally, the data will be used to show how the structural parameters influence the relatedness of the stimuli in an equivalence class.
Table 2 Effects of Directionality of Training on the Likelihood of Equivalence Class Formation Retarded 3-5 yr. Normal 8-10 yr. Directionality Adolescent Preschool Normal Gifted 1 Sa |right aarrow~ 4 Co's 1:8 4:6 2:2 2:2 4 Sa's |right aarrow~ 1 Co 8:8 5:5 1:1 2:2 Note. Each entry has an X:Y format where Y is the number of subjects studied, and X is the number of subjects who formed classes. (Data supplied by J.E. Spradlin.)
Directionality of Training
Spradlin and Saunders (1986) and Saunders, Wachter, and Spradlin (1988) studied the global effects of directionality of training on the likelihood of class formation with children who differed in terms of chronological and mental age. The children were trained using one of two protocols. In the multiple sample-single comparison protocol, BA, CA, DA, and EA were trained. Thus B, C, D, and E were used as samples and A was used as the comparison. In the single sample-multiple comparison protocol, AB, AC, AD, and AE were trained. Thus B, C, D, and E were used as comparisons and A was always used as the sample. Both training sets, then, shared the same nodal structure and distribution of singles; they differed only in terms of directionality of training. The multiple sample-single comparison protocol has also been called the many-to-one procedure (Urcuioli, Zentall, Jackson-Smith, & Steirn, 1989; Zentall, Steirn, Sherburne, & Urcuioli, 1991) and the comparison-as-node procedure (Saunders, Saunders, Williams, & Spradlin, 1993: this issue). The single sample-multiple comparison protocol has also been called the one-to-many procedure (Zentall, Steirn, Randall, Roper, & Urcuioli, in press) and the sample-as-node procedure (Saunders et al., 1993: this issue).
Table 2 summarizes the results for the four different subject populations. Mentally retarded adolescents and normal 3-5 year old preschool children were more likely to form equivalence classes when training was conducted with the multiple sample-single comparison protocol rather than the single sample-multiple comparison protocol. No difference was found, however, with 8-10 year old normal and 8-10 year old gifted children. Thus the likelihood of class formation was influenced both by directionality of training and chronological or mental age.
Directionality of training also describes the formal characteristics of specific emergent relations within an equivalence class. An example is illustrated in Figure 2. When a class is formed by training BA, AC, and DA, both BC and BD are 1-node relations. For BC, the directionality in the prerequisite BA and AC relations is the same (B |right arrow~ A |right arrow~ C); thus BC is a transitive relation. For BD, the directionality of training in the prerequisite relations BA and DA are in opposition (B |right arrow~ A |left arrow~ D); thus BD is an equivalence relation. The linkage of B to C and B to D, then, differs only in terms of directionality of training. Because the formal linkage of B to C and D is different, will a behavioral function trained to the B stimulus transfer differentially to the C and D stimuli?
De Rose, Mcllvane, Dube, Galpin, and Stoddard (1988) conducted an experiment that fits this analytic scheme. Nonsense shapes were used as stimuli. First, the choice of B1 in a B1/B2 simultaneous discrimination task was reinforced until subjects always chose B1. Conditional discrimination training was then used to establish the conditional relations, BA, AC, and DA. Tests for emergent relations showed that two 4-member classes had been formed. These tests were conducted in the absence of B1/B2 discrimination training trials. Finally transfer tests were conducted with D1/D2 and C1/C2. These tests were conducted in the absence/presence of the B1/B2 trials. Subjects were more likely to choose C1 in the C1/C2 test than D1 in the D1/D2 test. More transfer of discriminative function occurred between the stimili in the transitive relation than in the equivalence relation. Amount of transfer was thus correlated with the directionality of training.
The performances occasioned by the BC emergent relations test demonstrated transitivity. Thus only one property was involved in the transfer of discriminative control from B to C. The performances occasioned by the BD emergent relation demonstrated both transitivity and symmetry. Thus two properties were involved in the transfer of discriminative control from B to D. Amount of transfer, then, was an inverse function of the number of properties that link the stimuli used in the transfer tests. Because the directionality of training indexed the number of linking properties, it predicted amount of transfer.
It is commonly assumed that the stimuli in an equivalence class are substitutable for each other. By definition, substitutability means that stimuli are related to each other by equality. One stimulus can thus be substituted for another in any test, with no change in test performance. The transfer data presented by de Rose et al. (1988) show that the discriminative function acquired by one stimulus in a class transfers differentially to other stimuli in the class. Therefore the stimuli in the class could not have been substitutable for each other or equally related to each other. Instead, the stimuli in the previously established equivalence class must be unequally substitutable for each other or differentially related to each other. The relatedness of stimuli in an equivalence class, then, is a function of directionality of training. Additional data from other tests would be needed to assess the generality of this notion.
Nodal distance is another structural parameter that has been explored experimentally. A five-member equivalence class can be formed with the written stimuli DOS, TWO, ZWEI, 2, and 0010 by training four stimulus-stimulus relations such as DOS-TWO, TWO-ZWEI, ZWEI-2, and 2-0010 as illustrated in Figure 3. The test pairs 0010-DOS, 0010-TWO, and 0010-ZWEI are of particular interest because each contains stimuli that were not used together in training. The stimuli 0010 and DOS are separated by three intervening nodal stimuli: the numeral 2, ZWEI, and TWO. The stimuli 0010 and TWO are separated by two nodes: the numeral 2 and ZWEI. The stimuli 0010 and ZWEI are separated by one node: the numeral 2. Nodal distance is the minimum number of nodal stimuli in a training set that separates any two stimuli in a class. Nodal distance provides a formal means of describing emergent relations.
Fields et al. (1984) and Fields and Verhave (1987) proposed that the relatedness of any two stimuli in equivalence classes is an inverse function of nodal distance that separates the stimuli. This proposal grew out of the results of the derived list experiments (Ebbinghaus, 1913), serial learning experiments (Slamecka, 1985), the analysis of semantic memory networks (Collins & Loftus, 1975; Collins & Quillian, 1969), and fragmentary data culled from some early studies of equivalence class formation (Lazar, Davis-Lang, & Sanchez, 1984; McDonagh, Mcllvane, & Stoddard, 1984; Saunders et al., 1988; Sidman, Kirk, & Willson-Morris, 1985). The notion of differential associative strength or the partial substitutability of the stimuli in a class implies that many different tests would all yield concordant performances that varied systematically with nodal distance.
This proposal is evaluated with the results of eight studies. They are divided into experiments that measure emergent relations test performances and post-class formation transfer test performances.
Emergent Relations Tests
Frequency measures. The first study to systematically assess the effects of nodal distance in equivalence classes was reported by Fields, Adams, Verhave, & Newman (1990). Two 3-member classes of nonsense syllables were established by training AB and BC. After demonstrating the presence of all emergent relations, CD was trained. To assess the formation of the 4-member class, ABCD, the O-node symmetry test trials (DC), the 1-node test trials (DB and BD), and the 2-node test trials (DA and AD) were mixed together and presented in a test block. The block was repeated until the class-consistent comparisons were always chosen.
At the start of testing, the likelihood of choosing the class-consistent comparison was highest for the DC test, intermediate for the DB and BD tests, and lowest for the DA and AD tests. Thus the degree of class consistent performance occasioned by each emergent relations test was an inverse function of its nodal distance. With repeated testing, the class-consistent selections increased to 100% accuracy in an order that was a direct function of nodal distance.
Dube, Green, and Serna (1993) reported similar finding with 4-member classes of the same structure. Their classes, however, consisted of auditory stimuli instead of visual stimuli. In addition, the training and testing procedure differed from that used by Fields et al. (1990). Their findings extend the generality of the effects of nodal distance across sensory modalities and procedural variables.
Kennedy (1991) extended the findings presented by Fields et al. (1990) to a larger class size and a different class structure. Using nonsense shapes as class members, Kennedy established three 3-node, 7-member classes by training AB, AC, BD, BE, CF, and CG. The structure is illustrated in Figure 4. The emergent relations test blocks contained O-node symmetry tests, as well as 1-, 2-, and 3-node equivalence tests.
For each subject, in the initial tests, the likelihood of choosing class-consistent comparisons was greatest in the O-node symmetry tests and decreased systematically in the presence of the 1-, 2-, and 3-node tests, respectively. Thus performance was an inverse function of nodal distance. With repeated testing, all tests were eventually passed in an order that was a direct function of nodal distance. The results of these three studies showed the same effects of nodal distance regardless of stimuli used, class size, class structure, or the number of classes that were established simultaneously.
Reaction time measures. Nodal distance also influenced reaction times (RT) occasioned by emergent relations tests. Wulfert and Hayes (1988) established 3-member equivalence classes by training AB and AC. After the emergent relations tests were passed, reaction times were measured during the presentation of a mixed trial block. The reaction times were averaged across subjects. The mean reaction times for the trained relations did not differ from those occasioned by the BA and CA symmetry tests (0-node relations). These RTs were shorter than those occasioned by the AC transitivity and the CA equivalence tests (1-node relations). Thus the reaction times occasioned by various training and emergent relations were a direct function of the nodal distance that separated the stimuli in each relation.
The apparent sensitivity of reaction time to nodality led Bentall, Dickins, and Fox (1993) to measure reaction times occasioned by emergent relations during the formation of equivalence classes. Six 3-member classes were established by training AB and BC. Thereafter, the trained relations and all of the emergent relations were presented in a mixed block of trials that was presented twice. The class-consistent performances during the second block demonstrated class formation. Average reaction times for correct choices in training trials, O-node BA and CB symmetry tests, and 1-node AC transitivity and CA equivalence tests were measured in each block. In the initial test block, short RTs were occasioned by the trained relations and the symmetry tests. The reaction times occasioned by the transitivity tests were longer and the reaction times occasioned by the equivalence tests were the longest. By the second test block, the differences in reaction times decreased. Although repetition of testing attenuated the effect, reaction time was a direct function of nodal distance during class formation.
Post-Class Formation Transfer Tests
An equivalence class has been formed once comparison selections in all emergent relations tests are class-consistent. At that point, selection is not correlated with nodal distance. Nodal distance, however, still influences reaction time. Thus the effects of nodal distance still influence emergent performances after classes are established. How, then, would nodal distance influence performances in new post-class formation transfer tests?
Single-option response transfer tests. Fields, Newman, Adams, and Verhave (1993) established two 5-member equivalence classes by training AB, BC, CD, and DE. The simple-to-complex training/testing protocol was used (Adams, Fields, & Verhave, 1993). All emergent relations tests were passed immediately. Therefore test performances did not vary with nodal distance. Then a single-option response transfer test was conducted. First the A1 and A2 stimuli were established as discriminanda for different responses. Thereafter the A through E stimuli in both classes were presented singly, in a random order under extinction conditions. Testing was repeated until complete and accurate transfer occurred.
Nine subjects were studied and two classes were tested per subject, yielding test results for 18 classes. For some subjects, the two classes yielded different outcomes. The classes were therefore analyzed individually. For some classes, responding immediately transferred on a completely class-consistent basis to the B, C, D, and E stimuli. Because a measurement ceiling had been reached, it was not possible to assess any nodal distance effects using the data from these classes. For the other classes, the likelihood of evoking the response trained to the A member of the class was an inverse function of the number of nodal stimuli that separated B, C, D, and E from A. With repeated testing, the frequency of class-consistent responding increased to 100% in an order that was a direct function of the nodal distance separating the B, C, D, and E stimuli from the A stimulus in each class.
Within-class preference tests. Fields, Adams, and Verhave (1989, May) studied the effects of nodal distance in a post-class formation within-class preference test that pitted different emergent relations against each other. After training AB, BC, CD, and DE for two sets of stimuli, between-class emergent relations tests (e.g., D1 as sample with B1 and B2 as comparisons) were passed demonstrating the formation of two 3-node, 5-member equivalence classes. Then they conducted within-class conditional discrimination preference tests where the sample and BOTH comparison stimuli came from the same class. In such a test, each comparison stimulus was separated from the sample by a different number of nodes. For example, a test trial could contain E1 as the sample with C1 and A1 as comparisons. In this test, E1 was separated from C1 by one node and E1 was separated from A1 by three nodes. This test enabled us to assess preferences for comparisons separated from a sample by one and three nodes. Choice of C1 would indicate a preference for the comparison that was separated from the sample by a smaller nodal distance, whereas choice of A1 would indicate a preference for the comparison separated by a larger nodal distance.
Table 3 All of the Within-Class Preference Tests Sample Sample Stimuli and and Co-N Co-F Sa Co-N Co-F EQV-1 EQV-2 D1 B1 A1 E1 C1 B1 EQV-1 EQV-3 E1 C1 A1 EQV-2 EQV-3 E1 B1 A1 TNS-1 TNS-2 A1 C1 D1 B1 D1 E1 TNS-1 TNS-3 A1 C1 E1 TNS-2 TNS-3 A1 D1 E1 SYM EQV-1 C1 B1 A1 D1 C1 B1 E1 D1 C1 SYM EQV-2 D1 C1 A1 E1 D1 B1 SYM EQV-3 E1 D1 A1 SYM TNS-1 B1 A1 D1 C1 B1 E1 SYM TNS-2 B1 A1 E1 Note. The first column indicates the type of emergent relation that is composed of the sample and the "near" comparison. EQV is an equivalence test, TNS is a transitivity test, and SYM is a symmetry test. The number after EQV and TNS specifies the number of nodes separating the stimuli in the relation. The second column indicates the type of emergent relation that is composed of the sample and the "far" comparison. The particular stimuli used as samples and comparisons in each trial are listed symbolically in the remaining columns. Co-N means the comparison that is nearer to the sample based on nodal distance. Co-F means the comparison that is farther from the sample based on nodal distance.
All of the within-class preference tests are listed in Table 3. The first column indicates the type of emergent relation that is composed of the sample and the "near" comparison. The second column indicates the type of emergent relation that is composed of the sample and the "far" comparison. The particular stimuli used in each test are listed symbolically in the remaining columns.
In each test, each of the eight subjects chose the comparison that was separated from the sample by fewer nodal stimuli. Thus the comparison selections on the within-class preference tests were an inverse function of nodal distance. The performances on each test occurred immediately and were maintained throughout testing. This preparation thus provided a steady-state measure of the effects of nodal distance after the formation of equivalence classes.
Barnes (1992, April) used a similar approach to study the effects of nodal distance on within-class preference test performances. The relations, AB, BC, CD, and DE, were trained with eight subjects. The stimuli represented by A-E were nonsense syllables. In separate trials that used lines of differing lengths as samples and comparisons, three shapes signaled choice of the comparison that was closest to the length of the sample line, most different from the sample line, or the line that was neither the most similar nor the most different relative to the sample line. The shapes were then used as instructional stimuli in the subsequent preference tests. Each preference trial contained a sample and three comparisons; all four stimuli were from the same class. The comparisons differed in terms of nodal distance from the sample. In addition, one of the three instructional cues was present. After extensive testing, comparison selection was correlated with nodal distance and the prevailing instructional stimulus. For instance, subjects chose the nonsense syllable that was nodally farthest from the nonsense syllable sample when the trial was presented with the shape that occasioned reinforcement for the choice of a comparison line that differed maximally in length from a sample line. These results replicate and extend our findings using within-class preference tests.
Multiple-option response transfer tests. The steady-state effects of nodal distance have also been demonstrated using a multiple-option response transfer test (Barnes, 1992, April). After training AB, BC, CD, and DE, a high rate of responding was reinforced in the presence of A1 and a low rate of responding was reinforced in the presence of El. The rates trained to the A2 and E2 stimuli were reversed. The differential response rates trained to the A and E stimuli within a class provided at least two response options for each of the remaining stimuli in a class. A response transfer test was then conducted by presenting each of the 10 stimuli alone. The rates of responding occasioned by the B, C, and D stimuli for a given class became stable and were an inverse function of the nodal distance that separated the B, C, and D stimuli from the A stimuli in the same class.
Summary of Nodal Distance Effects
Nodal distance influenced the performances occasioned by many different types of tests. The initial accuracy of performances on emergent relations tests, the initial degree of response transfer in postclass-formation transfer tests, preferences for one class member over others in post-class-formation within-class preference tests, the degree of class-consistent response transfer, and response rate in post-class-formation response transfer tests were all inverse functions of the nodal distance that separated the stimuli in an equivalence class. In addition, the order in which emergent relations reached criterion performance levels, the reaction times occasioned by emergent relations tests, and the order in which response transfer became completely class-consistent were all direct functions of nodal distance. Thus nodal distance influenced a wide range of performances occasioned by the stimuli in equivalence classes.
After the effects of nodal distance were no longer observed in the emergent relations tests, they still influenced the performances occasioned by subsequent transfer tests. Finally, the effects of nodal distance were transient within some testing conditions but were maintained within other testing conditions. These results show that the effects of nodal distance can become behaviorally silent but do not disappear as potential sources of behavioral control (Hearst, 1988). Thus, the effects of nodal distance may be essentially permanent. Because nodal distance is specified by the conditional relations that are trained to establish an equivalence class, the effects of nodal distance may be imparted during the training of equivalence classes. What factors, then, govern the transient and maintained effects of nodal distance?
Transient vs. Steady-State Effects of Nodal Distance
Steady-state performances correlated with nodal distance were produced by within-class preference tests, and the multiple-option response transfer test. Transient performances correlated with nodal distance were produced by emergent relations tests and single-option response transfer tests. What functional parameters of these training and testing procedures were responsible for these differences in performance?
Within-class preference tests. The within-class preference tests used two or more comparison stimuli on each test trial; all belonged to the same class as the sample. This required subjects to respond differentially to the stimuli within a class on each test trial. Thus the effects of a within-class parameter, nodal distance, could be assessed. Because there were no comparisons from a class that differed from the sample, subjects did not have the option of selecting a comparison stimuli indicative of a between-class discrimination. Thus between-class discriminative processes could not interfere with the within-class effects of nodal distance. As a result, performances correlated with nodal distance were maintained.
Multiple-option response transfer tests. After being trained to respond at different rates to the A and E stimuli in one class, subjects were presented with the A, B, C, D, and E stimuli in a transfer test. In the transfer test, each stimulus in the class was presented alone. The response rates occasioned by the B, C, and D stimuli were correlated with nodal distance. The different rates of responding that had been reinforced in the presence of the A and the E stimuli served as behavioral anchors that provided a means of scaling the B, C, and D stimuli over the A and E stimuli. As with the preference tests, the availability of at least two response options per class enabled steady-state performances that were correlated with the effects of nodal distance. Although present with the A and E stimuli, there were no between-class contingencies in tests with the B, C, and D stimuli; thus, between-class discriminative performances did not interfere with the performances occasioned by the B, C, and D stimuli that were governed by nodal distance.
Emergent relations tests. On an emergent relations test trial, only one comparison stimulus was presented from any class; the remaining comparisons were from other classes. It was not possible to make selections between stimuli in the same class on any test trial. In addition, between-class discriminative performances were reinforced on the training trials used to establish the baseline conditional relations for the classes. The comparison stimuli available on a test trial and the historical contingencies favored between-class discriminative performances relative to performances influenced by nodal distance. When emergent relations test performances were class-consistent from the start, the between-class factors completely overshadowed the effects of nodal distance. When effects of nodal distance were transient, they were seen at the start of testing and then dissipated with test repetition. The decline in the effect of nodal distance on performance reflected an increase in control exerted by a set of class-based discriminations. Between-class contingencies thus increasingly overshadowed the effects of nodal distance.
Single-option response transfer tests. The performances observed during the single-option response transfer tests either were not correlated with nodal distance or were correlated on a transient basis. Before the response transfer tests were conducted, only one response was trained to a stimulus in a class. In the response transfer tests, subjects could emit different responses for different classes, but they could not emit different responses to the stimuli in the same class. The available response options and the pretransfer discrimination training history strongly supported between-class responding on the transfer tests. Effects of nodal distance either did not appear during the transfer tests or appeared early and were then overshadowed by the performances controlled by the current between-class contingencies and the history of responding on a between-class basis. If test conditions minimize the effects of interfering variables, then performance will be correlated with nodal distance.
To summarize, tests that allowed a subject to make comparisons between the stimuli within a class yielded sustained performances that were correlated with nodal distance. Tests that supported discriminations between classes yielded performances that were correlated with nodal distance on a transient basis, or immediate criterion level performances that could not be correlated with nodal distance. The effects of nodal distance then interacted with the conditions of training and testing. Its effects were maintained when at least two responses could be evoked by the stimuli in a given class. Its effects were transient when only one response could be evoked by the stimuli in a given class. Thus, a comprehensive characterization of an equivalence class requires a specification of its structural parameters as well as the functional parameters used for training and for testing.
Substitutability and the Relatedness of Stimuli
Equivalence classes have been characterized as being sets of stimuli that are interchangeable or substitutable for each other (Devaney, Hayes, & Nelson, 1986; Dixon & Spradlin, 1976; Saunders & Green, 1992; Sidman, 1990; Sidman & Tailby, 1982; Spradlin & Dixon, 1976; Spradlin & Saunders, 1984, 1986). Substitutability has one operational referent; on a given test trial, any comparison stimulus is selected as long as it is from the same class as the prevailing sample stimulus.
The term substitutability is also used as an intervening variable (Sidman, 1990). Used in this fashion, substitutability implies that all of the stimuli in an equivalence class are related to each other on the basis of equality. Any function acquired by one stimulus in a class should thus transfer equally to all of the remaining class members. This implies that structural parameters should not influence the performances occasioned by the stimuli in an equivalence class. The data cited in this paper show that test performances are influenced by two structural parameters of equivalence classes, nodal distance and directionality of training. Thus the stimuli in an equivalence class are unequally or partially substitutable for each other. More simply stated, the stimuli in an equivalence class are differentially related to each other. The relatedness of the stimuli is an inverse function of nodal distance and is also influenced by directionality of training.
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