Empirically, generalization between a training and a test stimulus falls off in close approximation to an exponential decay function of distance between the two stimuli in the "stimulus space" obtained by multidimensional scaling. Mathematically, this result is derivable from the assumption that an individual takes the training stimulus to belong to a "consequential" region that includes that stimulus but is otherwise of unknown location, size, and shape in the stimulus space (Shepard, 1987). As the individual gains additional information about the consequential region-by finding other stimuli to be consequential or nOl-the theory predicts the shape of the generalization function to change toward the function relating actual probability of the consequence to location in the stimulus space. This paper describes a natural connectionist implementation of the theory, and illustrates how implications of the theory for generalization, discrimination, and classification learning can be explored by connectionist simulation. 1 THE THEORY OF GENERALIZATION Because we never confront exactly the same situation twice, anything we have learned in any previous situation can guide us in deciding which action to take in the present situation only to the extent that the similarity between the two situations is sufficient to justify generalization of our previous learning to the present situation. Accordingly, principles of generalization must be foundational for any theory of behavior. In Shepard (1987) nonarbitrary principles of generalization were sought that would be optimum in any world in which an object, however distinct from other objects, is generally a member of some class or natural kind sharing some dispositional property of potential consequence for the individual.

Spherical Units can be used to construct dynamic reconfigurable consequential regions, the geometric bases for Shepard's (1987) theory of stimulus generalization in animals and humans. We derive from Shepard's (1987) generalization theory a particular multi-layer network with dynamic (centers and radii) spherical regions which possesses a specific mass function (Cauchy). This learning model generalizes the configural-cue network model (Gluck & Bower 1988): (1) configural cues can be learned and do not require pre-wiring the power-set of cues, (2) Consequential regions are continuous rather than discrete and (3) Competition amoungst receptive fields is shown to be increased by the global extent of a particular mass function (Cauchy). We compare other common mass functions (Gaussian; used in models of Moody & Darken; 1989, Krushke, 1990) or just standard backpropogation networks with hyperplane/logistic hidden units showing that neither fare as well as models of human generalization and learning.

Empirically, generalization between a training and a test stimulus falls off in close approximation to an exponential decay function of distance between the two stimuli in the "stimulus space" obtained by multidimensional scaling. Mathematically, this result is derivable from the assumption that an individual takes the training stimulus to belong to a "consequential" region that includes that stimulus but is otherwise of unknown location, size, and shape in the stimulus space (Shepard, 1987). As the individual gains additional information about the consequential region-by finding other stimuli to be consequential or nOl-the theory predicts the shape of the generalization function to change toward the function relating actual probability of the consequence to location in the stimulus space. This paper describes a natural connectionist implementation of the theory, and illustrates how implications of the theory for generalization, discrimination, and classification learning can be explored by connectionist simulation. 1 THE THEORY OF GENERALIZATION Because we never confront exactly the same situation twice, anything we have learned in any previous situation can guide us in deciding which action to take in the present situation only to the extent that the similarity between the two situations is sufficient to justify generalization of our previous learning to the present situation. Accordingly, principles of generalization must be foundational for any theory of behavior. In Shepard (1987) nonarbitrary principles of generalization were sought that would be optimum in any world in which an object, however distinct from other objects, is generally a member of some class or natural kind sharing some dispositional property of potential consequence for the individual.

Spherical Units can be used to construct dynamic reconfigurable consequential regions, the geometric bases for Shepard's (1987) theory of stimulus generalization in animals and humans. We derive from Shepard's (1987) generalization theory a particular multi-layer network with dynamic (centers and radii) spherical regions which possesses a specific mass function (Cauchy). This learning model generalizes the configural-cue network model (Gluck & Bower 1988): (1) configural cues can be learned and do not require pre-wiring the power-set of cues, (2) Consequential regions are continuous rather than discrete and (3) Competition amoungst receptive fields is shown to be increased by the global extent of a particular mass function (Cauchy). We compare other common mass functions (Gaussian; used in models of Moody & Darken; 1989, Krushke, 1990) or just standard backpropogation networks with hyperplane/logistic hidden units showing that neither fare as well as models of human generalization and learning.

Empirically, generalization between a training and a test stimulus falls off in close approximation to an exponential decay function of distance between the two stimuli in the "stimulus space" obtained by multidimensional scaling. Mathematically, thisresult is derivable from the assumption that an individual takes the training stimulus to belong to a "consequential" region that includes that stimulus but is otherwise of unknown location, size, and shape in the stimulus space (Shepard, 1987). As the individual gains additional information about the consequential region-by finding other stimuli to be consequential or nOl-the theory predicts the shape of the generalization function to change toward the function relating actual probability of the consequence to location in the stimulus space. This paper describes a natural connectionist implementation of the theory, and illustrates how implications of the theory for generalization, discrimination, and classification learning can be explored by connectionist simulation. 1 THE THEORY OF GENERALIZATION Because we never confront exactly the same situation twice, anything we have learned in any previous situation can guide us in deciding which action to take in the present situation only to the extent that the similarity between the two situations is sufficient to justify generalization of our previous learning to the present situation. Accordingly, principles of generalization must be foundational for any theory of behavior. In Shepard (1987) nonarbitrary principles of generalization were sought that would be optimum in any world in which an object, however distinct from other objects, is generally a member of some class or natural kind sharing some dispositional property of potential consequence for the individual. A newly encountered plant or animal might be edible or 665 666 Shepard and Kannappan poisonous.

Spherical Units can be used to construct dynamic reconfigurable consequential regions, the geometric bases for Shepard's (1987) theory of stimulus generalization in animals and humans. We derive from Shepard's (1987) generalization theory a particular multi-layer network with dynamic (centers and radii) spherical regions which possesses a specific mass function (Cauchy). This learning model generalizes the configural-cue network model (Gluck & Bower 1988): (1) configural cues can be learned and do not require pre-wiring the power-set of cues, (2) Consequential regions are continuous rather than discrete and (3) Competition amoungst receptive fields is shown to be increased by the global extent of a particular mass function (Cauchy). We compare other common mass functions (Gaussian; used in models of Moody & Darken; 1989, Krushke, 1990) or just standard backpropogation networks with hyperplane/logistic hidden units showing that neither fare as well as models of human generalization and learning.