Acquisition and Use of Mental Operators: Multinomial Modeling versus ACT-R

Knowledge about operators, about the conditions of their applicability, and about their effects is essential for effective interaction with the physical world. By combining two defining dimensions of this knowledge – abstractness of content and directionality of access – we can distinguish four classes of representational units: rules, instances, episodes, and structures. We present a multinomial model that measures the characteristics of these units. This model was applied to an experiment on the acquisition and use of alphabet-arithmetic operators (Müller & Gehrke, 2002). The multinomial model could be fitted very well to the data and allows calculating the proportions of the different kinds of mental operators. To compare these findings with a simulation of the specific cognitive processes, we developed an ACT-R model. Separating four cases of information processing in correspondence to the knowledge units in the multinomial model confirmed the estimates of the multinomial analysis. This finding demonstrates the usefulness of multinomial modeling as a statistical tool to investigate cognitive processes. Also, it provides converging evidence for the use of different kinds of knowledge, even in simple tasks. Units of Mental Operators Knowledge about causes and their effects is essential for our successful interaction with the physical world. More specifically, knowledge about operations, about the preconditions of their applicability and the resulting effects is crucial for successful planning. The term mental operator refers to this kind of knowledge throughout this paper. The content of a mental operator can be separated into three structurally different parts, namely the preconditions of applicability, the representation of the referred operation(s), and the effect or consequences of the application of the operator. Accordingly, three main types of knowledge use can be identified that require the specification of one of these parts if information about the other two parts is given: 1) Prognosis tasks, which require that the effect of operator application has to be predicted. An important instance of this kind of use is finding a way to solve a problem by forward chaining in problem space. 2) Retrognosis tasks, which require that the preconditions of operator application be identified. This kind of knowledge use is important if one tries to find a solution in problem space by backward chaining. 3) A third class of tasks requires the identification of an operation by comparing information about the initial state with information about the resulting effects. This kind of knowledge use is important if one has to select an appropriate action to achieve an intended effect. It is also involved in diagnosis tasks that require the identification of an operation as cause of an observed effect. In the literature on problem solving and skill acquisition, different formats are proposed as representational units of mental operators. Four main classes of mental operators can be distinguished: production rules (e.g. Anderson & Lebiere, 1998), instances (e.g. Logan, 1988), episodes (= chunks in SOAR, Newell, 1990), and conceptual structures (e.g. Müller, 1999). These representational units can be classified by crossing directionality of access and abstractness of content as two essential characteristics of mental operators (see Table 1). Table 1: Units of Mental Operators Ordered by Abstractness of Content and Directionality of Access.