Abstract
In experimental psychology, artificial grammars, generated by directed graphs, are used to test the ability of subjects to implicitly learn the structure of complex rules. We introduce the necessary notation and mathematics to view an artificial grammar as the sequence space of a dynamical system. The complexity of the artificial grammar is equated with the topological entropy of the dynamical system and is computed by finding the largest eigenvalue of an associated transition matrix. We develop the necessary mathematics and include relevant examples (one from the implicit learning literature) to show that topological entropy is easy to compute, well defined, and intuitive and, thereby, provides a quantitative measure of complexity that can be used to compare data across different implicit learning experiments.
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REFERENCES
Adler, R., Konheim, A., & McAndrew, M. (1965). Topological entropy. Transactions of the American Mathematical Society, 114, 309-319.
Blahut, R. E. (1988). Principles and practice of information theory. Reading, MA: Addison-Wesley.
Cohen, A., Ivry, R. I., & Keele, S. W. (1990). Attention and structure in sequence learning. Journal of Experimental Psychology: Learning, Memmory, and Cognition, 16, 17-30.
Gantmacher, F. R. (1974). The theory of matrices, Volume II. New York, NY: Chelsea Publishing, Co.
Higham, P. A. (1997). Dissociations of Grammaticality and Specific Similarity Effects in Artificial Grammar Learning. Journal of Experimental Psychology: Learning, Memory, and Cognition, 23, 1029-1045.
Mathews, R. C., Buss, R. R., Stanley, W. B., Blanchard-Fields, F., Cho, J. R., & Druhan, B. (1989). Role of implicit and explicit processes in learning from examples: a synergistic effect. Journal of Experimental Psychology: Learning, Memory, and Cognition, 15, 1083-1100.
Reber, A. (1989). Implicit learning and tacit knowledge. Journal of Experimental Psychology: General, 118, 219-235.
Reber, A. & Allen, R. (1978). Analogic and abstraction strategies in synthetic grammar learning: A functionalist interpretation. Cognition, 6, 193-221.
Reed, J. M. & Johnson, P. J. (1994). Assessing implicit learning with indirect tests: Determining what is learned about sequence structure. Journal of Experimental Psychology: Learning, Memory, and Cognition, 20, 585-594.
Reed, J. M. & Johnson, P. J. (1998). Implicit Learning: Methodological Issues and Evidence of Unique Characteristics. In Stadler, M. A. & Frensch, P. A. (Eds.), Handbook of Implicit Learning Thousand Oaks, CA: Sage Publications, Inc. 261-294.
Robinson, C. (1995). Dynamical systems: Stability, symbolic dynamics, and chaos. Ann Arbor, MI: CRC Press.
Shannon, C. E. (1948). The mathematical theory of Communications. Bell Systems Technical Journal, 27, 379-423.
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Bollt, E.M., Jones, M.A. The Complexity of Artificial Grammars. Nonlinear Dynamics Psychol Life Sci 4, 153–168 (2000). https://doi.org/10.1023/A:1009524428448
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DOI: https://doi.org/10.1023/A:1009524428448