Cognitive Complexity in Matrix Reasoning Tasks Marco Ragni (ragni@cognition.uni-freiburg.de) Department of Cognitive Science, University of Freiburg, Germany Philip Stahl (stahl@informatik.uni-freiburg.de) Department of Computer Science University of Freiburg, Germany Thomas Fangmeier (fangmeier@cognition.uni-freiburg.de) Department of Cognitive Science University of Freiburg, Germany Abstract Reasoning difficulty for items in IQ-tests is generally deter- mined empirically: The item difficulty is measured by the number of reasoners who are able to solve the problem. Al- though this method has proven successful, (nearly all IQ-Tests are designed this way) – it is desirable to have an inherent for- mal measure reflecting the reasoning complexity involved. In this article, we analyze and classify geometrical analogy rea- soning problems. Based on the types of functions necessary to solve these problems, a complexity measure is introduced, which reliably captures human reasoning difficulty. Finally, our complexity measure is compared to the empirical difficulty ranking as determined by Cattells Culture Fair Test and Evans Analogy problems. Keywords: Cognitive Complexity; Analogical Reasoning; Geometric Analogies Introduction For the past hundred year human intelligence has mostly been tested by use of IQ-tests (Binet & Simon, 1905). Geomet- rical analogy problems (cf. Fig. 1) are part of a number of IQ-tests, for example the Hamburg-Wechsler-Intelligence- Test (Wechsler, Hardesty, Lauber, & Bondy, 1961). A signifi- cant number of IQ-tests even consist exclusively of such geo- metrical reasoning problems, e.g., Cattell’s Culture Fair Test (K. Cattell & Cattell, 1959) or Ravens’ Standard Progressive Matrices (Raven, Raven, & Court, 2000) and Advanced Pro- gressive Matrices (Raven, 1962). Such problems are some- times classified as culture fair (R. Cattell, 1968) as they re- quire less declarative knowledge than for instance word anal- ogy problems. While the success in solving word analogy problems can depend on additional knowledge, geometrical reasoning problems can be modeled using mathematical func- tions exclusively. For this reason these problems are more accessible in formal terms than other analogy problems. An individuals intelligence is always measured by determining the deviation of his or her performance on a given set of rea- soning problems, from a particular group (specific age and educational status, etc.). Problems in turn are classified em- pirically as simple or challenging, based on whether a given population is able to solve most or only a limited number of similar problems. While it is possible to empirically capture the human reasoning difficulty – it seems more desirable to identify the characteristics of such problems formally. Such a formal characterization may elucidate future test develop- ment and can then form a formal foundation of reasoning complexity. An analysis of the IQ-Test problems of Raven has been conducted by Carpenter, Just, and Shell (1990). Fig- ure 1 is an example, variations of which can be found in pop- ular literature (e.g., Eysenck, 1962; Russell & Carter, 1994). ? Figure 1: An example of a geometric reasoning problem. The task is to fill in one of the four answers below. The correct solution is the third one. All figures and problems in the fol- lowing were designed by the authors to protect the security of IQ-tests. A side-effect of such a formal classification is that mental operations or functions must be classified as easier or more difficult for the human reasoner to apply. Another aspect is the creation of new, different reasoning problems with the same reasoning difficulty. A formal classification in turn re- quires a computational model. This approach, cognition as computation, was coined and introduced by Newell and Si- mon (1972). Once problems are formally represented and functionally classified, the computational requirements nec- essary to solve them can be calculated. In this article we put In: Kokinov, B., Karmiloff-Smith, A., Nersessian, N. J. (eds.) European Perspectives on Cognitive Science. © New Bulgarian University Press, 2011 ISBN 978-954-535-660-5