Any mathematical pattern can be the generation principle for number series. In con-
trast to most of the application fields of artificial neural networks (ANN) a successful
solution does not only require an approximation of the underlying function but to correctly
predict the exact next number. We propose a dynamic learning approach and evaluate our
method empirically on number series from the Online Encyclopedia of Integer Sequences.
Finally, we investigate research questions about the performance of ANNs, structural prop-
erties, and the adequate architecture of the ANN to deal successfully with number series.
Solving number series poses a challenging problem for humans and Artificial Intelligence
Systems. The task is to correctly predict the next number in a given series, in accordance
with a pattern inherent to that series. We propose a novel method based on Artificial
Neural Networks with a dynamic learning approach to solve number series problems. Our
method is evaluated on an own experiment and over 50.000 number series from the Online
Encyclopedia of Integer Sequences (OEIS) database.
Human reasoners have an impressive ability to solve analogical reasoning problems and they still outperform computational systems. Analogical reasoning is relevant in dealing with intelligence tests. There are two kinds of approaches: to solve IQ-test problems in a way similar to humans (i.e., a cognitive approach) or to solve these problems optimally (i.e., the AI approach). Most systems can be associated with one of these approaches. Detailed systems solving geometrical intelligence tests, explaining cognitive operations based on working memory and producing precise predictions and results such as error rates and response times have not been developed so far. We present a system implemented in the cognitive architecture ACT-R, able to solve analogously developed problems of Raven’s Standard and Advanced Progressive Matrices. The model solves 66 of the 72 tested problems of both tests. The model’s predicted error rates correlate to human performance with r = .8 for the Advanced Progressive Matrices and r = .7 for all problems together.
Positioning. The article, published at the European Conference on Artificial Intelligence 2012, presents an ACT-R model that is able to solve Raven’s Matrix Test.
Research Question. Is it possible to explain the limitations in human reasoning in IQ- tests based on working memory limitations?
Method. Cognitive modeling; formal analysis
Results. The cognitive model in ACT-R, is able to solve analogously developed problems of Raven’s Standard and Advanced Progressive Matrices. The model solves 66 of the 72 tested problems of both tests. The model’s predicted error rates correlate to human performance with r = .8 for the Advanced Progressive Matrices and r = .7 for all problems together.
Ragni, M., & Neubert, S. (2012). Solving Raven’s IQ-tests: An AI and Cognitive Modeling Approach. In L. D. Raedt et al. (Eds.), Proceedings of the 20th European Conference on Artificial Intelligence (Vol. 242, pp. 666–671). Amsterdam: IOS Press.
Positioning. The article, published as a book chapter in the Proceedings of the German Conference on Artificial Intelligence 2011, proposes to use Artificial Neural Networks to solve number series problems.
Research Question. Is it possible to develop a cognitive system able to solve number series problems of intelligence test or the 50 000 problems in the Online Encyclopedia of Integer Series?
Method. Dynamic training method for Artificial Neural Network
Results. Using a dynamic learning approach the approach can solve 26 951 of 57 524 number series.
Ragni, M., & Klein, A. (2011). Predicting numbers: An AI approach to solving number series. In S. Edelkamp & J. Bach (Eds.), KI-2011. Springer LNAI, Heidelberg.
Positioning. The article, published at the International Joint Conference on Computational Intelligence (IJCI 2011), investigates if we can solve classical intelligence test problems and what parameters fit the best.
Research Question. What kind of architectural and formal properties can have an influence on successful artificial neural networks solving number series?
Method. Formal analysis of artificial neural networks
Results. Systematically testing the best parameters (number of input nodes, hidden nodes, and learning rate) shows that the structure of the Artificial Neural Networks can determine the success of solving a number sequence: 2-4 input nodes and about 5-6 hidden nodes provide the best framework to solve number series. By allowing approximations (deviations of ± 5) be improved to solve about 39 000 number series of the Online Encyclopedia.
Ragni, M., & Klein, A. (2011b). Solving Number Series – Architectural Properties of Successful Artificial Neural Networks. In K. Madani, J. Kacprzyk, & J. Filipe (Eds.), NCTA 2011 – Proceedings of the International Conference on Neural Computation Theory and Applications(pp. 224–229). SciTePress.
Positioning. The article, published in the Journal Cognitive Systems Research, presents an overview and characterization of complex cognition.
Content. Complex cognition addresses research on: “(a) high-level cognitive processes – mainly problem solving, reasoning, and decision making – and their interaction with more basic processes such as perception, learning, motivation and emotion and (b) cognitive processes which take place in a complex, typically dynamic, environment.” The article presents an overview about past and current research and a precision of the definition of complex cognition from both an AI and psychological perspective. There is a great emphasis on the challenges for cognitive systems. Complex cognition goes far beyond simple cognitive processes and requires all possible methods from cognitive science research: from analytical, empirical, to engineering methods. The article finally presents challenges for complex problem solving, dynamic decision making, and finally learning of concepts, skills and strategies.
Schmid, U., Ragni, M., Gonzalez, C., & Funke, J. (2011). The challenge of complexity for cognitive systems. Cognitive Systems Research, 12(3-4), 211-218.
Positioning. The article, published in the Journal Cognitive Systems Research, investigates an important complex cognition task: the dynamic stocks and flows task.
Method. Formal analysis; artificial neural networks; cognitive and computational modeling.
Results. The dynamic stock and flow task – an exploration problem – is first formally generalized to a general stock and flow task with an arbitrary number of tanks that have to be controlled for. As the system requires learning and a highadaptability it proposes to use a computational approach based on artificial neural networks combined with heuristics. The system is evaluated on all problems from the DSF-challenge and yields satisfactorily results.
Ragni, M., Steffenhagen, F., & Klein, A. (2011). Generalized dynamic stock and flow systems: An AI approach. Cognitive Systems Research, 12(3-4), 309-320.
Positioning. The article, published at the Proceedings of the 33rd Annual Conference of the Cognitive Science Society, develops an ACT-R model for the four problem classes for the Tower of London task mentioned in Kaller et al. (2011).
Method. Cognitive modeling; heuristic analysis
Results. The model can replicate the empirical results of Kaller et al. (2011) satisfactorily and introduces structural patterns for the first time. Additionally, representational aspects can be responsible for the used heuristics of the participants. This provides later the foundation for the more complex Rush Hour problem and another visual representation aspect: the Gestalt principles.
Albrecht, R., Brüssow, S., Kaller, C., & Ragni, M. (2011). Using a Cognitive Model for an In-Depth Analysis of the Tower of London. In L. Carlson, C. Hoelscher, & T. F. Shipley (Eds.), Proceedings of the 33rd Annual Conference of the Cognitive Science Society (pp. 693–698). Austin, TX: Cognitive Science Society.
Positioning. The article, published at the cognitive science conference 2011, investigates if it is possible to find a complexity measure reflecting the human reasoning difficulty in solving planning problems such as Rush Hour.
Method. Formal analysis; psychological experiment
Results. A measure of cognitive complexity must take operational aspects of human information processing into account. The article proposes a structural complexity measure that is based on the number and connectedness of subgoals necessary to solve a problem. It discusses several measures for assessing the goodness of a solution. The best fitting formula correlates for about r = .77 (p < .0001).
Ragni, M., Steffenhagen, F., & Fangmeier, T. (2011). A Structural Complexity Measure for Predicting Human Planning Performance. In L. Carlson, C. Hoelscher, & T. Shipley (Eds.), Proceedings of the 33rd Annual Conference of the Cognitive Science Society (pp. 353–2358). Austin, TX: Cognitive Science Society.
Positioning. The article, to be submitted to the Journal Spatial Reasoning and Computation, investigates human solutions by parametrized cognitive agents and tries to replicate the human behavior based on specific characteristics such as working memory limit etc.
Method. Computational analysis; cognitive modeling
Results. The article proposes to combine methods from bioinformatics to compute the relative similarity between the agent and humans play. The cognitive agents can predict per player to reach a relative similarity of 44% and at maximum a mean similarity of up to 51 %. In some single tasks, agents reached an absolute similarity of up to 17 sequential states resulting in a relative similarity of 94 %. Some aspects that could not be explained by the cognitive agent may depend again on Gestalt principles.
Steffenhagen, F., & Ragni, M. (submitted). Analyzing Preferred Planning Strategies in Spatial Problem Solving: Rush Hour as an Example. Spatial Reasoning and Computation.