The Neural Algebra and its Impact on Design and Test of Intelligent Systems

Abstract

The Graph Model of Combinatory Logic (Engeler, 1981) is also a mathematical model for "how does the brain think". It attempts to explain how complex scripts of behavior and conceptual content can reside in, combine, and interact on large neural networks (Engeler 2019). This has an impact on building intelligent systems that interact with humans. Intelligent systems should employ the same kind of concepts humans do; otherwise, their actions remain incomprehensible and erratic to human users, concepts can be represented in the Graph Model by using the "Lambda-Theorem" found be Barendregt in 1977.Both the Graph Model and the mathematical model for the human brain have been published as part of Theoretical Computer Science and remain thus out of the reach of normal AI engineers. Nevertheless, they suggest solutions for today's problems with Intelligent Systems, such as autonomous vehicles mastering the traffic in Palermo, or robots caring for people and working together with them. Intelligent Systems need to adhere to concepts quite similar to humans that follow certain rules in their behavior.The paper explains what a “Concept” is in AI, how to state requirements for AI, and how to test them. Intelligent Systems using concepts behave similar to humans, following rules but are still able to break the rules when need arises, and can be certified for safety and security; solving certain difficulties for Learning Machines that can learn and unlearn.ReferencesH. P. Barendregt, "The Type-Free Lambda-Calculus," in Handbook of Math. Logic, vol. 90, J. Barwise, Ed., Amsterdam, North Holland, 1977, pp. 1091 -1132.E. Engeler, "Algebras and Combinators," Algebra Universalis, vol. 13, pp. 389-392, 1981. E. Engeler, "Neural algebra on "How does the brain think?"," Theoretical Computer Science, vol. 777, pp. 296-307, 2019. T. M. Fehlmann, Autonomous Real-time Testing – Testing Artificial Intelligence and Other Complex Systems, Berlin, Germany: Logos Press, 2020. H. Curry and R. Feys, Combinatory Logic, Vol. I, Amsterdam: North-Holland, 1958. K. Bimbó, Combinatory Logic - Pure, Applied and Typed, Boca Raton, FL: CRC Press, 2012.

Keywords: Combinatorial Logic, The Graph Model, Neural Algebra, Artificial Intelligence, Learning Machines, Intelligent Systems.

DOI: 10.54941/ahfe1004475