## Project CEMAPRE internal

Title | Combinatorial Game Theory: Simplicity rule of order n |

Participants | Alda Carvalho (Principal Investigator), Carlos Santos |

Summary | Dedekind cuts are a method of construction of the real numbers from the rationals. An irrational Dedekind cut is a partition of the rational numbers into two non-empty sets A and B, such that all elements of A are less than all elements of B, A contains no greatest element among the rationals, and B contains no smallest element among the rationals. A cut of that kind defines a unique irrational number which, loosely speaking, fills the «gap» between A and B. It is well-known that, in Combinatorial Game Theory, dyadics are constructed day-by-day, recursively, from the previously born dyadics; Conway's approach only needs the empty set and an appropriate order relation. Using analogy, a combinatorial form G is a dyadic if it fills the gap between its options G^R_i and G^L_j. Given a short form G such that all options are dyadics and G^L_i < G^R_j, there is an infinite number of dyadics that fill the gap (G is a short form and, because of that, it has finite sets of options). The well-known simplicity rule indicates how to choose "the one", being a fundamental concept for the development of Combinatorial Game Theory [1, 2, 3, 4]. There are combinatorial rulesets that seem to need generalizations of the simplicity rule. Variations of the classical Hackenbush are examples of that. This project proposes to study the mathematical properties of this kind of generalizations, as well as their applications in game practice. References: [1] J. Conway. On Numbers and Games. Academic Press, 1976. [2] E. Berlekamp, J. Conway, R. Guy. Winning Ways. Academic Press, London, 1982. [3] M. Albert, R. Nowakowski, D. Wolfe. Lessons in Play: An Introduction to Combinatorial Game Theory. A. K. Peters, 2007. [4] A. N. Siegel. Combinatorial Game Theory, American Math. Soc., 2013. |