Research projects

Project CEMAPRE internal

TitleCombinatorial Game Theory: Simplicity rule of order n
ParticipantsAlda Carvalho (Principal Investigator), Carlos Santos
SummaryDedekind 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

[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.