Ordinal sums of impartial games

• Autor(a) principal: Carvalho, Alda
• Data de Publicação: 2018
• Outros Autores: Neto, João, Santos, Carlos
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10400.21/9108
• topic: Combinatorial game theory; Gin sum; Impartial games; Minimum excluded value; Normal-play; OAK; Ordinal sum

In an ordinal sum of two combinatorial games G and H, denoted by G : H, a player may move in either G (base) or H (subordinate), with the additional constraint that any move on G completely annihilates the component H. It is well-known that the ordinal sum does not depend on the form of its subordinate, but depends on the form of its base. In this work, we analyze g(G : H) where G and H are impartial forms, observing that the g-values are related to the concept of minimum excluded value of order k. As a case study, we introduce the ruleset OAK, a generalization of GREEN HACKENBUSH. By defining the operation gin sum, it is possible to determine the literal forms of the bases in polynomial time. (C) 2017 Elsevier B.V. All rights reserved.