We investigate the problem of type isomorphisms in the presence of higher-order references. We first introduce a finitary programming language with sum types and higher-order references, for which we build a fully abstract games model following the work of Abramsky, Honda and McCusker. Solving an open problem by Laurent, we show that two finitely branching arenas are isomorphic if and only if they are geometrically the same, up to renaming of moves (Laurent's forest isomorphism). We deduce from this an equational theory characterizing isomorphisms of types in our language. We show however that Laurent's conjecture does not hold on infinitely branching arenas, yielding new non-trivial type isomorphisms in a variant of our language with natural numbers.

• 68Q45 - Formal languages and automata
• 68Q60 - Specification and verification (program logics, model checking, etc.)
• 68Q85 - Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
• 91A05 - 2-person games
• 91A80 - Applications of game theory