Polarisation describes in logic the presence of an evaluation order by
distinguishing positive connectives and negative connectives at a
formal level. This approach has recently been characterised with
duploids [1], a generalisation of categories where the associativity
of composition is relaxed.
Following Melliès [2], we understand the Blass problem of game
semantics (lack of associativity) as a polarisation of games. In this
talk we exhibit duploid structures at work in Rideau and Winskel's
concurrent games [3]. To this end we introduce “duploid situations”,
an equivalent but more verbose characterisation of duploids, and
exhibit such structures in concurrent games.
Previous notions of polarised games force the evaluation order
indirectly by restricting to a call-by-push-value format, and then
exhibit the Blass phenomenon via some form of CPS translation. This
amounts to constructing a duploid but the construction is a
completion. In contrast, the new duploids are a conservative extension
of structures found in concurrent games. In other words, we describe
the polarisation of games directly without forcing sequentiality
artificially, and the new notion is a convenient description to help
recognise duploids as they occur in practice.
[1] G. Munch-Maccagnoni, “Models of a Non-Associative Composition,” in
Proc. FoSSaCS, 2014, vol. 8412, pp. 397–412.
[2] P.-A. Melliès, “Asynchronous Games 3 An Innocent Model of Linear
Logic,” Electr. Notes Theor. Comput. Sci., vol. 122, pp. 171–192,
2005.
[3] S. Rideau and G. Winskel, “Concurrent Strategies,” in LICS, 2011,
pp. 409–418.