On Probabilistic Distributed Strategies

Abstract

Abstract. In a distributed game we imagine a team Player engaging a team Opponent in a distributed fashion. No longer can we assume that moves of Player and Opponent alternate. Rather the history of a play more naturally takes the form of a partial order of dependency between occurrences of moves. How are we to define strategies within such a game, and how are we to adjoin probability to such a broad class of strategies? The answer yields a surprisingly rich language of probabilistic distributed strategies and the possibility of programming (optimal) probabilistic strategies. Along the way we shall encounter solutions to: the need to mix probability and nondeterminism; the problem of parallel causes in which members of the same team can race to make the same move, and why this leads us to invent a new model for the semantics of distributed systems.

Full abstraction for the quantum lambda-calculus

“…This is an immediate consequence of the monotone condition for quantum strategies, which in the absence of quantum spaces boils down to the conditions on probabilistic strategies [Winskel 2015]. Using this, we prove convergence by exploiting that we can łtrace outž any strategy:…”

“…is performed monomial per monomial, so this amounts to the condition in ] applied separately for each monomial. This adapts and extends the inclusion-exclusion principle used for probabilistic strategies [Winskel 2015], the reader is directed to for more details and intuitions. Using the compact closed structure of CPM, any Q (x) ∈ CPM(H (x A ), H (x B )) can be reorganised as a map in CPM(H (σx) − , H (σx) + ), from the Hilbert space corresponding to the negative events to those for the positive events.…”