Computational complementarity

Interactivity generates paradox in that the interactive control by one systemC of predicates about another system-under-studyS may falsify these predicates. We formulate an “interactive logic” to resolve this paradox of interactivity. Our construction generalizes one, the Galois connection, used by Von Neumann for the similar quantum paradox. We apply the construction to atransition system, a concept that includes general systems, automata, and quantum systems. In some (classical) automataS, the interactive predicates aboutS show quantumlike complementarity arising from interactivity: The interactive paradox generates the quantum paradox. Some classicalS's have noncommutative algebras of interactively observable coordinates similar to the Heisenberg algebra of a quantum system. SuchS's are “hidden variable” models of quantum theory not covered by the hidden variable studies of Von Neumann, Bohm, Bell, or Kochen and Specker. It is conceivable that some quantum effects in Nature arise from interactivity.