Synthesis, Alternation and Algebra: Exploring Regular Systems and Games

Licentiate thesis, 2026

This thesis explores the role of alternation, the structured interplay between existential and universal choice, in synthesis and formal reasoning about reactive systems. The first contribution shows that the supervisory control synthesis problem reduces to solving B¨uchi games via a compact and uniform reduction, improving the known complexity bound and establishing a formal bridge to reactive synthesis. The second contribution introduces Alternating Kleene Algebra (AKA), a sound and complete equational axiomatization for regular expressions with intersection, resolving an open problem of Antimirov and achieving optimal proof complexity. Together, the two contributions argue that alternation enables computational efficiency in synthesis and games, and that its algebraic foundations can potentially offer a path to compositional reasoning in reactive systems.