FREC delivrable 12: Stone duality in semantics vs. Stone duality in variety theory

This report is the 12th delivrable of the FREC project. This report contains the description of a denotational semantics for λY calculus that captures precisely weak Monadic Second Order Logic on the Böhm trees defined by λY -terms. Exploiting Stone duality we have been able to turn properties expressed by those models into an inference system. Moreover we exploited internal dualities of the model so as to exhibit an original type system that refutes properties of λY -terms. We give a denotational semantics of λY -calculus that is capable of computing properties expressed in weak monadic second-order logic (WMSO). λY -calculus faithfully models control in higher-order programs. Our model construction allows to reduce verification to evaluation. The model that we propose presents a lot of symmetries that reflect elegantly the dualities of weak alternating automata at all orders. Following the domain in logical form approach, we derive a type system for verifying WMSO properties.