Typing weak MSOL properties

International audienceWe consider non-interpreted functional programs: the result of the execution of a program is its normal form, that can be seen as the tree of calls to built-in operations. Weak monadic second-order logic (wMSO) is well suited to express properties of such trees. This is an extension of first order logic with quantification over finite sets. Many behavioral properties of programs can be expressed in wMSO. We use the simply typed lambda calculus with the fixpoint operator, $\lambda Y$-calculus, as an abstraction of functional programs that faithfully represents the higher-order control flow. We give a type system for ensuring that the result of the execution of a $\lambda Y$-program satisfies a given wMSO property. The type system is an extension of a standard intersection type system with both: the least-fixpoint rule, and a restricted version of the greatest-fixpoint rule. In order to prove soundness and completeness of the system we construct a denotational semantics of $\lambda Y$-calculus that is capable of computing properties expressed in wMSO. The model presents many symmetries reflecting dualities in the logic and has also other applications on its own. The type system is obtained from the model following the domain in logical form approach