Formalizing synthesis in TLA

This report proposes a TLA+ definition for the problem of constructing a strategy that implements a temporal property. It is based on a note by Lamport [1] that outlines a formalization of realizability in TLA. The modified definition proposed here is expressed axiomatically in TLA+. Specifying what function is acceptable as a strategy requires care, so that a function with empty domain be avoided, while ensuring that the strategy will not need to have a domain too large to be a set. We prove that initial conditions should appear in assumptions only, unless an initial predicate is added to the definition of a realization. We show that a specification should include an assumption about a set of initial values to ensure that realizability does not become unprovable. We discuss what form of open-system properties expressed with the “while- plus” operator -+-> are realizable. We formalize the notions of interleaving and disjoint-state behaviors, based on definitions given by Lamport and Abadi, and consider the notion of interleaving for an open-system property. We give examples of expressing different forms of games in TLA+ using the proposed definition, including games with partial information.