Open Implication A New Relation on Specifications Given in Linear Temporal Logic

The wish to automatically construct correct systems from specifications has been around for about half a century. Church was the first one to describe this idea in 1962. Our work is closely related to this idea. We will consider systems which react to the environment denoted as open modules and our specification language is linear temporal logic (LTL). Synthesis is the problem of automatically constructing correct open modules from an LTL specification. In 1992, it was shown to be 2EXP complete. This result discouraged many researchers to continue working on the problem since it seemed to be intractable. Only in recent years the topic was picked up again and new results leaded to new ideas. The core idea of this work is a new relation on LTL formulas. An LTL formula open implies another formula, if every open module realizing the first formula also realizes the second formula. Hence open implication can be used to improve specifications or to synthesize smaller solutions. Compared to trace inclusion, open implication is harder to calculate and weaker. Thus there are formulas such that one formula open implies the other but trace inclusion does not hold. In this work we will formally define open implication. Then we will give an algorithm to calculate open implication for full LTL. The algorithm meets the lower bound of 2EXP in the first argument and PSPACE in the second. Subsequently, we explain an efficient algorithm for a subset of LTL, that of General Reactivity of Rank 1 (GR(1)). We implemented the algorithm in a tool called Anzu which will also be discussed together with the results of our case study.