A Weakness Measure for GR(1) Formulae

Abstract

When dealing with unrealizable specifications in reactivesynthesis, finding the weakest environment assumptions that ensurerealizability is often considered a desirable property. However,little effort has been dedicated to defining or evaluating thenotion of weakness of assumptions formally. The question of whetherone assumption is weaker than another is commonly interpreted byconsidering the implication relationship between the two or,equivalently, their language inclusion. This interpretation fails toprovide any insight into the weakness of the assumptions whenimplication (or language inclusion) does not hold. To ourknowledge, the only measure that is capable of comparing twoformulae in this case is entropy, but even it cannot distinguish theweakness of assumptions expressed as fairness properties. In thispaper, we propose a refined measure of weakness based on combiningentropy with Hausdorff dimension, a concept that captures the notionof size of the omega -language satisfying a linear temporal logicformula. We focus on a special subset of linear temporal logicformulae which is of particular interest in reactive synthesis,called GR(1). We identify the conditions under which this measureis guaranteed to distinguish between weaker and stronger GR(1)formulae, and propose a refined measure to cover cases when twoformulae are strictly ordered by implication but have the sameentropy and Hausdorff dimension. We prove the consistency betweenour weakness measure and logical implication, that is, if oneformula implies another, the latter is weaker than the formeraccording to our measure. We evaluate our proposed weaknessmeasure in two contexts. The first is in computing GR(1) assumptionrefinements where our weakness measure is used as a heuristic todrive the refinement search towards weaker solutions. The second isin the context of quantitative model checking where it is used tomeasure the size of the language of a model violating a lineartemporal logic formula.