A Weakness Measure for GR(1) Formulae

Abstract When dealing with unrealizable specifications in reactive synthesis, finding the weakest environment assumptions that ensure realizability is often considered a desirable property. However, little effort has been dedicated to defining or evaluating notion of weakness formally. The question whether one assumption weaker than another commonly interpreted by considering implication relationship between two or, equivalently, their language inclusion. This interpretation fails provide any insight into when (or inclusion) does not hold. To our knowledge, only measure capable comparing formulae this case entropy, but even it cannot distinguish expressed as fairness properties. In paper, we propose refined based on combining entropy Hausdorff dimension, concept captures size ω -language satisfying linear temporal logic formula. We focus special subset which particular interest called GR(1). identify conditions under guaranteed and stronger GR(1) formulae, cover cases are strictly ordered have same dimension. prove consistency logical implication, is, if formula implies another, latter former according measure. evaluate proposed contexts. first computing refinements where used heuristic drive refinement search towards solutions. second context quantitative model checking violating