KRW Composition Theorems via Lifting

One of the major open problems in complexity theory is proving super-logarithmiclower bounds on the depth of circuits (i.e., P⊈NC¹). Karchmer et al. (Comput Complex 5(3/4):191–204, 1995) suggested to approach thisproblem by proving that depth complexity behaves “as expected”with respect to the composition of functions f◊g. They showedthat the validity of this conjecture would imply that P⊈NC¹.Several works have made progress toward resolving this conjectureby proving special cases. In particular, these works proved the KRWconjecture for every outer function f, but only for few innerfunctions g. Thus, it is an important challenge to prove the KRWconjecture for a wider range of inner functions.In this work, we extend significantly the range of inner functionsthat can be handled. First, we consider the monotone versionof the KRW conjecture. We prove it for every monotone inner function gwhose depth complexity can be lower-bounded via a query-to-communicationlifting theorem. This allows us to handle several new and well-studiedfunctions such as the s-t-connectivity, clique,and generation functions.In order to carry this progress back to the non-monotone setting,we introduce a new notion of semi-monotone composition, whichcombines the non-monotone complexity of the outer function f withthe monotone complexity of the inner function g. In this setting,we prove the KRW conjecture for a similar selection of inner functions g,but only for a specific choice of the outer function f.