A lower bound for the complexity of Craig's interpolants in sentential logic

For any sentenceα (in sentential logic) letdα be the delay complexity of the boolean functionfα represented byα. We prove that for infinitely manyd (and starting with somed<620) there exist valid implicationsα→β withdα,dβ≦d such that any Craig's interpolantx has its delay complexitydχ greater thand+(1/3)·log(d/2). This is the first (non-trivial) known lower bound on the complexity of Craig's interpolants in sentential logic, whose general study may well have an impact on the central problems of computation theory.