Alas, Philosophy, Medicine, Law, and unfortunately also Theology, have I studied in detail, and still remained a fool, not a bit wiser than before. Magister and even Doctor am I called, and for a decade am I sick and tired of pulling my pupils by the nose and understanding that we can know nothing.

Summary: This appendix briefly surveys some attempts at proving lower bounds on the complexity of natural computational problems. In the first part, devoted to circuit complexity, we describe lower bounds on the size of (restricted) circuits that solve natural computational problems. This can be viewed as a program whose long-term goal is proving that P ≠ NP. In the second part, devoted to proof complexity, we describe lower bounds on the length of (restricted) propositional proofs of natural tautologies. This can be viewed as a program whose long-term goal is proving that NP ≠ coNP.

We comment that while the activity in these areas is aimed toward developing proof techniques that may be applied to the resolution of the “big problems” (such as P versus NP), the current achievements (though very impressive) seem very far from reaching this goal. Current crown-jewel achievements in these areas take the form of tight (or strong) lower bounds on the complexity of computing (resp., proving) “relatively simple” functions (resp., claims) in restricted models of computation (resp., proof systems).