Guest Column

Abstract

We survey lower-bound results in complexity theory that have been obtained via newfound interconnections between propositional proof complexity, boolean circuit complexity, and query/communication complexity. We advocate for the theory of total search problems (TFNP) as a unifying language for these connections and discuss how this perspective suggests a whole programme for further research.

Strong co-nondeterministic lower bounds for NP cannot be proved feasibly

“…There are several results which mitigate this situation. For example, 'lifting' theorems provide a method for deriving monotone circuit lower bounds from lower bounds for weak proof systems [11]. In the algebraic setting, superpolynomial lower bounds for CNF tautologies in the Ideal Proof System imply that the Permanent does not have p-size arithmetic circuits [12].…”