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Proofs, Circuits, and Communication

“…First we instantiate this model and consider a dag-like version of decision trees, which we call conjunction-dags following [GGKS20]. This model and close variations on it have been studied in the literature in a number of separate works prior to [GGKS20]; notably, by Pudlák [Pud00], and then by Atserias and Dalmau [AD08], under the name Prover-Adversary games. Definition 3.2.…”

“…If we are given a resolution proof Π, then we can simply replace each clause in Π with the negation of that clause (i.e., a conjunction); it is easy to verify that the resulting dag satisfies the three properties of a conjunction-dag. The converse direction is not much harder; we refer to [GGKS20] for a proof. Theorem 3.3.…”

“…Razborov's original definition used the language of TFNP, particularly as a "communication version" of the complexity class PLS (see Section 5 for more details on this). The model was subsequently studied by Krajíček [Kra97] and simplified by Sokolov [Sok17], although we will use the following version from [GGKS20]. As we will be primarily interested in monotone circuit complexity, we define it purely for the mKW game.…”

“…If so, this would allow us to translate lower bounds for resolution proofs (which is a very well-developed theory, e.g., [Hak85,BW01]) into lower bounds for monotone boolean circuits, for which we essentially have only one technique, the method of approximations with its variations [Raz85a, Juk97, HR00, BU99]. Indeed, such a result was proven by Göös, Garg, Kamath and Sokolov [GGKS20], who crucially used the search viewpoint described above in their proof.…”

“…In recent years there has been a large number of new results in both propositional proof complexity and boolean circuit complexity. These results include: optimal 2 Ω(n) lower bounds on the size of monotone boolean formulas computing an explicit boolean function in NP [PR17], the refinement of the mon-AC i hierarchy from the mon-NC i hierarchy and new tradeoffs for cutting planes proofs [dRNV16], a new family of techniques for proving lower bounds on cutting planes proofs and monotone circuit size [GGKS20], and exponential lower bounds on the size of cutting planes proofs for random CNF formulas [FPPR17,HP17]. All of these new results have been enabled, either directly or indirectly, by two tools:…”