currently there’s a sentence in this post part of which says, literally, “Pick out the right vertexes (one per word), wire them together so that there are no dangling unconnected edges, and viola!”

As typos go, this one is funny. ]]>

So, if you look at the “big picture”, the upper bound is exponential. But if you look at it locally, the introduction of new rules and premises makes it combinatorial.

]]>I believe my statement is correct. It grows exponentially w.r.t. the proof length, that is what Marcus Hutter’s famous paper The Fastest and Shortest Algorithm for All Well-Defined Problems says. He means the length of the binary representation of the proof, but I think it applies here too. There is a finite number of rules and axioms, ultimately a inference tree of size S can be turned into a binary string of length O(S).

Maybe you mean w.r.t. to something else. For instance the complexity w.r.t. the length of the theorem is unbounded.

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