By Todorova G., Vitillaro E.

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5 (Wrathall 1978). LinH = RUD Proof (sketch). Using the natural coding of computations of machines by 0-1 strings one verifies that Eo" c_ RUD, from which LinH CRUD follows immediately. The opposite inclusion is obvious. D. Basic complexity theory 20 The possibility of coding in Do(N) merits further discussion. 4. 6 (Bennett 1962). The graph of exponentiation {(x, y, z) I xl' = z} is rudimentary. 7 (Wrathall 1978). All context free languages are rudimentary and hence in DO(N). 4. The term TimeSpace(f(n), g(n)) denotes the class of languages recognized by a Turing machine working simultaneously in time f (n) and space g(n).

15 the sequent S has a size m 0(1) LK-proof. D. 3 has a much stronger lower bound for the sequent S. The next proposition will show that every treelike proof can be balanced with only a polynomial increase in size. The lemma itself looks rather technical so let us first motivate it. , Sk (additional axioms: initial sequents). Let a be a truth assignment not satisfying S. , Zt of sequents such that Zo = S, each Zi is false under a, Z;+i is a hypothesis of the inference yielding Z;, and Z, is initial.

5 to the assumption that the sequent has a short LK-derivation. 9. There is a sequent S of size I S1 = m such that every cut free, treelike LK proof of S has at least 2"(" sequents. Moreover, the sequent S has an L K proof of size m 00). 2. Let 7r be a cut-free, treelike LK-proof of S. ,q, Basic propositional logic 38 where Dr are disjunctions of literals p;j or -p;j and qs are atoms among pig. , D (we identify a disjunction of literals with the clause consisting of those literals) with at most k resolution inferences.