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A note on relative efficiency of axiom systems. (English) Zbl 0812.03026
The authors introduce a notion of relative efficiency for axiom systems. Given an axiom system $$A_ \beta$$ for a theory $$T$$ consistent with $$S^ 1_ 2$$, they show that the problem of deciding whether an axiom system $$A_ \alpha$$ for the same theory is more efficient than $$A_ \beta$$ is $$\Pi_ 2$$-hard. Several possibilities of speed-up of proofs are examined in relation to pairs of axiom systems $$A_ \alpha$$, $$A_ \beta$$, with $$A_ \alpha \supseteq A_ \beta$$, both in the case that $$A_ \alpha$$ and $$A_ \beta$$ have the same language, and in the case that the language of $$A_ \alpha$$ extends that of $$A_ \beta$$; in the latter case, letting $$\text{Pr}_ \alpha$$, $$\text{Pr}_ \beta$$ denote the theories axiomatized by $$A_ \alpha$$, $$A_ \beta$$, respectively, and assuming $$\text{Pr}_ \alpha$$ to be a conservative extension of $$\text{Pr}_ \beta$$, the authors show that if $$A_ \alpha - A_ \beta$$ contains no non-logical axioms, then $$A_ \alpha$$ can only be a linear speed-up of $$A_ \beta$$; otherwise, given any recursive function $$g$$ and any $$A_ \beta$$, there exists a finite extension $$A_ \alpha$$ of $$A_ \beta$$ such that $$A_ \alpha$$ is a speed-up of $$A_ \beta$$ with respect to $$g$$.

##### MSC:
 03F20 Complexity of proofs 03F30 First-order arithmetic and fragments
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##### References:
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