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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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