Many-one reducibility with realizability
arxiv(2024)
摘要
In this article, we propose a new classification of Σ^0_2 formulas
under the realizability interpretation of many-one reducibility (i.e., Levin
reducibility). For example, Fin, the decision of being eventually zero
for sequences, is many-one/Levin complete among Σ^0_2 formulas of the
form ∃ n∀ m≥ n.φ(m,x), where φ is decidable. The
decision of boundedness for sequences BddSeq and posets PO_
top are many-one/Levin complete among Σ^0_2 formulas of the form
∃ n∀ m≥ n∀ k.φ(m,k,x), where φ is
decidable. However, unlike the classical many-one reducibility, none of the
above is Σ^0_2-complete. The decision of non-density of linear order
NonDense is truly Σ^0_2-complete.
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