We investigate the theory IΔ0+Ω1 and strengthen [Bu86, Theorem 8.6] to the following: if NP ≠ co-NP, then Σ-completeness for witness comparison foumulas is not provable in bounded arithmetic. i.e.,
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200019514/resource/name/S0022481200019514_Uequ1.gif?pub-status=live)
Next we study a “small reflection principle” in bounded arithmetic. We prove that for all sentences φ
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The proof hinges on the use of definable cuts and partial satisfaction predicates akin to those introduced by Pudlák in [Pu86].
Finally, we give some applications of the small reflection principle, showing that the principle can sometimes be invoked in order to circumvent the use of provable Σ-completeness for witness comparison formulas.