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We prove several results showing that the algebraic 
$K$-theory of valuation rings behaves as though such rings were regular Noetherian, in particular an analogue of the Geisser–Levine theorem. We also give some new proofs of known results concerning cdh descent of algebraic 
$K$-theory.
We give model theoretic criteria for the existence of ∃∀ and ∀∃- formulas in the ring language to define uniformly the valuation rings 
${\cal O}$ of models 
$\left( {K,\,{\cal O}} \right)$ of an elementary theory Σ of henselian valued fields. As one of the applications we obtain the existence of an ∃∀-formula defining uniformly the valuation rings 
${\cal O}$ of valued henselian fields 
$\left( {K,\,{\cal O}} \right)$ whose residue class field k is finite, pseudofinite, or hilbertian. We also obtain ∀∃-formulas φ2 and φ4 such that φ2 defines uniformly k[[t]] in k(t) whenever k is finite or the function field of a real or complex curve, and φ4 replaces φ2 if k is any number field.
