__Research article:__

** Samuel R. Buss and Jan Krajicek and Gaisi Takeuti.
"Provably total functions in the bounded arithmetic theories R ^{i}_{3},
U^{i}_{2}, and V^{i}_{2}."
In Proof Theory, Arithmetic, and Complexity, P. Clote and J.
Krajicek (eds), Oxford University Press, 1993, pp. 116-161. **

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** Abstract: **This paper investigates the provably
total functions of fragments of first- and second-order Bounded Arithmetic. The (strongly)
$\Sigma^b_i$-definable functions of $S^{i-1}_3$ and $R^i_3$ are precisely the (strong)
$\FPthreewp {i-1}$ functions. The $\Sigma^{1,b}_i$-definable functions of $V^{i-1}_2$ and
$U^i_2$ are the $\EXwp {i-1}$ functions and the $\Sigma^{1,b}_i$-definable functions of
$V^i_2$ are the $\EX i$-functions. We give witnessing theorems for these theories and
prove conservation results for $R^i_3$ over $S^{i-1}_3$ and for $U^i_2$ over $V^{i-1}_2$.