A comparison of polynomial time reducibilities

On the limitations of locally robust positive reductions

“…Also, it is easy to see as a corollary of previous work on disjunctive reductions that (i) there exist recursive sets A and B such that A ≤ P ptt B but A ≤ P btt B, and (ii) there exist recursive sets A and B such that A ≤ P pbtt B but A ≤ P m B [15]. Though locally robust positive reductions are in general more flexible than globally robust positive reductions, the following theorem shows that local robustness does not add extra power for the special case of positive bounded truth-table reductions.…”

“…Positive reducibility was first studied for polynomial-time truth-table reductions in [15]. Selman, in [19], extended the definition to Turing reductions.…”

“…are the elements in the list f (x). b Definition 3.4 [15] A ≤ P pbtt C if A ≤ T C by some polynomial-time, globally positive machine M , and a polynomial-time computable function f : {0, 1} * → {c, 0, 1, } * such that M on input x makes queries only from the list f (x). Moreover the number of elements in the list f (x) is bounded by some constant independent of x. M above can be equivalently represented by a polynomial-time evaluator e such that for all oracles C, e(x, χ C (y 1 ), χ C (y 2 ), .…”

“…b In [15] the first argument of e is α(x), however without loss of generality we can take this to be x and let e do the (polynomial-time) computation required to obtain α. Locally robust reductions can now be defined with respect to reductions involving locally robust machines.…”

Bi-Immunity Separates Strong NP-Completeness Notions

“…The question of whether various completeness notions for NP are distinct has a very long history [LLS75], and has always been of interest because of the surprising phenomenon that no natural NP-complete problem has ever been discovered that requires anything other than many-one reducibility for proving its completeness. This is in contrast to the situation for NP-hard problems.…”