its a generally open TCS problem subject to ongoing research whether & the exact conditions Cook & Karp reductions are equivalent and is apparently closely related to the open NP=?coNP question & other complexity class separations eg E=?NE (wrt sparse languages).

here are two research papers on the subject & further leads on tcs.se via similar question:

• Are Cook and Karp Ever the Same? Beigel, Fortnow

• Cook vs. Karp-Levin: Seperating Completeness Notions If NP Is Not Small (1995) Lutz, Mayordomo

• many one reductions vs Turing reductions to define NPC, tcs.se

its a generally open TCS problem subject to ongoing research whether & the exact conditions Cook & Karp reductions are equivalent and is apparently closely related to the open NP=?coNP question & other complexity class separations eg E=?NE (wrt sparse languages).

here are two research papers on the subject & further leads on tcs.se via similar question:

• Are Cook and Karp Ever the Same? Beigel, Fortnow

• Cook vs. Karp-Levin: Seperating Completeness Notions If NP Is Not Small (1995) Lutz, Mayordomo

• many one reductions vs Turing reductions to define NPC, tcs.se