The Mishnah in Sanhedrin 40 says that Rebbi Yochanan Ben Zakai searched for a contradicition in the Bedikos with the goal of saving the suspected murderer, an act that the Mishnah considered praiseworthy (apparently a fulfillment of v'Hitzilu ha'Eidah).
What is accomplished if a contradiction in the Bedikos is discovered since the Gemara here says that the suspect is given Kipah anyway?
J. Daniel
In the Sugya on Daf 40a, Rav Chisda rejects the idea that all Bedikos can disqualify Edim and stresses that only Bedikos which refer to the murder weapon can disqualify the testimony. Ben Zakai argues and holds that all Bedikos are like Chakiros and disqualify the testimony. (See Rabbeinu Chananel.) According to Ben Zakai the suspect would indeed be saved and not sent to the Kipah, and that is how the Beis Din ruled when Ben Zakai came up with his method of saving the defendant.
However since in Rav Chisda's opinion the contradiction in Bedikos is not enough to disqualify testimony, he uses this on Daf 81b to explain how Kipah could work, since although Ben Zakai may have used them to disqualify testimony, Rav Chisda considers the testimony to be true, and he should rule that the defendant is killed. Nevertheless, he suggests not to punish with Misas Beis Din in such a situation because of the dissenting view of Ben Zakai. That is why Rav Chisda suggests using the Kipah in this situation. Ben Zakai is quoted in the Gemara on 81b only to explain why the Mishnah cannot give Misas Beis Din, even though the contradiction in Bedikos, does not disqualify the testimony in Rav Chisda's opinion.
Yoel Domb
The Halachah l'Moshe mi'Sinai of Kipah applies to one who kills but there are no witnesses to incriminate him, as the Mishnah says on 81b. But how can Beis Din know that the suspected murderer killed, if there are no witnesses to bring testimony about the murder before Beis Din?
It therefore seems that Kipah is a discretionary measure, which is only employed when the Beis Din is convinced, based on circumstantial evidence, that the suspected murderer is indeed guilty of the crime but he is being exempted from Misas Beis Din due to a technicality. This is included in what Chazal refer to as judging a Din Emes l'Amito (see Tosfos in Megilah 15b) - Beis Din must determine the logical, and not just Halachic, truth.
Thus, Ben Zakai (40a) exonerated the defendant in a case where the court suspected the witnesses of lying. In his case, the defendant indeed was left free. Rav Chisda (81b) situation is different. He remarks that if the court is convinced that the defendant is guilty, but they cannot kill him due to a minor contradiction in the testimony of the witnesses - we do not let the killer escape. Rather, we finish him off using the Kipah measure.
Best regards,
Mordecai Kornfeld