What exactly is the logical relation between the links in the twelvefold chain of Pratītyasamutpāda?

I'm struggling hard to make final sense of the chain of "dependent origination". In the oldest texts in Pali and Sanskrit, already, there are different explanations. The *Mahānidānasutta*, where the chain has just nine links, the general formula is > x-paccayā y meaning > through the condition of x, y (is). Now the question arises, what kind of condition this is thought to be: a sufficient condition? A necessary condition? Or both of them? The second part, states quite clearly, that if x is not there, then y is also not there, implying necessity of x for y. Now, a general rule (e.g. Majjhima Nikaya 38, 19) states: > imasmiṃ sati idaṃ hoti, imassuppādā idaṃ uppajjati which translates as: > If that is, this is, through arising of that, this arises This is also consistent with a parallel in the Sanskrit *Catuṣpariṣatsūtra* but not with the view, that we are dealing with mere necessary conditions. They need to be sufficient, for this general statement to apply. Now here things become even more complicated, since another general rule, actually a seemingly extended version of the former states (MN 79, 7, also in SN and Udana) > imasmiṃ sati idaṃ hoti, imassuppādā idaṃ uppajjati; imasmiṃ asati idaṃ na hoti, imassa nirodhā idaṃ nirujjhati which now means > When that is, this is, through arising of that, this arises; when that is not, this is not, with cessation of that, this ceases. implying biconditionality. How are the different versions explained? What is, in the end, the relation between the links? What can be thought to be the Buddha's original version and intention?