user16: 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?

user295: As I see it some of the links are bidirectional while some links like between sensations and craving (the place you can break the cycle) unidirectional.
If craving there should be sensations but in a liberated person thought he experiences sensations there is no craving.

user5513: The Pratītyasamutpāda theory can be applied for different conceptual levels of life. The relation of the chain also differ accordingly.
Life can be defined as

a physical process.
a mental process.
a hybrid of both above.
a conceptual phenomenon.
a combination of all these.
as three life -- past, present and future.   
a moment of thought. Etc.

The original meaning is not as sufficient, necessary, or bi conditions. 
Pratītyasamutpāda is  ‘THE RELATION’  that lord Buddha use to explain the worldly phenomena. 

user18: According to Theravada, the actual conditionality is more complicated even than is explained by PS. For a true understanding of conditionality (apart from attaining Buddhahood), a study of the Mahapatthana is probably more enlightening, intellectually speaking, since it outlines 24 types of conditionality:

Hetupaccayo , ārammaṇapaccayo, adhipatipaccayo, anantarapaccayo, samanantarapaccayo, sahajātapaccayo, aññamaññapaccayo, nissayapaccayo, upanissayapaccayo, purejātapaccayo, pacchājātapaccayo, āsevanapaccayo, kammapaccayo, vipākapaccayo, āhārapaccayo, indriyapaccayo, jhānapaccayo, maggapaccayo, sampayuttapaccayo, vippayuttapaccayo, atthipaccayo, natthipaccayo, vigatapaccayo, avigatapaccayoti.

Reading the Paccayaniddeso, it seems clear that the type of conditionality depends on the link in question. For example, 

rūpāyatanaṃ cakkhuviññāṇadhātuyā taṃsampayuttakānañca dhammānaṃ ārammaṇapaccayena paccayo.

Basically, forms share conditionality with the eye by virtue of being their object, etc.
I'm not an expert in Theravada Abhidhamma by any means, but it certainly doesn't seem like one can give a categorical answer or take PS as a strictly literal chain of linear causation; it seems much more an exegetical teaching meant to summarize the nature of samsara. In Theravada it is further complicated by the idea of relating to three lives. Read the article by P.A. Payutta for explanation of that concept (and for more insight into PS in general):
http://www.abuddhistlibrary.com/Buddhism/B%20-%20Theravada/Teachers/Ven%20Payutto/Dependent%20Origination/Dependent%20Origination%20The%20Buddhist%20Law%20of%20Conditionality.htm
EDIT: Here's a source that compares PS and Patthana:
http://stylomilo.com/files/mv/YMBADip/Abhi/Patthana%20naya%20-%20LTY.pdf
It points out the difference in style between PS and Patthana; as I said, PS is conventional and Patthana is dealing with ultimate realities. It also states that Patthana:

examines in greater detail cause and effect, as PS does not explain how
  the cause becomes the effect, or the relationships between cause and
  effect. For example, no single cause can produce an effect and a cause
  does not produce only a single effect. Therefore, it is a collection of
  causes which produces a collection of effects. PS looks at the chief causeand the prominent effect only

Seems like the article is worth reading, but I just found it on Google, so no assurances :)

user745: The conditional relation of the links of dependent origination are necessary conditions, not sufficient conditions. This is clear when we compare the links in dependent origination with other teachings. For example, according to the twelve links, the sixfold sense-base is a condition for the arising of contact. But in many places in the Suttas it is taught that:

Dependent on the eye & forms there arises consciousness at the eye.
  The meeting of the three is contact. (Similarly with the other senses)

which indicates that the sense base isn't sufficient because contact also depends on an object and on the arising of consciousness.
Also I don't see how 

When that is, this is, through arising of that, this arises; when that
  is not, this is not, with cessation of that, this ceases.

Implies biconditionality. It sounds like straightforward necessary causation to me.

user145: In pratityasamutpada there are actually eleven links, as each causal link occurs between one of twelve states.
The relation is described as "yā tatra tathatā avitathatā anaññathatā idappaccayatā"—“that wherein is reality (tathatā), not unreality (avitathatā), not otherness (anaññathatā), specific conditionality: [that is called dependent origination]”, and in detail is explained as this:

Because particular states are produced by particular conditions,
  neither  less nor more, it is called reality (suchness, tathata).
  Because once the conditions have  met in combination there is no
  non-producing, even for an instant, of the states  they generate, it
  is called not unreality (not unsuchness). Because there is no arising
  of one state with another state’s conditions, it is called not
  otherness.  Because there is a condition, or because there is a total
  of conditions, for these  states beginning with ageing-and-death as
  already stated, it is called specific  conditionality. (Visuddhimagga)

So, this is both sufficient and necessary conditions. But, it's important to know that pratityasamutpada is a brief explanation, almost enumeration, and not all conditions are named, only so-called representative cause and fruit. The most detailed explanation of the paticcasamuppada you can read in Visuddhimagga (pdf).

user43: As I understand, the relationship is that of mutual implication. This meaning is reflected in the name pratitya-samutpada, that is often translated as "dependent co-arising" and such.
Mutual implication means one of the categories serves as context for the other and the other way around. Top is defined against bottom and vice verse. Life is defined against death and vice verse. This (upclose) is defined against That (far). Subjective is defined against Objective.
The twelve nidanas are meant to explain the emergence of the notion of Death by implication from Birth, of Birth by implication from Identifying with Separate Being, of Identifying with Separate Being by implication from Goal-making, of Goal-making by implication from Temporal Projection, of Temporal Projection by implication from Sensation as Result, of Sensation as Result by implication from Contact, of Contact by implication from Doors onto the world, of Doors onto the world by implication from Objects, of Objects by implication from Recognition, of Recognition by implication from Imprints, of Imprints by implication from Ignorance.

16: <p>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.</p>

<p>The <em>Mahānidānasutta</em>, where the chain has just nine links, the general formula is</p>

<blockquote>
  <p>x-paccayā y</p>
</blockquote>

<p>meaning</p>

<blockquote>
  <p>through the condition of x, y (is).</p>
</blockquote>

<p>Now the question arises, what kind of condition this is thought to be: a sufficient condition? A necessary condition? Or both of them?</p>

<p>The second part, states quite clearly, that if x is not there, then y is also not there, implying necessity of x for y.</p>

<p>Now, a general rule (e.g. Majjhima Nikaya 38, 19) states:</p>

<blockquote>
  <p>imasmiṃ sati idaṃ hoti, imassuppādā idaṃ uppajjati</p>
</blockquote>

<p>which translates as:</p>

<blockquote>
  <p>If that is, this is, through arising of that, this arises</p>
</blockquote>

<p>This is also consistent with a parallel in the Sanskrit <em>Catuṣpariṣatsūtra</em> but not with the view, that we are dealing with mere necessary conditions. They need to be sufficient, for this general statement to apply.</p>

<p>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)</p>

<blockquote>
  <p>imasmiṃ sati idaṃ hoti, imassuppādā idaṃ uppajjati; imasmiṃ asati idaṃ na hoti, imassa nirodhā idaṃ nirujjhati</p>
</blockquote>

<p>which now means</p>

<blockquote>
  <p>When that is, this is, through arising of that, this arises; when that is not, this is not, with cessation of that, this ceases.</p>
</blockquote>

<p>implying biconditionality.</p>

<p>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?</p>
 295: <p>As I see it some of the links are bidirectional while some links like between sensations and craving (the place you can break the cycle) unidirectional.</p>

<p>If craving there should be sensations but in a liberated person thought he experiences sensations there is no craving.</p>
 5513: <p>The Pratītyasamutpāda theory can be applied for different conceptual levels of life. The relation of the chain also differ accordingly.
Life can be defined as</p>

<ol>
<li>a physical process.</li>
<li>a mental process.</li>
<li>a hybrid of both above.</li>
<li>a conceptual phenomenon.</li>
<li>a combination of all these.</li>
<li>as three life -- past, present and future.   </li>
<li>a moment of thought. Etc.</li>
</ol>

<p>The original meaning is not as sufficient, necessary, or bi conditions. 
Pratītyasamutpāda is  ‘<a href="https://buddhism.stackexchange.com/questions/10789/what-is-the-relation-between-prat%C4%ABtyasamutp%C4%81da-and-four-noble-truths/10965#10965">THE RELATION</a>’  that lord Buddha use to explain the worldly phenomena. </p>
 18: <p>According to Theravada, the actual conditionality is more complicated even than is explained by PS. For a true understanding of conditionality (apart from attaining Buddhahood), a study of the Mahapatthana is probably more enlightening, intellectually speaking, since it outlines 24 types of conditionality:</p>

<blockquote>
  <p>Hetupaccayo , ārammaṇapaccayo, adhipatipaccayo, anantarapaccayo, samanantarapaccayo, sahajātapaccayo, aññamaññapaccayo, nissayapaccayo, upanissayapaccayo, purejātapaccayo, pacchājātapaccayo, āsevanapaccayo, kammapaccayo, vipākapaccayo, āhārapaccayo, indriyapaccayo, jhānapaccayo, maggapaccayo, sampayuttapaccayo, vippayuttapaccayo, atthipaccayo, natthipaccayo, vigatapaccayo, avigatapaccayoti.</p>
</blockquote>

<p>Reading the Paccayaniddeso, it seems clear that the type of conditionality depends on the link in question. For example, </p>

<blockquote>
  <p>rūpāyatanaṃ cakkhuviññāṇadhātuyā taṃsampayuttakānañca dhammānaṃ ārammaṇapaccayena paccayo.</p>
</blockquote>

<p>Basically, forms share conditionality with the eye by virtue of being their object, etc.</p>

<p>I'm not an expert in Theravada Abhidhamma by any means, but it certainly doesn't seem like one can give a categorical answer or take PS as a strictly literal chain of linear causation; it seems much more an exegetical teaching meant to summarize the nature of samsara. In Theravada it is further complicated by the idea of relating to three lives. Read the article by P.A. Payutta for explanation of that concept (and for more insight into PS in general):</p>

<p><a href="http://www.abuddhistlibrary.com/Buddhism/B%20-%20Theravada/Teachers/Ven%20Payutto/Dependent%20Origination/Dependent%20Origination%20The%20Buddhist%20Law%20of%20Conditionality.htm" rel="nofollow">http://www.abuddhistlibrary.com/Buddhism/B%20-%20Theravada/Teachers/Ven%20Payutto/Dependent%20Origination/Dependent%20Origination%20The%20Buddhist%20Law%20of%20Conditionality.htm</a></p>

<p>EDIT: Here's a source that compares PS and Patthana:</p>

<p><a href="http://stylomilo.com/files/mv/YMBADip/Abhi/Patthana%20naya%20-%20LTY.pdf" rel="nofollow">http://stylomilo.com/files/mv/YMBADip/Abhi/Patthana%20naya%20-%20LTY.pdf</a></p>

<p>It points out the difference in style between PS and Patthana; as I said, PS is conventional and Patthana is dealing with ultimate realities. It also states that Patthana:</p>

<blockquote>
  <p>examines in greater detail cause and effect, as PS does not explain how
  the cause becomes the effect, or the relationships between cause and
  effect. For example, no single cause can produce an effect and a cause
  does not produce only a single effect. Therefore, it is a collection of
  causes which produces a collection of effects. PS looks at the chief causeand the prominent effect only</p>
</blockquote>

<p>Seems like the article is worth reading, but I just found it on Google, so no assurances :)</p>
 745: <p>The conditional relation of the links of dependent origination are necessary conditions, not sufficient conditions. This is clear when we compare the links in dependent origination with other teachings. For example, according to the twelve links, the sixfold sense-base is a condition for the arising of contact. But in many places in the Suttas it is taught that:</p>

<blockquote>
  <p>Dependent on the eye &amp; forms there arises consciousness at the eye.
  The meeting of the three is contact. (Similarly with the other senses)</p>
</blockquote>

<p>which indicates that the sense base isn't sufficient because contact also depends on an object and on the arising of consciousness.</p>

<p>Also I don't see how </p>

<blockquote>
  <p>When that is, this is, through arising of that, this arises; when that
  is not, this is not, with cessation of that, this ceases.</p>
</blockquote>

<p>Implies biconditionality. It sounds like straightforward necessary causation to me.</p>
 145: <p>In <em>pratityasamutpada</em> there are actually <em>eleven</em> links, as each causal link occurs <em>between</em> one of twelve states.</p>

<p>The relation is described as "<em>yā tatra tathatā avitathatā anaññathatā idappaccayatā</em>"—“<em>that wherein is reality (tathatā), not unreality (avitathatā), not otherness (anaññathatā), specific conditionality: [that is called dependent origination]</em>”, and in detail is explained as this:</p>

<blockquote>
  <p><b>B</b>ecause particular states are produced by particular conditions,
  neither  less nor more, it is called reality (suchness, <em>tathata</em>).
  <b>B</b>ecause once the conditions have  met in combination there is no
  non-producing, even for an instant, of the states  they generate, it
  is called not unreality (not unsuchness). <b>B</b>ecause there is no arising
  of one state with another state’s conditions, it is called not
  otherness.  <b>B</b>ecause there is a condition, or because there is a total
  of conditions, for these  states beginning with ageing-and-death as
  already stated, it is called specific  conditionality. (Visuddhimagga)</p>
</blockquote>

<p>So, this is both <em>sufficient</em> and <em>necessary</em> conditions. But, it's important to know that <em>pratityasamutpada</em> is a <em>brief</em> explanation, almost enumeration, and not all conditions are named, only so-called <em>representative cause and fruit</em>. The most detailed explanation of the <em>paticcasamuppada</em> you can read in Visuddhimagga (<a href="http://www.accesstoinsight.org/lib/authors/nanamoli/PathofPurification2011.pdf" rel="nofollow">pdf</a>).</p>
 43: <p>As I understand, the relationship is that of <em>mutual implication</em>. This meaning is reflected in the name pratitya-samutpada, that is often translated as "dependent co-arising" and such.</p>

<p>Mutual implication means one of the categories serves as context for the other and the other way around. Top is defined against bottom and vice verse. Life is defined against death and vice verse. This (upclose) is defined against That (far). Subjective is defined against Objective.</p>

<p>The twelve nidanas are meant to explain the emergence of the notion of Death by implication from Birth, of Birth by implication from Identifying with Separate Being, of Identifying with Separate Being by implication from Goal-making, of Goal-making by implication from Temporal Projection, of Temporal Projection by implication from Sensation as Result, of Sensation as Result by implication from Contact, of Contact by implication from Doors onto the world, of Doors onto the world by implication from Objects, of Objects by implication from Recognition, of Recognition by implication from Imprints, of Imprints by implication from Ignorance.</p>
