Is dependent co-arising an "IF AND ONLY IF A THEN B" relation?

No.

The 'iff and only iff' is redundant.

This leaves 'if A then B'

This is nonsense when it comes to dependent arising. It's not simple-minded (and flawed logic) but a complex description of the phenomenal world.

You have asked a good question, but why ask about the Arahant mind that is not the right direction for your question. I think it's better to ask how your mind operates within the framework of pratītyasamutpāda... because the Arahat knows and because of his knowledge he is librated, So if A is knowledge -> then B which is libration.

But your mind is ignorant, so with what condition can it acquire knowledge of libration when its condition is ignorance. If pratītyasamutpāda as you put it applies to everything then you will never be liberated. But it is not so, your volution / Karma is free to follow another path.

Please read my questions in the link below. The answers provided by @Saptha Visuddhi from Theravada perspective and the bold responses by @Mishu 米殊 should be sufficient to clear your confusion.

https://buddhism.stackexchange.com/questions/20792/how-does-human-will-fit-with-pratītyasamutpāda

This will explain only the first part.

This is called Noble Theory.

When this exists, that is; due to the arising of this, that arises. When this doesn’t exist, that is not; due to the cessation of this, that ceases. Dasabala Sutta SN 12.21

Because of A (existence of A),         B is(B exist).
If A arises(formally A is not there),  B will arise.
If A doesn't exist,                    B is not (exist).
Cessation of A (currently exist)       B ceases.

This is the basis of every link of cause effect theory. All the things happening in this world is due to this theory.

It is not "if you get A you get B". It is because of A, you get B.

For example: Suppose, we have a solar panel in our roof and a motor connected to it inside the house. Because if sunlight, motor works. If no sunlight (at night) motor is not working. When sunlight appears (in the morning) formally stopped motor starts working, At evening (due to cease of sunlight) working motor get stop working.

"Cessation of A, B ceases." This is why one can get rid of suffering.

"IF AND ONLY IF A THEN B" means if you get B you get A, if you get A you get B, if you don't get B you don't get A, and if you don't get A you don't get B.

Does dependent co-arising have the same relation I've mentioned above?

Let us understand it by taking an example: We say with birth as a condition aging and death comes to be.

You say if you get B you get A which is not the right way to state it. Because the right statement would be :IF aging and death then birth must have occurred.

You say if you get A you get B. Right way to put it : IF Birth has occurred then aging and death comes to be...

You say if you don't get B you don't get A. Right way to put it: IF there is a cessation of aging and death then there must have been cessation of Birth.

You say if you don't get A you don't get B. Right way to put it : IF there is cessation of birth then there is cessation of aging and death.

The main flaw in your statement is the usage of you get and you dont get.

Dependent origination only discusses conditionality and doesn't discuss who gets it or doesn't get it.

If this arises then that arises. If this does not arise then that does not arise.

If the answer is "yes" to the above question, as we know an Arahant has ceased the craving without remainder. Then do consciousness, name-&-form, six sense media, contact, and feeling not present in an Arahant? If these present as effects of past birth's causes, how one can escape from this cycle?

As I said dependent origination only discusses conditionality of the whole mass of suffering... it does not discuss who (whether Arahant or somebody else) suffers.

In simple dependent origination is origination of individuality or birth. So, from my perspective, 'Dependent Cessation' is cesession of every trace of individuality, with final knowledge as 'i am not', 'not mine'. So, origination and cessation are dependent of 'Universal Base', the uncreate, sublime. For simple understanding like this; A man on further side of lake watches his own mirage on water, thinks as 'my image, lost in thinking'. But when his interest ceases, he let go of mirage and know, 'This is only mirage, not me, I am not in water'. So, released. The one who ignorantly comes as me and mine, the same one goes and released. (Tathagata).

Tath+agat= The one who comes and thinks as mine and Tatha+gat= the one who wakes up and released by right knowledge, both are the same.

Part of the reason for (purpose of) explaining "Dependent Origination" is to explain how it happens and (more usefully or practically) how to stop it.

Because it can be stopped (interrupted), it is not, "B WILL ALWAYS CERTAINLY HAPPENS IF A HAPPENS, AND ONLY IF A". Instead, it is, "B POSSIBLY (or USUALLY) HAPPENS IF A HAPPENS, AND ONLY IF A"

Correct me if I wrong but I think that the "link" which people can break is between "feeling" and "craving". The training includes to not "delight" in "pleasant feelings" -- see for example SN 36.6 and SN 22.53.

I guess that means that when "craving" is stopped then all the subsequent nidanas are stopped too -- i.e. attachment, becoming, birth, and death -- and maybe ignorance too.

I imagine that the other nidanas would or must remain -- they correspond to the five skandhas -- but no longer as objects of attachment (nor of delusion) ... they remain until parinibbāna.

"IF AND ONLY IF A THEN B"

yes

means if you get B you get A

yes. if you get B you have A

if you get A you get B

not necessarily. the cycle of dependent origination can be cut/stopped at various places, such as at nama-rupa (MN 19) or at feeling (MN 38; SN 12.44) or even at craving so there is no attachment (MN 148).

if you don't get B you don't get A

not necessarily. A can arise without B arising

and if you don't get A you don't get B.

yes

Then do consciousness, name-&-form, six sense media, contact, and feeling not present in an Arahant?

the above are present in an arahant however there are not these phenomena when arising, affected or polluted by ignorance

refer to SN 22.48 & other suttas in another answer

OP: Does dependent co-arising have the same relation I've mentioned above?

Yes.

OP: If the answer is "yes" to the above question, as we know an Arahant has ceased the craving without remainder. Then do consciousness, name-&-form, six sense media, contact, and feeling not present in an Arahant?

Please see the quote from SN 22.48 below on the five (non-clinging) aggregates and the five clinging aggregates.

In an Arahant, ignorance ceases, hence the five clinging aggregates also cease, thus collapsing dependent origination nidanas. This means that the nidanas of consciousness, name-and-form, six sense media, contact and feeling that are tainted with clinging, cease in an Arahant.

However, the five (non-clinging) aggregates remain and continue functioning. This is also implied by Nibbana element with fuel remaining in Iti 44. This is also supported by Ven. Thanissaro's commentary quoted below.

When the Arahant passes away, the five (non-clinging) aggregates cease. This is also implied by Nibbana element without fuel remaining in Iti 44.

OP: If these present as effects of past birth's causes, how one can escape from this cycle?

I don't quite understand this question. From my understanding, the way to escape suffering is by uprooting ignorance through cultivating wisdom through the Noble Eightfold Path.

From SN 22.48 (trans. Ven. Bodhi):

“And what, bhikkhus, are the five aggregates? Whatever kind of form there is, whether past, future, or present, internal or external, gross or subtle, inferior or superior, far or near: this is called the form aggregate. Whatever kind of feeling there is … this is called the feeling aggregate. Whatever kind of perception there is … this is called the perception aggregate. Whatever kind of volitional formations there are … these are called the volitional formations aggregate. Whatever kind of consciousness there is, whether past, future, or present, internal or external, gross or subtle, inferior or superior, far or near: this is called the consciousness aggregate. These, bhikkhus, are called the five aggregates.

“And what, bhikkhus, are the five aggregates subject to clinging? Whatever kind of form there is, whether past, future, or present … far or near, that is tainted, that can be clung to: this is called the form aggregate subject to clinging. Whatever kind of feeling there is … that is tainted, that can be clung to: this is called the feeling aggregate subject to clinging. Whatever kind of perception there is … that is tainted, that can be clung to: this is called the perception aggregate subject to clinging. Whatever kind of volitional formations there are … that are tainted, that can be clung to: these are called the volitional formations aggregate subject to clinging. Whatever kind of consciousness there is, whether past, future, or present, internal or external, gross or subtle, inferior or superior, far or near, that is tainted, that can be clung to: this is called the consciousness aggregate subject to clinging. These, bhikkhus, are called the five aggregates subject to clinging.”

From Iti 44 (trans. Ven. Thanissaro):

This was said by the Blessed One, said by the Arahant, so I have heard: "Monks, there are these two forms of the Unbinding property. Which two? The Unbinding property with fuel remaining, & the Unbinding property with no fuel remaining.

And what is the Unbinding property with fuel remaining? There is the case where a monk is an arahant whose fermentations have ended, who has reached fulfillment, finished the task, laid down the burden, attained the true goal, ended the fetter of becoming, and is released through right gnosis. His five sense faculties still remain and, owing to their being intact, he is cognizant of the agreeable & the disagreeable, and is sensitive to pleasure & pain. His ending of passion, aversion, & delusion is termed the Unbinding property with fuel remaining.

And what is the Unbinding property with no fuel remaining? There is the case where a monk is an arahant whose fermentations have ended, who has reached fulfillment, finished the task, laid down the burden, attained the true goal, ended the fetter of becoming, and is released through right gnosis. For him, all that is sensed, being unrelished, will grow cold right here. This is termed the Unbinding property with no fuel remaining."

Commentary on Iti 44 by Ven. Thanissaro:

With fuel remaining (sa-upadisesa) and with no fuel remaining (anupadisesa): The analogy here is to a fire. In the first case, the flames are out, but the embers are still glowing. In the second, the fire is so thoroughly out that the embers have grown cold. The "fuel" here is the five aggregates. While the arahant is still alive, he/she still experiences the five aggregates, but they do not burn with the fires of passion, aversion, or delusion. When the arahant passes away, there is no longer any experience of aggregates here or anywhere else.

From comments:

OP: Five non-clinging aggregates are also conditioned. What is the cause for five non-clinging aggregates? Does that cause lie within the dependent co-arising?

The origin of the five (non-clinging) aggregates can be found in SN 35.145:

"Now what, monks, is old kamma? The eye is to be seen as old kamma, fabricated & willed, capable of being felt. The ear... The nose... The tongue... The body... The intellect is to be seen as old kamma, fabricated & willed, capable of being felt. This is called old kamma.

"And what is new kamma? Whatever kamma one does now with the body, with speech, or with the intellect: This is called new kamma.

If the next question is "whose old kamma is it?", the answer comes from SN 12.17.