You want to have a very high certainty that the system is:
- going to react accordingly to what you intended.
- arbitrarily not likely to be triggered after a false alarm.
- reliable and valid.
- arbitrarily not likely to be revealed to the world so its utility is corrupted.
Human and dead man trigger
The solution here is what was said in an earlier response: you just lock people in a basement with no other purpose than pushing a button at regular intervals. This seems nice because it's a direct reflect of what would happen in the countries near by. If no one pushes the button then it's almost certain that a disastrous event happened which prevented humans to come pushing the button... or is it? I mean, after all humans are humans. Especially if you select untrained regular people to be your "dead-men" you're facing a big problem. People are easily corrupted, manipulated and they are likely to leak out secret information that would help some enemies to prevent the automatic strike back to occur.
Let's imagine the following scenario: one of the people involved into the "dead man" process leaks out crucial information about how it works. This information are eventually revealed to an unfriendly country. This enemy send some people at different locations to infiltrate your system in order to be able to replicate its signal. The enemy finally unleash its nuclear power then simulate the signals until he finds out a way to permanently disable the threat.
This is mere one (and maybe not the most clever) among all the possibilities one could imagine to break in your system.
One could think of an automated process as highly reliable because it lacks human common flaws and does not feel anything. It just does its job and that's all but there are several limitations to a machine doing this kind of job. First of all it highly depends on the implementation of the machine by the creators. Since you said they are very cautious one could assume they would not make much mistakes but still... they would make a few at least.
Making a few mistakes could lead to a systematic bias in the system which could lead to a catastrophic result.
An automated process lacks the human intelligence which would allow someone to decide that the launch of nuclear missile has been triggered by error and has to be cancelled. If the sensors which are saying to the central system whether a attack has been made or not is deficient and works in a bad way the central system has no way to figure it out. This is therefore very dangerous and I doubt seriously that such an automated implementation would be chosen by very smart scientists.
Plus, no matter how secure your system is, it can still be infiltrated and corrupted to react in a way you didn't expect. For example an enemy could trigger your secret auto-defensive system intentionally in order to build himself a very nice reason to strike you back and destroy you without being bothered by any other countries.
EDIT: To answer the "mail solution" in an answer below (because it raises an interesting problem).
To solve the problem of false alarm one could say:
- Send mail to people and wait for response.
- Put some people in a confined box and wait for signals.
But those are just some implementations of an abstract conceptual solution to a more general problem: how do I check the existence of something?
No matter how clever this system is, this is just a ping. So... how can I be certain that what I see is what it is? How can I be certain that the answer comes from where I think it comes? How can I be certain that I waited long enough?
Any signals can be rerouted, modified, blocked, etc.
Let's do this with an Artificial Intelligence?
So what to do? Since we have human which are not reliable but flexible, cheap and "intelligent enough" to make proper decisions. Machine are reliable but the threat of a systematic bias, mechanical flaws or bugs is very likely and would result in an uncontrolled launch of nuclear missile.
One could think of artificial intelligence in order to have the best of the two worlds, a very docile and unbreakable mind which could "evaluate" a good behaviour in any unpredicted situation... this machine could also verify its own integrity and fix any bias and bugs it would find. Like a learning machine would do. We give this AI the goal : "fire a strike back if someone attacks us". And we let it implements the most optimised solution.
This is an even worse idea. This goal could be subject to perverse instantiation : the machine will not do what we had in mind.
My conclusion is that building a weapon which can decide to kill on its own is a pretty bad idea unless you find a way to be certain that the system is safe. Then whatever the system you chose, it will work since you designed it to be unbreakable. They way does not matter because I think this is not achievable with hard-science level of rigour.
Just assume your system works :) for example you can say that in your world, your sensors are overwhelmingly reliable. You can push the probability as far as you want by checking into your system with arbitrarily good precision.