Some of the comments on this thread are so far off base I feel motivated to make my annual pprune post.
First, the mechanics of flight automation. Certainly there are many challenges, and if you lack a doctorate and / or several years of experience in the field you're probably not able to identify many of them. In my experience a higher degree gets you to the point of truly understanding how little you know about a topic. So here are a few opinions from someone who knows just a little about flight automation.
Perhaps 80% of the technical problems of commercial flight automation have already been solved, some of them for many years. The other 20% are harder or more esoteric, but are not necessarily what you would expect. For example taxiing to the gate is one of the more challenging activities (for humans too, sometimes). Problem solving and mitigation is a curious mix of the blindingly obvious and the complex, and has been an active field of research and development for decades.
Many doubts have been expressed about high reliability software. This a reasonably mature field and more a matter of development time and cost than research. Many modern aircraft are already fully reliant on sophisticated software for flight safety. As a pilot you'll be aware of some of these systems, but the majority are silent and embedded throughout the avionics and other aircraft systems. Increasingly there is no such thing as manual handling - software is always in the loop. In the last couple of decades we've become quite good at designing software in such a way that the inevitable bugs do not become critical problems. Bug identification in avionic systems is relatively rare (but the systems are large so the report rate certainly isn't zero). Bugs that are causal or even a factor in a serious incident are exceptionally rare - far more so than pilot finger trouble or mechanical failure.
Please stop talking about a model of the human brain being a gate to flight automation. This is a fallacy. While neural networks are utilised in many fields of pattern recognition (and are present in aircraft systems today, heavily so in military applications), the sophistication of human reasoning is far from a prerequisite of automation. Many disciplines are involved and the catch all term of "AI" could mean anything from a state machine to a knowledge engine. We do not need to wait for the realisation of science fiction for this field to progress.
Flight automation is generally considered a technically easier problem to solve than road vehicle automation. Self driving vehicle developers will claim that their field is a lot harder (but they would). This is self evident to anyone who considers that commercial drones (autonomous, remote piloted and hybrid) have been on the market for years, and flying as research projects for many decades. Self driving cars are still some way off, although dozens of research vehicles are on public roads right now with a human minder. Although cars lack the degrees of freedom of an aircraft, their basic operation is nonetheless more complex to automate due to the complexity of terrain, a less predictable environment and the immediacy of criticality with many failure modes.
It was mentioned that you can't just stop an aircraft when it has a problem - true, and that's a good thing. If you can maintain a sensible attitude you may have seconds, minutes or even hours to solve a problem. Given that the human strength for problem solving is in troubleshooting unplanned issues, there remains the possibility of human fall-back in such events, even if this is remote. Automation is already more appropriate than human handling for short decision-time events. That the immediate response of an AP to a problem is to disengage is a reflection of a design approach developed in the 1950s. In more recent times we've been flying research aircraft with unconventional control and sensor inputs for many years and it's safe to claim that automation would have a greater chance of recovering an aircraft with damaged control response than a human pilot.
Of course some accidents that might be mitigated by a human may have a less satisfactory outcome with automation. It seems likely, however, that a greater number of accidents may be avoided through the application of automation. As such I believe that many of the systems required for automation of the complete flight profile will be incorporated into human piloted aircraft anyway, as part of the natural development of avionic systems. Actually this has been happening for years, and would happen more quickly if there were greater cost benefit (aircraft don't crash often enough to offset the cost).
Which brings us on to the motivation to remove the pilot. If automation can be deployed with an improvement (or at least no reduction) in accident rates then it's my belief that safety will not be a factor. Neither will it be a motivator for the reason just mentioned, unless the cost starts to be offset by insurance savings. Crew reduction, however, is certainly a factor. It's been said repeatedly that the cost of the flight crew is insufficient to justify the cost and complexity of automation, but that was not the case with navigation and flight engineering. Those roles are redundant for commercial benefit and this was made possible by the development of automation. That automation is not as complex as an unmanned flight deck, but 40 years ago the challenges were nonetheless substantial.
For me, the most relevant predictor of flight automation is the rail industry. Let's be clear about the technology - all aspects of rail operation can be and have been fully automated. Even so, the majority of modern rail systems have an on-board human driver. In some cases the human is fully redundant and is not actually driving the train, but is present for passenger perception and/or commercial concerns. The most commonly cited justification is for emergency response, as if one man can effectively marshal several hundred passengers in the event of an accident. Even if this were the case, a train manager would be a cheaper and more appropriate role to execute this function.
Surely flight automation will face this same hurdle. Perhaps few outside of the industry believe it right now, but I'm confident that the technology could be feasible within the next couple of generations of avionics. It could happen more quickly if there was demand. However I find it hard to believe that passengers and trade unions will accept this development easily. I think it likely that we'll see assisted and then fully automated road cars before we see an unmanned flight deck. Although the road car problem is harder to solve, the benefit is visible and huge - a large proportion of road deaths can and will be prevented. Perhaps that will reassure passengers that public transportation can also benefit from automation.
Until that happens the role of the pilot is going to become increasingly dull, as the reach of flight automation continues to grow. Perhaps the role of co-pilot will become the first casualty once automation and captain can be considered each other's redundancy.
All IMHO.