Explaining the Ubiquity of Phase Transitions in Decision Problems

I present an analytic approach to establishing the presence of phase transitions in a large set of decision problems. This approach does not require extensive computational study of the problems considered. The set – that of all paddable problems over even-sized alphabets satisfying a condition similar to not being sparse – shown to exhibit phase transitions contains many “practical” decision problems, is very large, and also contains extremely intractable problems.