Explaining the Ubiquity of Phase Transitions in Decision Problems
Andrew Jackson
TL;DR
The paper addresses why phase transitions are a pervasive feature of decision problems and introduces a non-computational framework based on paddability and RoughP to certify their existence across broad problem classes. It formalizes phase-transition notions via the acceptance fraction $A[\mathcal{S}^{\gamma}_{n}]$, and constructs a canonical parameter $\Gamma(x)=\mathcal{Q}(x)\sqrt{N_{\phi}(x)}$ using a P-isomorphism $\phi$ and a discriminator $\mathcal{Q}$. The main result shows that every paddable not-anywhere-exponentially-unbalanced language over an even-sized alphabet exhibits a phase transition, with rigorous bounds: for $\tau>0$, $\mathcal{A}'[\mathcal{S}^{\Gamma}_{\tau}]$ is near $1$, and for $\tau<0$, it is near $0$, while input density away from the threshold grows exponentially; the inverted form of the transition is shown equivalent to the canonical form. This framework explains the ubiquity of phase transitions and provides a general method to analyze broad classes of decision problems, though current results are restricted to even-sized alphabets and paddable languages.
Abstract
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.