Complexity Classes Without Machines: On Complete Languages for UP

This paper develops techniques for studying complexity classes that are not covered by known recursive enumerations of machines. Often, counting classes, probabilistic classes, and intersection classes lack such enumerations. Concentrating on the counting class UP, we show that there are relativizations for which $UP^{A}$ has no complete languages and other relativizations for which $P^{B} \neq UP^{B} \neq NP^{B}$ and $UP^{B}$ has complete languages. Among other results we show that $P \neq UP$ if and only if there exists a set $S$ in $P$ of Boolean formulas with at most one satisfying assignment such that $S \bigcap SAT$ is not in $P$. $P \neq UP \bigcap coUP$ if and only if there exists a set $S$ in $P$ of uniquely satisfiable Boolean formulas such that no polynomial-time machine can compute the solutions for the formulas in $S$. If $UP$ has complete languages then there exists a set $R$ in $P$ of Boolean formulas with at most one satisfying assignment so that $SAT \bigcap R$ is complete for $UP$. Finally, we indicate the wide applicability of our techniques to counting and probabilistic classes by using them to examine the probabilistic class $BPP$. There is a relativized world where $BPP^{A}$ has no complete languages. If $BPP$ has complete languages then it has a complete language of the form $B \bigcap MAJORITY$, where $B \in P$ and $MAJORITY = \{f | f$ is true for at least half of all assignments\} is the canonical $PP$-complete set