This is a post separated from Consequences of UP equals NP, and also a follow-up question to Semantic vs. Syntactic Complexity Classes.In the above post we learned about the semantic and syntactic classes. Briefly stated, when a class can be characterized as a leaf language class $\mathsf{L}[L_1|L_2]$, then a class is syntactic if $L_1 \cup L_2 = \Sigma^*$, that is, accepting language $L_1$ is the complement of rejecting language $L_2$; otherwise we called it a semantic class.One can see that $\mathsf{P}$, $\mathsf{NP}$ and $\mathsf{PP}$ are syntactic classes,while classes like $\mathsf{BPP}$ and $\mathsf{IP}$ are semantic classes.Classical result like $\mathsf{PSPACE} = \mathsf{IP}$ and conjecture $\mathsf{P} \stackrel{?}{=} \mathsf{BPP}$ both can be view as semantic classes turns out to have syntactic characterizations. It seems to me that the syntactic classes are easier to handle, since they have natural complete problems. Also techniques like diagonalization is easier to be applied on syntactic classes, since they have a natural machine enumeration. But still $\mathsf{BPP}$ as a semantic class seems to have much more nice properties than the syntactic class $\mathsf{PP}$. What benefits do we have if we have a syntactic representation of a semantic class, or vice versa? Are there results or proof techniques only applied to syntactic/semantic classes?
Benefits for syntactic and semantic classes
cc.complexity theory;complexity classes;big picture