##
**Gap-definable counting classes.**
*(English)*
Zbl 0802.68051

Summary: The function class \(\# {\mathbf P}\) lacks an important closure property: it is not closed under subtraction. To remedy this problem, we introduce the function class \(\text{Gap} {\mathbf P}\) as a natural alternative to \(\# {\mathbf P}\). \(\text{Gap} {\mathbf P}\) is the closure of \(\# {\mathbf P}\) under subtraction and has all the other useful closure properties of \(\# {\mathbf P}\) as well. We show that most previously studied counting classes, including \({\mathbf {PP}}\), C\(_ = {\mathbf P}\), and \(\text{Mod}_ k {\mathbf P}\), are “gap-definable,” i.e., definable using the values of \(\text{Gap} {\mathbf P}\) functions alone. We show that there is a smallest gap-definable class, \({\mathbf {SPP}}\), which is still large enough to contain Few. We also show that \({\mathbf S}\)PP consists of exactly those languages low for \(\text{Gap} {\mathbf P}\), and thus \({\mathbf S}\)PP languages are low for any gap-definable class. These results unify and improve erlier disparate results of J. Cai and L. Hemachandra [Math. Syst. Theory 23, No. 2, 95-106 (1990; Zbl 0718.68038)] and J. Köbler et al. [J. Comput. System Sci. 44, No. 2, 272-286 (1992; Zbl 0757.68056)]. We show further that any countable collection of languages is contained in a unique minimum gap-definable class, which implies that the gap-definable classes form a lattice under inclusion. Subtraction seems necessary for this result, since nothing similar is known for the \(\# {\mathbf P}\)-definable classes.

### MSC:

68Q15 | Complexity classes (hierarchies, relations among complexity classes, etc.) |

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