## Turing machines with few accepting computations and low sets for PP.(English)Zbl 0757.68056

Complexity classes, defined using the idea of bounding the number of accepting paths of a nondeterministic Turing machine, were introduced with the hope that perhaps they represent more tractable subclasses of the class $$NP$$. The paper investigates some properties of the path- restricted class $$Few$$, which includes also other known classes of this type, i.e. $$UP$$ and $$FewP$$.
It is shown that for every language in this class there exists a polynomial time nondeterministic machine that has exactly $$f(x) + 1$$ accepting paths for strings in the language, and $$f(x)$$ accepting paths otherwise (for a polynomial time computable function $$f$$). This result is then used to prove that $$Few$$ is low for the complexity classes $$PP$$, $$\oplus P$$, and exact counting, i.e. an oracle from $$Few$$ does not increase computational power of machines from these classes. Lowness for $$PP$$ is shown also for sets from the class $$BPP$$ and sparse sets in $$NP$$.

### MSC:

 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 03D15 Complexity of computation (including implicit computational complexity)
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### References:

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