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**A complexity theory for feasible closure properties.**
*(English)*
Zbl 0798.68060

Summary: The study of the complexity of sets encompasses two complementary aims: (1) establishing – usually via explicit construction of algorithms – that sets are feasible, and (2) studying the relative complexity of sets that plausibly might be feasible but are not currently known to be feasible (such as the NP-complete sets and the PSPACE-complete sets). For the study of the complexity of closure properties, a recent flurry of results has established an analog of (1); these papers explicitly demonstrate many closure properties possessed by PP and \(\text{C}_ =\text{P}\) (and the proofs implicitly give closure properties of the function class #P). The present paper presents and develops, just for function classes such as #P, SpanP, OptP, and MidP, and analog of (2): a general theory of the complexity of closure properties. In particular, we show that subtraction is hard for the closure properties of each of these classes: each is closed under subtraction if and only if it is closed under every polynomial-time operation. Previously, no property – natural or unnatural – had been known to have this behavior. We also prove other natural operations hard for the closure properties of #P, SpanP, OptP, and MidP, and we explore the relative complexity of operations that seem not to be #P-hard, such as maximum, minimum, decrement, and median. Moreover, for each of #P, SpanP, OptP, and MidP, we give a natural complete characterization – in terms of the collapse of complexity classes – of the conditions under which that class has every feasible closure property.

### MSC:

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

03D15 | Complexity of computation (including implicit computational complexity) |

68Q25 | Analysis of algorithms and problem complexity |

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