TY - JOUR

T1 - A complexity theory for feasible closure properties

AU - Ogiwara, Mitsunori

AU - Hemachandra, Lane A.

N1 - Funding Information:
* Work done in part while at the Tokyo Institute of Technology. + Research supported in part by the National Science Foundation under Grants NSF-INT-9116781/JSPS-ENG-207, CCR-8957604, and CCR-8996198 and by the International Information Science Foundation under Grant 90-l-3-228.

PY - 1993/6

Y1 - 1993/6

N2 - 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 C=P (and the proofs implicitly give closure properties of the function class #P). The present paper presents and develops, for function classes such as #P, SpanP, OptP, and MidP, an 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.

AB - 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 C=P (and the proofs implicitly give closure properties of the function class #P). The present paper presents and develops, for function classes such as #P, SpanP, OptP, and MidP, an 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.

UR - http://www.scopus.com/inward/record.url?scp=0000321801&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000321801&partnerID=8YFLogxK

U2 - 10.1016/0022-0000(93)90006-I

DO - 10.1016/0022-0000(93)90006-I

M3 - Article

AN - SCOPUS:0000321801

VL - 46

SP - 295

EP - 325

JO - Journal of Computer and System Sciences

JF - Journal of Computer and System Sciences

SN - 0022-0000

IS - 3

ER -