## Complete Sets and Structure in Subrecursive Classes (1998)

Venue: | In Proceedings of Logic Colloquium '96 |

Citations: | 14 - 1 self |

### BibTeX

@INPROCEEDINGS{Buhrman98completesets,

author = {Harry Buhrman and Leen Torenvliet},

title = {Complete Sets and Structure in Subrecursive Classes},

booktitle = {In Proceedings of Logic Colloquium '96},

year = {1998},

pages = {45--78},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completeness of several problems for NP by Cook [Coo71] and Levin [Lev73] and for many other problems by Karp [Kar72], the interest in completeness notions in complexity classes has tremendously increased. Virtually every form of reduction known in computability theory has found its way to complexity theory. This is usually done by imposing time and/or space bounds on the computational power of the device representing the reduction. Early on, Ladner et al. [LLS75] categorized the then known types of reductions and made a comparison between these by constructing sets that are reducible to each other via one type of reduction and not reducible via the other. They however were interested just in the rela...

### Citations

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Reducibility Among Combinatorial Problems
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- 1972
(Show Context)
Citation Context ...e induced by various notions of reductions. 1 Introduction After the demonstration of the completeness of several problems for NP by Cook [Coo71] and Levin [Lev73] and for many other problems by Karp =-=[Kar72]-=-, the interest in completeness notions in complexity classes has tremendously increased. Virtually every form of reduction known in computability theory has found its way to complexity theory. This is... |

774 | The Complexity of Theorem-Proving Procedures
- COOK
- 1971
(Show Context)
Citation Context ...iew of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completeness of several problems for NP by Cook =-=[Coo71]-=- and Levin [Lev73] and for many other problems by Karp [Kar72], the interest in completeness notions in complexity classes has tremendously increased. Virtually every form of reduction known in comput... |

235 |
Some connections between nonuniform and uniform complexity classes
- Karp, Lipton
- 1980
(Show Context)
Citation Context ... sets with low instance complexity everywhere, as in the corollary, are in P=poly. Assume for a contradiction that the corollary is not true, it now follows by a well-known theorem of Karp and Lipton =-=[KL80]-=- that if EXP has asp T -hard set that is sparse, that then EXP = \Sigma P 2 . Furthermore, \Sigma P 2 contains self-reducible Turing complete sets and hence by Theorem 33 above it follows that \Sigma ... |

162 |
On isomorphisms and density of NP and other complete sets
- Berman, Hartmanis
- 1977
(Show Context)
Citation Context ...PjEXP) 6= 0. Does thesp 1\Gammatt -complete degree on NPcoincide with thesp m-complete degree? 14 4 Isomorphism Special among the complete degrees are the degrees of isomorphism. Berman and Hartmanis =-=[BH77]-=- proved that all natural NP-complete sets are interreducible via length-increasing 1-1 reductions and could therefore, via a polynomial time analog of the Cantor-Bernstein-Myhill theorem show that the... |

58 | New collapse consequences of np having small circuits
- Kobler, Watanabe
- 1998
(Show Context)
Citation Context ...this question seems particularly interesting for EXP. (Until now the smallest class known not to be computable by polynomial size circuits is MA(exp).) This result is due to Buhrman and Thierauf. See =-=[KW95]-=-) There are some steps set along this path however. Theorem 20 ([Fu93]) 1. For ff ! 1, allsp n ff \GammaT -hard sets for EXP are exponentially dense. 2. For ff ! 1 4 , allsp n ff \GammaT -hard sets fo... |

50 | Sparse sets - Hartmanis, Immerman, et al. - 1985 |

49 | Measure on small complexity classes with applications for BPP
- Allender, Strauss
- 1994
(Show Context)
Citation Context ... -complete degree for E has measure 0 in E [ASNT94]. The question remains open forsp T -complete sets for E. Some progress has been made however. Allender and Straus showed the following. Theorem 15 (=-=[AS94]-=-) For almost every set A in EXP, BPP A = P A . This theorem shows that if BPP would be equal to EXP then the Turing complete sets for EXP would not have measure 0. Hence a proof that the Turing comple... |

46 | The complexity and distribution of hard problems - Juedes, Lutz - 1995 |

44 |
Polynomial Reducibilities and Complete Sets
- Berman
- 1977
(Show Context)
Citation Context ... closely related to the number of strings that a set has per length. It therefore comes as no big surprise that a sparse set cannot besp m-complete for EXP. This follows from an old theorem by Berman =-=[Ber77]-=-. For NP this question is a lot harder (if P = NP then any set in P issp m-hard for NP so also the sparse ones). It was answered by Mahaney [Mah82], building on earlier 18 work of Fortune [For79], who... |

41 | The isomorphism conjecture fails relative to a random oracle
- Kurtz, Mahaney, et al.
- 1995
(Show Context)
Citation Context ... unsuccessful work in trying to prove isomorphism of allsp m-complete sets in NP, and oracle proof that the conjecture might not hold (most prominent, the conjecture fails relative to a random oracle =-=[KMR89]-=-) opinion shifted against the idea that the isomorphism conjecture might be true. It was not until 1992 that Fenner, Fortnow and Kurtz [FFK92] proved the existence of an oracle relative to which the i... |

36 |
How hard are sparse sets
- Hemachandra, Ogiwara, et al.
- 1992
(Show Context)
Citation Context ...written on the structure of complete sets in complexity classes. The first were published in 1990 [Hom90, KMR90]. On complete sets with special structure (sparse sets) a survey was published in 1992 (=-=[HOW92]-=-). The present authors presented a survey in 1994 ([BT94]) in a paper that has roughly the same structure as this paper. The field is however rapidly expanding and needs surveys such as this and as [H... |

32 | Two queries
- Buhrman, Fortnow
- 1996
(Show Context)
Citation Context ...et with a dense subset of high generalized Kolmogorov complexity, Watanabe and Tang [WT89] show that the many-one and Turing complete degrees differ on PSPACE. From recent work of Buhrman and Fortnow =-=[BF96]-=- it follows that there exists a relativized world in which thesp m and thesp 1\Gammatt complete on PSPACE differ. On NP most problems remain open. With respect tosp m - andsp 1\Gammatt - complete degr... |

30 | Reductions in circuit complexity: An isomorphism theorem and a gap theorem
- AGRAWAL, ALLENDER, et al.
- 1998
(Show Context)
Citation Context ...ne and 1-tt, which coincide. (See Theorem 17) In connection with the isomorphism problem (See Theorem 13), degrees defined by reducibilities even stronger thansp m have been studied. In particular in =-=[AAR96]-=- the AC 0 degree is shown to collapse to the NC 0 degree for every complexity classe C that is closed under NC 1 reductions. On the class NP all relations of polynomial time complete degrees necessari... |

29 | The structure of complete degrees - Kurtz, Mahaney, et al. - 1990 |

28 |
P-mitotic sets
- Ambos-Spies
- 1984
(Show Context)
Citation Context ...n precisely the information of the original set. Ladner showed that the two seemingly different, but apparently related notions of auto-reducibility and mitoticity coincide for r.e. sets. Ambos-Spies =-=[AS84]-=- was the first to carry over the notions of autoreducibility and mitoticity to the realm of complexity theory. The autoreducibility notion translates into: Definition 39 A set A is polynomially autore... |

28 | Genericity and measure for exponential time
- Ambos-Spies, Neis, et al.
- 1996
(Show Context)
Citation Context ...hesp m -complete degree for E has measure 0 in E [JL93, May94a]. 2. For all k, thesp k\Gammatt -complete degree for E has measure 0 in E [BM95]. 3. Thesp btt -complete degree for E has measure 0 in E =-=[ASNT94]-=-. The question remains open forsp T -complete sets for E. Some progress has been made however. Allender and Straus showed the following. Theorem 15 ([AS94]) For almost every set A in EXP, BPP A = P A ... |

28 |
A note on sparse complete sets
- FORTUNE
- 1979
(Show Context)
Citation Context ...rman [Ber77]. For NP this question is a lot harder (if P = NP then any set in P issp m-hard for NP so also the sparse ones). It was answered by Mahaney [Mah82], building on earlier 18 work of Fortune =-=[For79]-=-, who showed that sparse sets could not besp m - complete unless NP = P. For P the question needs reformulation. Under p m -reductions any set is complete in P, but underslogspace m -reductions this i... |

28 |
Some remarks on witness functions for nonpolynomial and noncomplete sets
- Joseph, Young
- 1985
(Show Context)
Citation Context ... conjecture might be true. It was not until 1992 that Fenner, Fortnow and Kurtz [FFK92] proved the existence of an oracle relative to which the isomorphism conjecture holds. In 1985, Joseph and Young =-=[JY85]-=- constructed unnatural sets, the so-called k-creative sets for which every 1-1 polynomial time computable honest function is a productive function. Hence these sets are very unlikely to be isomorphic ... |

27 | Selective self-reducible sets: A new characterization of P
- Buhrman, Helden, et al.
- 1993
(Show Context)
Citation Context ...]) Allsp T -complete sets for all levels of the Polynomial Hierarchy and PSPACE are autoreducible. Unfortunately the techniques in [BF92] only apply to sets within PSPACE. Extending the techniques in =-=[BT96]-=- it can be shown that also the complete sets for EXP are autoreducible. Theorem 46 ( [BFT95]) Allsp T -complete sets for PSPACE and EXP are autoreducible. It can also be shown that allsp 2\Gammatt -co... |

26 | On being incoherent without being very hard
- Beigel, Feigenbaum
- 1992
(Show Context)
Citation Context ...ere made for mitoticity are true for autoreducibility but thesp T - complete sets seem to behave differently. Theorem 44 ( [BF92, BT96]) Everysp T -complete set for NP is autoreducible. 28 In fact in =-=[BF92]-=- it is shown that allsp T -degrees that contain a selfreducible set are completely autoreducible hence: Theorem 45 ( [BF92]) Allsp T -complete sets for all levels of the Polynomial Hierarchy and PSPAC... |

26 | The isomorphism conjecture holds relative to an oracle
- Fenner, Fortnow, et al.
- 1996
(Show Context)
Citation Context ...ominent, the conjecture fails relative to a random oracle [KMR89]) opinion shifted against the idea that the isomorphism conjecture might be true. It was not until 1992 that Fenner, Fortnow and Kurtz =-=[FFK92]-=- proved the existence of an oracle relative to which the isomorphism conjecture holds. In 1985, Joseph and Young [JY85] constructed unnatural sets, the so-called k-creative sets for which every 1-1 po... |

25 | A first-order isomorphism theorem
- Allender, Balcazar, et al.
- 1997
(Show Context)
Citation Context ...er 1-L reductions are all p-isomorphic. The first order projection (defined by Valiant in [Val82]) is another example of a very strong form of reduction. Allender, Balc'azar and Immerman 15 showed in =-=[ABI93]-=- that for the first order projections an isomorphism theorem holds Theorem 12 ([ABI93]) Let C be a nice complexity class, e.g. P, NP, PSPACE. All sets complete for C under first order projections are ... |

25 |
On log-tape isomorphisms of complete sets
- Hartmanis
- 1978
(Show Context)
Citation Context ... but underslogspace m -reductions this is only true if P = LOG. An analogous question about the density of logspace m -complete sets in P can thus be posed. Indeed, it was conjectured by Hartmanis in =-=[Har78]-=- that no sparse set can be complete for P under logspace reductions unless P = LOG. It was not until recently that this conjecture was proven true. Theorem 18 ([Ogi95, CS95]) If a sparse set S is hard... |

24 | Using autoreducibility to separate complexity classes
- Buhrman, Fortnow, et al.
(Show Context)
Citation Context ...al. [OKSW94]. 3 Set Structure: Post's Program Revisited Autoreducibility is a special form of reducibility introduced by Trakhtenbrot [Tra70]. This notion has received considerable attention recently =-=[BFT95]-=- because of its potential to discover answers to the fundamental questions. Autoreducibility is a structural notion that complete sets in some complexity classes do and complete sets in other complexi... |

22 | Superpolynomial circuits, almost sparse oracles and the exponential hierarchy
- Buhrman, Homer
- 1992
(Show Context)
Citation Context ...lowing questions. ffl For any k: are thesp n k \GammaT complete sets for EXP exponentially dense? ffl For any k: are thesp n k \GammaT weakly complete sets for EXP exponentially dense? ffl Related to =-=[BH92a]-=-, are thesp T -complete sets for NEXP not sparse unless NEXP = \Sigma P 2 ? 7 Redundant Information As noted in the previous section, some complexity classes do not allow for sparse complete sets unde... |

22 | On the structure of complete sets
- Buhrman, Torenvliet
- 1994
(Show Context)
Citation Context ...asses. The first were published in 1990 [Hom90, KMR90]. On complete sets with special structure (sparse sets) a survey was published in 1992 ([HOW92]). The present authors presented a survey in 1994 (=-=[BT94]-=-) in a paper that has roughly the same structure as this paper. The field is however rapidly expanding and needs surveys such as this and as [Hom96] for constant update. 2 Preliminaries All sets (and ... |

18 | Resource Bounded Reductions
- Buhrman
- 1993
(Show Context)
Citation Context ...ete sets are all different and, moreover, when not obviously ordered by inclusion, incomparable. In [BST93b] also the degrees of query-bounded reductions are compared and it turns out that Theorem 9 (=-=[BST93a]-=-) For C 2 fE; EXP;NE;NEXPg 1. For any ks2,sp k\Gammac -, andsP k\Gammad -completeness are incomparable on C 2. For any k and l, with k ! ls2 k \Gamma 2,sp k\GammaT - andsp l\Gammatt -completeness are ... |

17 | An excursion to the Kolmogorov random strings - Buhrman, Mayordomo - 1997 |

17 | On using oracles that compute values - Fenner, Homer, et al. - 1993 |

16 | Splittings, robustness, and structure of complete sets
- Buhrman, Hoene, et al.
- 1998
(Show Context)
Citation Context ...lete. The natural question to ask next after the result is obtained for manyone reductions is: "How do complete sets under weaker types of reductions behave?" The answer to this question was=-= given in [BHT93]-=-. They show 20 that the observation on this structural aspect of complete sets is not limited to many one completeness or to deterministic exponential time. Theorem 23 Given a recursive non-decreasing... |

15 |
Isomorphisms and 1-L reductions
- Allender
- 1988
(Show Context)
Citation Context ...ern versions of the isomorphism conjecture are more in the direction of stronger reductions. Most of the sets known to besp m-complete in NP are also complete under much stronger reductions. Allender =-=[All88]-=- was the first to show that sets in PSPACE, complete under 1-L reductions (which is a function computable by a logspace bounded Turing machine that has a one-way input head) are polynomial time isomor... |

15 | Structural properties of nondeterministic complete sets - Homer - 1990 |

14 | Structural properties of complete problems for exponential time
- Homer
- 1997
(Show Context)
Citation Context ...2]). The present authors presented a survey in 1994 ([BT94]) in a paper that has roughly the same structure as this paper. The field is however rapidly expanding and needs surveys such as this and as =-=[Hom96]-=- for constant update. 2 Preliminaries All sets (and languages) in this paper are subsets of \Gamma , where \Gamma = f0; 1g, and are denoted by capital letters A, B, C etc. Strings are elements of \Gam... |

13 | The resolution of a Hartmanis conjecture - Cai, Sivakumar - 1995 |

12 |
A comparison of weak completeness notions
- Ambos-Spies, Mayordomo, et al.
- 1996
(Show Context)
Citation Context ...et on exponential time, the weakly complete set. A set A is weakly hard under reductionsr , if a non-significant part (i.e., a class with non-zero p-measure) of E reduces to A. It was not known until =-=[ASMZ96]-=- whether the degrees of weakly complete sets under various reduction types are different. As it turns out, these degrees behave the same as the classical complete degrees. All are different except for... |

12 | With Quasi-linear Queries EXP is Not Polynomial-time Turning Reducible to Sparse Sets
- Fu
- 1995
(Show Context)
Citation Context ...mallest class known not to be computable by polynomial size circuits is MA(exp).) This result is due to Buhrman and Thierauf. See [KW95]) There are some steps set along this path however. Theorem 20 (=-=[Fu93]-=-) 1. For ff ! 1, allsp n ff \GammaT -hard sets for EXP are exponentially dense. 2. For ff ! 1 4 , allsp n ff \GammaT -hard sets for E are exponentially dense. An incomparable theorem dealing with the ... |

11 |
Polynomial isomorphism of 1-L complete sets
- Agrawal, Biswas
- 1993
(Show Context)
Citation Context ...nd that these sets are not necessarily isomorphic under 1-L computable isomorphisms. The following theorem by Agrawal and Biswas, is the most general theorem known for the 1-L reductions. Theorem 11 (=-=[AB93]-=-) Let C be a complexity class that is closed under lin-log reductions, e.g. P, NP, PSPACE. The sets complete for C under 1-L reductions are all p-isomorphic. The first order projection (defined by Val... |

10 | Random strings make hard instances
- Buhrman, Orponen
- 1996
(Show Context)
Citation Context ...ly Fortnow and Kummer [FK95] showed the instance complexity conjecture in a special case 24 Theorem 35 ([FK95]) Every tally set not in P has P-hard instances. Shifting to complete sets it is shown in =-=[BO94]-=- that the conjecture is correct forsp m-complete sets for E and polynomial time bounds t. Theorem 36 ( [BO94]) Let A be asp m -complete set for E. Then there exists an exponentially dense set C ` A, s... |

7 | L.: The relative power of logspace and polynomial time reductions
- Buhrman, Spaan, et al.
- 1993
(Show Context)
Citation Context ...of the p m -complete sets. 2. The degrees of thesp m -,sp btt -,sp c -,sp d -,sp tt - andsp T -complete sets are all different and, moreover, when not obviously ordered by inclusion, incomparable. In =-=[BST93b]-=- also the degrees of query-bounded reductions are compared and it turns out that Theorem 9 ([BST93a]) For C 2 fE; EXP;NE;NEXPg 1. For any ks2,sp k\Gammac -, andsP k\Gammad -completeness are incomparab... |

6 | Nonuniform Lower Bounds for Exponential Time Classes
- Homer, Mocas
- 1995
(Show Context)
Citation Context ...ty of exponential time computable sets being decidable with a fixed polynomial amount of advice. (Recall that if EXP has sparse complete sets then EXP ` P=poly.) They prove the following. Theorem 22 (=-=[HM93]-=-) for every k there exists a set A in EXP such that A is not in DTIME(2 n k )=ADVICE(n k ). Improving upon these results seems very interesting, but also seems very hard since it would require non rel... |

5 | A note on many-one and 1-truth table complete sets - Homer, Kurtz, et al. - 1993 |

4 |
The degree structure of 1-L reductions
- Burtschick, Hoene
- 1992
(Show Context)
Citation Context ... sets in PSPACE, complete under 1-L reductions (which is a function computable by a logspace bounded Turing machine that has a one-way input head) are polynomial time isomorphic. Burtschick and Hoene =-=[BH92b]-=- showed on the other hand that these sets are not necessarily isomorphic under 1-L computable isomorphisms. The following theorem by Agrawal and Biswas, is the most general theorem known for the 1-L r... |

4 | On complete sets for nondeterministic classes - Buhrman, Homer, et al. - 1991 |

4 |
Immunity of complete problems
- Homer, Wang
- 1994
(Show Context)
Citation Context ...omial? 2. If A is complete for EXP undersp tt orsp T , does A have P-hard instances ? 3. Is the set of hard instances in 1 or 2 dense? 4. Do many-one complete sets for NP have t-hard instances? 2 See =-=[HW94]-=- for an extensive study on immunity andsp m -complete sets for EXP and NEXP. 25 9 Post's Program Revisited The final structural property that we wish to address in this paper is that of auto-reducibil... |

3 | Functional oracle queries as a measure of parallel time - Alvarez, Jenner - 1991 |

1 |
Resource bounded instance complexity
- Fortnow, Kummer
- 1995
(Show Context)
Citation Context ...any polynomial t there exists a constant c and infinitely many x, such that, ic t (x : A)sK 2 cn (x) \Gamma c. More recently, Fortnow and Kummer gave more evidence for this conjecture. 23 Theorem 32 (=-=[FK95]-=-) Let A be a set not in E. Then for any c there exists a c 0 and d such that for infinitely many x, ic 2 cn (x : A)sK 2 c 0 n (x) \Gamma d. We will now shift our attention to complete sets. 8.1 Comple... |