## Bi-immunity Results for Cheatable Sets (1995)

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Venue: | Theoretical Computer Science |

Citations: | 10 - 6 self |

### BibTeX

@ARTICLE{Beigel95bi-immunityresults,

author = {Richard Beigel},

title = {Bi-immunity Results for Cheatable Sets},

journal = {Theoretical Computer Science},

year = {1995},

volume = {73},

pages = {73--3}

}

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### Abstract

An oracle A is k-cheatable if there is a polynomial-time algorithm to determine the answers to 2 k parallel queries to A from the answers to only k queries to some other oracle B. It is known that 1-cheatable sets cannot be bi-immune for P. In contrast, we construct 2-cheatable sets that are bi-immune for arbitrary time complexity classes. In addition, for each k, we construct a set that is (k + 1)-cheatable, but not k-cheatable; we show that this separation does not hold with biimmunity. We show that if a recursive set A is bi-immune for P then there exists an infinite 1-cheatable set that is polynomial-time mreducible to A. Consequently if NP contains a set that is bi-immune for P then NP contains a set that is not polynomial-time Turingequivalent to a self-reducible set. 1. Introduction Complexity theory deals with how hard problems are. Time, space, and alternation have served as measures of difficulty. Recently, researchers have Research supported by a Fannie and John Hertz ...

### Citations

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(Show Context)
Citation Context ... a string. We determine whether s 2 A by running the construction at stage log (jsj). The simulation of M i dominates the running time of the stage. The simulation can be performed in O(4 jsj ) steps =-=[22]-=-. This time is linear in the length of the input. We use our k parallel queries to determine the membership of the other k strings. We immediately obtain the following: Corollary 22 Let ks1. Then i. T... |

247 |
On the structure of polynomial time reducibility
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- 1975
(Show Context)
Citation Context ...ithm relative to B that determines whether x is in A by making only k parallel queries to B. The definitions above are equivalent to definitions made by Book and Ko [14] and Ladner, Lynch, and Selman =-=[25]-=-. Note that our definitions of polynomial-time k-Turing reducibility and polynomial time k-truth-table reducibility make sense if A or B or both are replaced by polynomial lengthbounded functions, ins... |

193 |
The complexity of optimization problems
- Krentel
- 1988
(Show Context)
Citation Context ...aduate student at Stanford University. 1 looked at the number of queries that must be made to an oracle by a polynomial time algorithm that solves a problem, as a measure of that problem's difficulty =-=[15, 24]. When we -=-consider number-of-queries as a complexity measure, it is natural to consider the "complexity classes" of functions and sets induced by that measure. We call these complexity classes bounded... |

162 |
HARTMANIS: On isomorphisms and density of NP and other complete sets
- BERMAN, JURIS
- 1977
(Show Context)
Citation Context ...nitely often, i.e., whether they cannot be bi-immune for P. In [3], Balcazar and Schoning formalized the notion of being easy infinitely often, which was previously considered by Berman and Hartmanis =-=[13]-=- and by Rabin [26]. Definition 10 ffl A set A is immune for a class C if A contains no infinite subset that belongs to C. ffl A set A is co-immune for a class C ifsA is immune for C. ffl A set A is bi... |

72 | Some connections between bounded query classes and nonuniform complexity
- Amir, Beigel, et al.
- 1990
(Show Context)
Citation Context ...l non-p-superterse sets. For example, all cheatable sets are recursive [10]; however non-p-superterse sets can be nonrecursive [10, 11]. Cheatability is preserved by polynomial-time Turing reductions =-=[1]-=-; non-p-superterseness is not [1]. A cheatable set cannot be self-reducible unless it is in P [1]; no corresponding result is known for non-p-superterse sets. All cheatable sets are computable in poly... |

68 | Bounded queries to SAT and the boolean hierarchy
- Beigel
- 1991
(Show Context)
Citation Context ... to compute more functions than only k \Gamma 1 serial queries to B for every set B. The reason for defining p-superterseness is that most proofs of p-terseness are in fact proofs of p-superterseness =-=[2, 9, 7, 4, 5]-=-; in addition k-query psuperterseness is the strongest of our sixteen ways of stating that k queries to an oracle enable us to compute more functions in polynomial time than only k \Gamma 1 queries. A... |

54 |
Relative to a random Oracle A, P A 6= NP A 6= coNP A , with probability 1
- Bennet, Gill
- 1981
(Show Context)
Citation Context ...ing-reducible to B is in P. iii. By Theorem 19.i,ii,v there exists a set B such that BsP m A, B = 2 P, and B is not polynomial-time Turing-equivalent to any self-reducible set. Since Bennett and Gill =-=[12]-=- have shown that NP contains a language that is bi-immune for P under almost all relativizations, our assumption that NP contains a language that is bi-immune for P is plausible. (Since Homer and Maas... |

49 |
Bi-immune sets for complexity classes
- Balc'azar, Schoning
- 1985
(Show Context)
Citation Context ...ains no infinite subset that belongs to C. ffl A set A is co-immune for a class C ifsA is immune for C. ffl A set A is bi-immune for a class C if A is immune for C and co-immune for C. The authors of =-=[3, 13, 26]-=- have noted that a set A is easy infinitely often if and only if A is not bi-immune for P. Amir and Gasarch [2] have found an elegant proof that no 1-cheatable set is bi-immune for P; a constructive p... |

44 | Query-Limited Reducibilities
- Beigel
- 1987
(Show Context)
Citation Context ...ely " cheatable sets are recursive; by analogy the Nonspeedup Theorem suggests that cheatable sets might be easy in some standard complexitytheoretic sense. Amir and Gasarch [2] and (later) ourse=-=lves [8]-=- have shown that 1-cheatable sets cannot be bi-immune for P, so membership in a 1cheatable set can be decided in polynomial time infinitely often. In contrast, we show that 2-cheatable sets can be bi-... |

34 |
Optimal algorithms for self-reducible problems
- SCHNORR
- 1976
(Show Context)
Citation Context ...putable set. Because A is co-immune for P, the second possibility is ruled out, so S " A is an infinite 1-cheatable set. Because A is immune for P, the set S " A cannot be in P. ii. Let B = =-=S " A. In [27]-=-, Schnorr defined self-reducibility. Definition 18 A set A is self-reducible if there is a polynomial time bounded oracle Turing machine M such that ffl All strings queried by M are strictly shorter t... |

31 | Terse, superterse, and verbose sets
- Beigel, Gasarch, et al.
- 1993
(Show Context)
Citation Context ...ften? If F A 2 k can be computed by asking k queries to some other oracle within unbounded computation time (i.e., (9B)[F A 2 ksk-T B]), then we might say that A is recursively cheatable. However, in =-=[10] we prove -=-the Nonspeedup Theorem, which states that A is "recursively cheatable" in this sense if and only if A is recursive. By analogy, we might conjecture that all cheatable sets are in P. However,... |

30 |
On sets truth-table reducible to sparse sets
- Book, Ko
- 1988
(Show Context)
Citation Context ...1 queries?" The basic question in the preceding paragraph has at least sixteen interpretations. Queries may be made in series as in a Turing reduction (serial queries have also been called "=-=adaptive" [14]-=- because each query is allowed to depend on the answers to the previous queries), or in parallel as in a truthtable reduction. It may be required that the k \Gamma 1 queries be made to the same oracle... |

29 |
Polynomial terse sets
- Amir, Gasarch
- 1988
(Show Context)
Citation Context ... to compute more functions than only k \Gamma 1 serial queries to B for every set B. The reason for defining p-superterseness is that most proofs of p-terseness are in fact proofs of p-superterseness =-=[2, 9, 7, 4, 5]-=-; in addition k-query psuperterseness is the strongest of our sixteen ways of stating that k queries to an oracle enable us to compute more functions in polynomial time than only k \Gamma 1 queries. A... |

29 |
A structural theorem that depends quantitatively on the complexity of SAT
- Beigel
- 1987
(Show Context)
Citation Context ... to compute more functions than only k \Gamma 1 serial queries to B for every set B. The reason for defining p-superterseness is that most proofs of p-terseness are in fact proofs of p-superterseness =-=[2, 9, 7, 4, 5]-=-; in addition k-query psuperterseness is the strongest of our sixteen ways of stating that k queries to an oracle enable us to compute more functions in polynomial time than only k \Gamma 1 queries. A... |

28 | NP-hard sets are p-superterse unless R = NP
- Beigel
- 1988
(Show Context)
Citation Context |

27 |
On helping by robust oracle machines
- Ko
- 1987
(Show Context)
Citation Context ...be forthcoming.) We think that the conclusions in this theorem are suggestive of the likely behavior of sets in NP. Parts (ii) and (iii) of this theorem partially answer Selman's question in [28]; Ko =-=[23]-=- has also made some progress on Selman's question. By closely examining the proof of the preceding theorem, we can obtain a slightly stronger, but more complicated result. Let A be a set in NP that is... |

18 |
Natural self-reducible sets
- Selman
- 1988
(Show Context)
Citation Context ...k-cheatable does not hold with biimmunity, and it provides a plausible condition under which NP intersects a non-self-reducible polynomial-time Turing degree (partially answering a question of Selman =-=[28]-=-). 2. Bounded Query Reductions We say that n queries to an oracle are made in parallel if a list of all n queries is formed before any of them is made. Otherwise we say that n queries are made in seri... |

17 |
Degree of difficulty of computing a function and a partial ordering of recursive sets
- Rabin
- 1960
(Show Context)
Citation Context ..., whether they cannot be bi-immune for P. In [3], Balcazar and Schoning formalized the notion of being easy infinitely often, which was previously considered by Berman and Hartmanis [13] and by Rabin =-=[26]-=-. Definition 10 ffl A set A is immune for a class C if A contains no infinite subset that belongs to C. ffl A set A is co-immune for a class C ifsA is immune for C. ffl A set A is bi-immune for a clas... |

15 | Bounded query classes and the difference hierarchy
- Beigel, Gasarch, et al.
- 1989
(Show Context)
Citation Context ... all non-p-superterse sets or else admit only weaker analogies for general non-p-superterse sets. For example, all cheatable sets are recursive [10]; however non-p-superterse sets can be nonrecursive =-=[10, 11]-=-. Cheatability is preserved by polynomial-time Turing reductions [1]; non-p-superterseness is not [1]. A cheatable set cannot be self-reducible unless it is in P [1]; no corresponding result is known ... |

10 |
A hierarchy theorem for almost everywhere complex sets with applications to polynomial complexity degrees
- Geske, Huyn, et al.
- 1987
(Show Context)
Citation Context ...of: Define tow(0) = 1 and tow(i + 1) = 2 tow(i) . Define log (0) x = x and log (i+1) x = log log (i) x. Define log x to be the greatest i such that log (i) xs1. By the extended time hierarchy theorem =-=[17]-=-, there is a set B in DTIME(17 2 n ) that is bi-immune for DTIME(3 2 n ). Let A = f0 x : 0 tow(log x) 2 Bg: Then A is produced by dividing 0 into blocks such that the block beginning with i contains 2... |

7 |
The Complexity of Optimization Functions
- Gasarch
- 1986
(Show Context)
Citation Context ...aduate student at Stanford University. 1 looked at the number of queries that must be made to an oracle by a polynomial time algorithm that solves a problem, as a measure of that problem's difficulty =-=[15, 24]. When we -=-consider number-of-queries as a complexity measure, it is natural to consider the "complexity classes" of functions and sets induced by that measure. We call these complexity classes bounded... |

6 |
Using self-reducibility to characterize polynomial time
- Goldsmith, Joseph, et al.
- 1993
(Show Context)
Citation Context ... DTIME((tow(n)) k ). (Consider f0 n : 0 tow(n) 2 Sg.) That is equivalent to assuming that [ k0 NTIME(2 (tow(n)) k ) 6= [ k0 DTIME(2 (tow(n)) k ). Some results like this were obtained independently in =-=[19]-=-. Using the techniques of this section, we have obtained similar results for near-testable sets [6]. 16 6. A Hierarchy of Cheatable Sets By Corollary 14, there exists a 2-cheatable set that is not 1-c... |

5 |
Oracel Dependent Properties of the Lattice NP-sets
- Homer, Maass
- 1983
(Show Context)
Citation Context ...ave shown that NP contains a language that is bi-immune for P under almost all relativizations, our assumption that NP contains a language that is bi-immune for P is plausible. (Since Homer and Maass =-=[21]-=- have constructed a relativized world in which P 6= NP but no language in NP is bi-immune for P, we do not expect a proof of our assumption to be forthcoming.) We think that the conclusions in this th... |

3 |
Oracles for deterministic versus alternating classes
- Gasarch
- 1987
(Show Context)
Citation Context ...ows from the preceding theorem and the definitions. iii. Follows from the proof of the preceding theorem and Theorem 4.i. Since some other separation results in complexity theory hold with biimmunity =-=[3, 16, 17]-=-, we might wonder if Corollary 22.i holds with biimmunity. In other words, does there exists a k-cheatable set A such that neither A norsA contains a (k \Gamma 1)-cheatable subset? To the contrary, we... |

2 |
SAT A is terse with probability 1
- Beigel
- 1987
(Show Context)
Citation Context |

1 |
A note on some open problems of
- Beigel
- 1988
(Show Context)
Citation Context ...(2 (tow(n)) k ) 6= [ k0 DTIME(2 (tow(n)) k ). Some results like this were obtained independently in [19]. Using the techniques of this section, we have obtained similar results for near-testable sets =-=[6]-=-. 16 6. A Hierarchy of Cheatable Sets By Corollary 14, there exists a 2-cheatable set that is not 1-cheatable. The following theorem uses diagonalization to extend this result. Theorem 21 If ks1 then ... |

1 |
A note on bi-immunity and p-closeness of p-cheatable sets
- Goldsmith, Joseph, et al.
(Show Context)
Citation Context ...n)s2 n . The theorem follows for smaller functions, because if a language is bi-immune for DTIME(2 n ) then it must be bi-immune for all subsets of DTIME(2 n ). Recently, Goldsmith, Joseph, and Young =-=[18]-=- have independently discovered another proof of our result. The next corollary implies the existence of 2-cheatable sets that are not 1-cheatable. Corollary 14 There exists a set A such that (8ns2)[F ... |

1 |
Sparse sets in NP \Gamma P: Exptime verse nexptime
- Hartmanis, Immerman, et al.
- 1985
(Show Context)
Citation Context ...d of assuming that NP contains a language that is bi-immune for P, it would have been sufficient to assume directly that the set S = f0 tow(n) : ns0g contains a subset B that is in NP \Gamma P. As in =-=[20]-=-, this is easily seen to be equivalent to assuming that there is a tally set in [ k0 NTIME((tow(n)) k ) \Gamma [ k0 DTIME((tow(n)) k ). (Consider f0 n : 0 tow(n) 2 Sg.) That is equivalent to assuming ... |