## Some observations about relativization of space bounded computations (1988)

Venue: | Bulletin of the EATCS |

Citations: | 4 - 1 self |

### BibTeX

@ARTICLE{Hartmanis88someobservations,

author = {Juris Hartmanis and Richard Chang and Jim Kadin and Stephen G. Mitchell},

title = {Some observations about relativization of space bounded computations},

journal = {Bulletin of the EATCS},

year = {1988},

volume = {35},

pages = {423--433}

}

### OpenURL

### Abstract

In this column we explore what relativization says about space bounded computations and what recent results about space bounded computations say about relativization. There is a strong belief in computational complexity circles that problems which can be relativized in two contradictory ways are very hard to solve. We believe that such problems can only be solved by proof techniques that do not relativize. For example, standard diagonalization methods are powerless in these cases [BGS75, HH76, Hop84]. So far only very simple problems with contradictory relativizations have been solved [Har85, Cha90]. Unfortunately, many important problems in complexity theory have contradictory relativizations and have to be viewed as inaccessible to our current proof techniques. The classic example is the P? = NP problem for which Baker, Gill and Solovay [BGS75] exhibited recursive oracles A and B such that P A = NP A and P B � = NP B. Since then, a large number of other computational complexity problems have been shown to have contradictory relativizations and so far none of them have been solved. 1 Today, a proof that a problem has contradictory relativizations is viewed as strong evidence that the problem cannot be solved by current proof techniques and such

### Citations

315 |
Relationships between nondeterministic and deterministic tape complexities
- Savitch
- 1970
(Show Context)
Citation Context ...mputations was initiated [SHL65, LSH65], it was clear that the reusability of space yields sharper hierarchy results than those obtainable for time bounded computations [HS65]. Also, Savitch’s result =-=[Sav70]-=- showed that nondeterminism for space bounded computations could at best save the square root in resources over deterministic computations. That is, for space constructible bounds L(n) ≥ log n : There... |

308 | Algebraic methods for interactive proof systems - Lund, Fortnow, et al. - 1992 |

234 | Nondeterministic space is closed under complementation - Immerman - 1988 |

188 |
On the computational complexity of algorithms
- Hartmanis, Stearns
(Show Context)
Citation Context ...n the study of space bounded computations was initiated [SHL65, LSH65], it was clear that the reusability of space yields sharper hierarchy results than those obtainable for time bounded computations =-=[HS65]-=-. Also, Savitch’s result [Sav70] showed that nondeterminism for space bounded computations could at best save the square root in resources over deterministic computations. That is, for space construct... |

110 | The method of forced enumeration for nondeterministic automata - Szelepcsényi - 1988 |

63 | Bounded queries to SAT and the boolean hierarchy
- Beigel
- 1991
(Show Context)
Citation Context ...AT, and one can easily show that this result fails to hold with probability 1 for random oracles [Kad88]. A surprising version of this result for a constant number of queries has been shown by Beigel =-=[Bei87]-=-. For all k ≥ 1, P SAT[k] = P SAT�[2k −1] Some other interesting results about various access mechanisms to SAT have been observed in [Wag88]. Let LOG SAT denote the set of languages accepted by a det... |

63 |
Relativized circuit complexity
- Wilson
- 1985
(Show Context)
Citation Context ...elativization behave very similarly when we deal with the possibility of separating deterministic and nondeterministic space bounded classes. 3 Relativized Separation of Space Bounded Computations In =-=[Wil85]-=- it was shown that LOG = NLOG ⇐⇒ (∀A)[ LOG D(A) = NLOG D(A) ]. This is a generalization of a result in [Sim77] and in [RS81]. A later publication of this result can also be found in [KL87]. See also [... |

56 |
The strong exponential hierarchy collapses
- Hemachandra
- 1989
(Show Context)
Citation Context ... different access mechanisms and the influence of the structure of information, is the result showing that polynomially many parallel queries to SAT can be replaced by O(logn) many sequential queries =-=[Hem87]-=-: P SAT[log n] = P SAT� . Clearly, this result depends on the structure of SAT, and one can easily show that this result fails to hold with probability 1 for random oracles [Kad88]. A surprising versi... |

45 |
Generalized Kolmogorov complexity and the structure of feasible computations
- Hartmanis
- 1983
(Show Context)
Citation Context ...ent if and only if there exists a sparse, easily computable subset of GAP in NLOG − LOG.sTo make “easily computable” precise, let Mu be a standard universal TM (with a one-way output tape) and define =-=[Har83]-=- KS[log m, log m] = { w | |w| = m and (∃y)[ |y| ≤ log m and Mu(y) = w is computed on log m tape ] }. Lemma 7 SPACE[n] = NSPACE[n] ⇐⇒ GAP ∩ KS[log m, log m] ∈ LOG. Proof: (⇐) Observe that, like NLOG, a... |

45 |
Space-bounded hierarchies and probabilistic computation
- Ruzzo, Simon, et al.
- 1984
(Show Context)
Citation Context ...ativization of space bounded computations presented a more complex problem than the time bounded case. For space bounded computations, there are several options for the access mechanism to the oracle =-=[LL76, RST84]-=-. For deterministic and nondeterministic (log n)-space bounded computations, we can allow the machines to use an additional one-way write-only oracle tape on which they can write n k -long queries. We... |

42 | IP = PSPACE - Shamir - 1992 |

26 |
Bounded query computations
- Wagner
- 1988
(Show Context)
Citation Context ... for a constant number of queries has been shown by Beigel [Bei87]. For all k ≥ 1, P SAT[k] = P SAT�[2k −1] Some other interesting results about various access mechanisms to SAT have been observed in =-=[Wag88]-=-. Let LOG SAT denote the set of languages accepted by a deterministic log n tape bounded machine with a one-way oracle tape. Then P SAT[log n] = LOG SAT = LOG SAT[log n] = LOG SAT� .s2 Relativization ... |

24 |
Relativization of questions about log space computability
- Ladner, Lynch
- 1976
(Show Context)
Citation Context ...h strings. This permits an A where { 1 n | (∃x)[ x ∈ A and |x| = n] } ∈ NLOG A − P A . Similarly, we can construct oracles B and C which, for this model of relativization, invalidate Savitch’s result =-=[LL76]-=-, NLOG B �⊆ SPACE B [(log n) 2 ], as well as the Immerman-Szelepcsényi result, NLOG C �= NLOG C . The construction of C is achieved by judiciously placing at most one string of each length in C so tha... |

14 | Independence results in computer science - Hartmanis, Hopcroft - 1976 |

14 | Lewis II, Hierarchies of memory limited computations - Stearns, Hartmanis, et al. - 1965 |

11 |
Restricted Turing Reducibilities and the Structure of the Polynomial Time Hierarchy
- Kadin
- 1988
(Show Context)
Citation Context ... sequential queries [Hem87]: P SAT[log n] = P SAT� . Clearly, this result depends on the structure of SAT, and one can easily show that this result fails to hold with probability 1 for random oracles =-=[Kad88]-=-. A surprising version of this result for a constant number of queries has been shown by Beigel [Bei87]. For all k ≥ 1, P SAT[k] = P SAT�[2k −1] Some other interesting results about various access mec... |

11 | Memory bounds for recognition of context-free and context-sensitive languages - Stearns, Hartmanis - 1965 |

10 | Solvable problems with conflicting relativizations - Hartmanis - 1985 |

8 |
On some subrecursive reducibilities
- Simon
- 1977
(Show Context)
Citation Context ...ministic space bounded classes. 3 Relativized Separation of Space Bounded Computations In [Wil85] it was shown that LOG = NLOG ⇐⇒ (∀A)[ LOG D(A) = NLOG D(A) ]. This is a generalization of a result in =-=[Sim77]-=- and in [RS81]. A later publication of this result can also be found in [KL87]. See also [Wil86]. Our extensions of this result are summarized in the following two theorems. The key ideas for the proo... |

6 |
Limitations on separating nondeterministic complexity classes
- Rackoff, Seiferas
- 1981
(Show Context)
Citation Context ...bounded classes. 3 Relativized Separation of Space Bounded Computations In [Wil85] it was shown that LOG = NLOG ⇐⇒ (∀A)[ LOG D(A) = NLOG D(A) ]. This is a generalization of a result in [Sim77] and in =-=[RS81]-=-. A later publication of this result can also be found in [KL87]. See also [Wil86]. Our extensions of this result are summarized in the following two theorems. The key ideas for the proofs, which are ... |

4 | Turing machines - Hopcroft - 1984 |

4 |
Parallel computation and the NC hierarchy relativized
- Wilson
- 1986
(Show Context)
Citation Context ...] it was shown that LOG = NLOG ⇐⇒ (∀A)[ LOG D(A) = NLOG D(A) ]. This is a generalization of a result in [Sim77] and in [RS81]. A later publication of this result can also be found in [KL87]. See also =-=[Wil86]-=-. Our extensions of this result are summarized in the following two theorems. The key ideas for the proofs, which are quite simple, will be given in the next section. Theorem 1 Consider only space con... |

3 | An example of a theorem that has contradictory relativizations and a diagonalization proof - Chang - 1990 |

1 |
Separation with the Ruzzo, Simon, and Tompa relativization implies DSPACE[log n] �= NSPACE[log n
- Kirsig, Lange
- 1987
(Show Context)
Citation Context ...tations In [Wil85] it was shown that LOG = NLOG ⇐⇒ (∀A)[ LOG D(A) = NLOG D(A) ]. This is a generalization of a result in [Sim77] and in [RS81]. A later publication of this result can also be found in =-=[KL87]-=-. See also [Wil86]. Our extensions of this result are summarized in the following two theorems. The key ideas for the proofs, which are quite simple, will be given in the next section. Theorem 1 Consi... |