## On Truth-Table Reducibility to SAT (2002)

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Citations: | 50 - 2 self |

### BibTeX

@MISC{Buss02ontruth-table,

author = {Samuel Buss and Louise Hay},

title = {On Truth-Table Reducibility to SAT},

year = {2002}

}

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### Abstract

We show that polynomial time truth-table reducibility via Boolean circuits to SAT is the same as logspace truth-table reducibility via Boolean formulas to SAT and the same as logspace Turing reducibility to SAT . In addition, we prove that a constant number of rounds of parallel queries to SAT is equivalent to one round of parallel queries.

### Citations

245 |
A.L.: A comparison of polynomial time reducibilities
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- 1975
(Show Context)
Citation Context ...one round of parallel queries. We give an oracle relative to which ∆ p 2 is not equal to the class of predicates polynomial time truth-table reducible to SAT. 1 Introduction Ladner, Lynch, and Selman =-=[17]-=- introduced the notion of polynomial time truth-table reducibility as an aid to the study of low level complexity. As noted by Ladner and Lynch [16] this is equivalent to the notion of polynomial ∗ Su... |

193 |
The Complexity of Optimization Problems
- Krentel
- 1988
(Show Context)
Citation Context ...of Ladner and Lynch. Let ≤ p tt(NP) denote the class of decision problems polynomial time truthtable reducible to NP; this paper presents several other characterizations of this class. First, Krentel =-=[14]-=- defines the class P SAT [O(log n)]; we show this is equal to ≤ p tt(NP). Cai and Hemachandra [7] defined a hierarchy ΣL k , above the finite levels of the difference hierarchy over NP; we show this Π... |

124 |
Bounded Arithmetic. Bibliopolis
- Buss
- 1986
(Show Context)
Citation Context ...e hierarchy over NP; we show this ΠL k hierarchy collapses with ΣL 2 =ΠL 2 =≤ p tt(NP). Furthermore, ≤ p tt(NP) is precisely the set of predicates definable by Σb 2 ∩ Πb 2 formulas as defined by Buss =-=[2]-=-. The class ≤ p tt(NP) is defined so that it contains the predicates which are computed in polynomial time by a Turing machine which is allowed one set of parallel queries to SAT. This paper shows tha... |

80 |
The boolean hierarchy I: Structural properties
- Cai, Gundermann, et al.
- 1988
(Show Context)
Citation Context ... difference) hierarchy over NP. The difference hierarchy was defined in an abstract setting by Hausdorff [10] and the Boolean hierarchy over NP has been extensively studied by a number of researchers =-=[22, 8, 13, 5, 6]-=-. Cai and Hemachandra [8] showed that there are oracles relative to which the Boolean hierarchy over has infinitely many levels, and oracles relative to which the Boolean hierarchy extends exactly k l... |

76 |
More complicated questions about maxima and minima, and some closures
- Wagner
(Show Context)
Citation Context ...olean formula. Accordingly, we shall define three different versions of polynomial time truth-table reducibilities. (These have already been considered by Ladner and Lynch and more recently by Wagner =-=[19]-=- and Köbler, Schöning and Wagner [13].) 2.1 Some Definitions and a Main Theorem As usual, P and NP denote the class of polynomial time and nondetermindenote polynomial istic polynomial time predicates... |

67 | Bounded queries to SAT and the Boolean hierarchy
- Beigel
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(Show Context)
Citation Context ...g the ‘census function’ method of Hemachandra [11]. Similar ‘counting’ techniques have been used recently by a number of authors, an early example is Jenner, Kirsig and Lange [12]; the work of Beigel =-=[1]-=- was particularly helpful to us in the original formulation of our results. The current paper is a slightly revised version of part of [4]; we wish to thank the referees for a number of suggestions wh... |

64 |
The circuit value problem is log space complete for P
- Ladner
- 1975
(Show Context)
Citation Context ... the code z but eval is required to be logspace computable. By the theorem of Lynch [18] that Boolean formulas can be evaluated in logspace, if A ≤ log bf B then A ≤log LL B ; and by Ladner’s theorem =-=[15]-=- that the circuit value problem is complete for P ,ifA≤ log LL B then A ≤log tt B . It is obvious that ≤ p tt(NP) ⊇≤ p bf (NP) ⊇≤p ftt (NP); complete problems for these classes are given by Köbler, Sc... |

57 |
The strong exponential hierarchy collapses
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- 1987
(Show Context)
Citation Context ...agner [21, 20]. In addition, one of the referees pointed out that alternative proofs of most of the results in sections 1 through 3 can be 2sobtained using the ‘census function’ method of Hemachandra =-=[11]-=-. Similar ‘counting’ techniques have been used recently by a number of authors, an early example is Jenner, Kirsig and Lange [12]; the work of Beigel [1] was particularly helpful to us in the original... |

31 |
The Boolean hierarchy: Hardware over NP
- Cai, Hemachandra
- 1986
(Show Context)
Citation Context ...le reducible to NP; this paper presents several other characterizations of this class. First, Krentel [14] defines the class P SAT [O(log n)]; we show this is equal to ≤ p tt(NP). Cai and Hemachandra =-=[7]-=- defined a hierarchy ΣL k , above the finite levels of the difference hierarchy over NP; we show this ΠL k hierarchy collapses with ΣL 2 =ΠL 2 =≤ p tt(NP). Furthermore, ≤ p tt(NP) is precisely the set... |

30 |
On the Boolean closure of NP
- Wagner, Wechsung
- 1985
(Show Context)
Citation Context ... difference) hierarchy over NP. The difference hierarchy was defined in an abstract setting by Hausdorff [10] and the Boolean hierarchy over NP has been extensively studied by a number of researchers =-=[22, 8, 13, 5, 6]-=-. Cai and Hemachandra [8] showed that there are oracles relative to which the Boolean hierarchy over has infinitely many levels, and oracles relative to which the Boolean hierarchy extends exactly k l... |

26 | Relativization of questions about log space computability
- Ladner, Lynch
- 1976
(Show Context)
Citation Context ...le to SAT. 1 Introduction Ladner, Lynch, and Selman [17] introduced the notion of polynomial time truth-table reducibility as an aid to the study of low level complexity. As noted by Ladner and Lynch =-=[16]-=- this is equivalent to the notion of polynomial ∗ Supported in part by NSF grants DMS-8511465, DMS-8701828 and DMS-8902480. † Work partially completed while on sabbatical leave at the University of Ca... |

25 | Log space recognition and translation of parenthesis languages
- Lynch
- 1977
(Show Context)
Citation Context ...led the eval function a ‘tt-condition evaluator’. For ≤ log LL reducibility there is no restriction on the format of the code z but eval is required to be logspace computable. By the theorem of Lynch =-=[18]-=- that Boolean formulas can be evaluated in logspace, if A ≤ log bf B then A ≤log LL B ; and by Ladner’s theorem [15] that the circuit value problem is complete for P ,ifA≤ log LL B then A ≤log tt B . ... |

22 |
conservation results for fragments of bounded arithmetic, in Logic and Computation, proceedings of a Workshop held Carnegie-Mellon
- Axiomatizations
- 1987
(Show Context)
Citation Context ...s in the class Σ b i or Π b i define precisely the predicates in the i-th level of the polynomial time hierarchy Σ p i or Πb i , respectively. The class Σ b 2 ∩ Π b 2 has already been studied by Buss =-=[3]-=- who showed that it is the 7slargest class of formulas for which S1 2 is known to prove PIND (polynomialor length-induction). For the reader who has not studied bounded arithmetic, we provide some alt... |

13 | On Truth-Table Reducibility to SAT and the Difference Hierarchy over NP
- Buss, Hay
- 1988
(Show Context)
Citation Context ... example is Jenner, Kirsig and Lange [12]; the work of Beigel [1] was particularly helpful to us in the original formulation of our results. The current paper is a slightly revised version of part of =-=[4]-=-; we wish to thank the referees for a number of suggestions which improved this paper. 2 Truth-Table Reducibility to SAT Polynomial time truth-table reducibility (≤ p tt) was first introduced by Ladne... |

11 |
Three results on the polynomial isomorphism of complete sets
- Goldsmith, Joseph
- 1986
(Show Context)
Citation Context ...rarchy and collapses PSPACE to the k-th level. ✷ An oracle A is a subset of {0, 1} ∗ ;ifσ∈A, then |σ| is the length of σ and σi denotes the i-th symbol in the string σ. Following Goldsmith and Joseph =-=[9]-=-, define a relativized Boolean formula to be of the form ΦA � � � � � � = φ(x1,...,xn) ∧ ∧ ρi ∈A 1≤i≤m τi /∈A 1≤i≤k where φ is in conjunctive normal form and each ρi and τi is a concatenation of the f... |

10 |
The logarithmic alternation hierarchy collapses
- Lange, Jenner, et al.
- 1987
(Show Context)
Citation Context ... 3 can be 2sobtained using the ‘census function’ method of Hemachandra [11]. Similar ‘counting’ techniques have been used recently by a number of authors, an early example is Jenner, Kirsig and Lange =-=[12]-=-; the work of Beigel [1] was particularly helpful to us in the original formulation of our results. The current paper is a slightly revised version of part of [4]; we wish to thank the referees for a ... |

5 |
query classes
- Bounded
- 1990
(Show Context)
Citation Context ...ual to NP relative to these oracles. Many of the results of this paper have been obtained independently by K. Wagner; in particular, Theorem 1 and many of the other results of section 2 are in Wagner =-=[21, 20]-=-. In addition, one of the referees pointed out that alternative proofs of most of the results in sections 1 through 3 can be 2sobtained using the ‘census function’ method of Hemachandra [11]. Similar ... |

4 |
The difference and truth-table hierarchies for NP, Informatique Theorique et Applications
- Köbler, Schöning, et al.
- 1987
(Show Context)
Citation Context ...act that all these definitions coincide provide evidence that this is a robust and interesting class for computational complexity. The results below refute a conjecture of Köbler, Schöning and Wagner =-=[13]-=-, but answer only a special case of the original questions of Ladner and Lynch. Let ≤ p tt(NP) denote the class of decision problems polynomial time truthtable reducible to NP; this paper presents sev... |

2 |
restricting the access to an NP oracle
- On
- 1988
(Show Context)
Citation Context ...ual to NP relative to these oracles. Many of the results of this paper have been obtained independently by K. Wagner; in particular, Theorem 1 and many of the other results of section 2 are in Wagner =-=[21, 20]-=-. In addition, one of the referees pointed out that alternative proofs of most of the results in sections 1 through 3 can be 2sobtained using the ‘census function’ method of Hemachandra [11]. Similar ... |

2 | The di#erence and truth-table hierarchies for NP, Informatique Theorique et Applications - obler, oning, et al. - 1987 |