## Algorithms for Boolean function query properties (2003)

### Cached

### Download Links

- [arxiv.org]
- [arxiv.org]
- [www.cs.berkeley.edu]
- [www.scottaaronson.com]
- DBLP

### Other Repositories/Bibliography

Venue: | SIAM J. COMPUT |

Citations: | 6 - 4 self |

### BibTeX

@ARTICLE{Aaronson03algorithmsfor,

author = {Scott Aaronson},

title = {Algorithms for Boolean function query properties},

journal = {SIAM J. COMPUT},

year = {2003},

volume = {32},

pages = {2003}

}

### OpenURL

### Abstract

We present new algorithms to compute fundamental properties of a Boolean function given in truth-table form. Specifically, we give an O(N 2.322 log N) algorithm for block sensitivity, an O(N 1.585 log N) algorithm for ‘tree decomposition, ’ and an O(N) algorithm for ‘quasisymmetry.’ These algorithms are based on new insights into the structure of Boolean functions that may be of independent interest. We also give a subexponential-time algorithm for the space-bounded quantum query complexity of a Boolean function. To prove this algorithm correct, we develop a theory of limited-precision representation of unitary operators, building on work of Bernstein and Vazirani.

### Citations

847 | A fast quantum mechanical algorithm for database search
- Grover
- 1996
(Show Context)
Citation Context ..., nontrivial lower bounds are more readily shown for the former measure than for the latter. Also, query complexity has proved to be a powerful tool for studying the capabilities of quantum computers =-=[2, 3, 5, 11]-=-. Numerous Boolean function properties relevant to query complexity have been defined, such as sensitivity, block sensitivity, randomized and quantum query complexity, and degree as a real polynomial.... |

482 | Quantum complexity theory
- Bernstein, Vazirani
- 1997
(Show Context)
Citation Context ...e search. For what follows, it will be convenient to extend the quantum oracle model to allow intermediate observations. With an unlimited workspace, this cannot decrease the number of queries needed =-=[6]-=-. In the space-bounded setting, however, it might make a larger di#erence. We define a composite algorithm # # to be an alternating sequence # 1 # D 1 #s# # t # D t . Each # i is a quantum query algor... |

267 | Quantum lower bounds by polynomials
- Beals, Buhrman, et al.
(Show Context)
Citation Context ...bject classifications. 68Q10, 68Q17, 68Q25, 68W01, 81P68. 1. Introduction. The query complexity of Boolean functions, also called blackbox or decision-tree complexity, has been well studied for years =-=[5, 7, 16]-=-. Counting how many queries are needed to evaluate a function is easier than counting how many computational steps are needed; thus, nontrivial lower bounds are more readily shown for the former measu... |

185 |
Constructing optimal binary decision trees is NPcomplete
- Hyafil, Rivest
- 1976
(Show Context)
Citation Context ...1 ### ; } return D[# n ]; That f is given as a truth table is crucial: if f is non-total and only the inputs for which f is defined are given, then deciding whether D(f) # k for some k is NP-complete =-=[14]-=-. 3.2. Certificate Complexity. Given an input X to f , a certificate for X is a constant-valued restriction that agrees with X on the fixed variables. The size of the certificate is the number of fixe... |

175 | Natural proofs
- Razborov, Rudich
- 1997
(Show Context)
Citation Context ...for studying algorithmic problems such as those considered in this paper. Much e#ort has been devoted to finding Boolean function properties that do not naturalize in the sense of Razborov and Rudich =-=[18]-=-, and that might therefore be useful for proving circuit lower bounds. In our view, it would help this e#ort to have a better general understanding of the complexity of problems on Boolean function tr... |

155 |
Quantum computations: algorithms and error correction
- Kitaev
- 1997
(Show Context)
Citation Context ...ithm, given a fixed accuracy that the algorithm needs to attain. An alternative approach to approximating SQ 2,m (f) would be to represent each unitary matrix as a product of elementary gates. Kitaev =-=[15]-=- and independently Solovay [21] showed that a 2 m 2 m unitary matrix can be represented with arbitrary accuracy # > 0 by a product of 2 O(m)polylog(1/#) unitary gates. But this yields a 2 2 O(m) polyl... |

146 |
On the degree of boolean functions as real polynomials
- Nisan, Szegedy
- 1994
(Show Context)
Citation Context ...Polynomial. Let deg(f) be the minimum degree of an nvariate real multilinear polynomial p such that, for all X # {0, 1} n , p(X) = f(X). Degree was introduced to query complexity by Nisan and Szegedy =-=[17]-=-, who observed the relationship deg(f) # D(f ). Later Beals et al. [5] related degree to quantum query complexity by showing that deg(f) # 2QE (f ). The following lemma, adapted from Lemma 4 of [7], i... |

145 | Quantum lower bounds by quantum arguments
- Ambainis
- 2000
(Show Context)
Citation Context ..., nontrivial lower bounds are more readily shown for the former measure than for the latter. Also, query complexity has proved to be a powerful tool for studying the capabilities of quantum computers =-=[2, 3, 5, 11]-=-. Numerous Boolean function properties relevant to query complexity have been defined, such as sensitivity, block sensitivity, randomized and quantum query complexity, and degree as a real polynomial.... |

122 | Complexity measures and decision tree complexity: A survey. Theor
- Buhrman, Wolf
(Show Context)
Citation Context ...bject classifications. 68Q10, 68Q17, 68Q25, 68W01, 81P68. 1. Introduction. The query complexity of Boolean functions, also called blackbox or decision-tree complexity, has been well studied for years =-=[5, 7, 16]-=-. Counting how many queries are needed to evaluate a function is easier than counting how many computational steps are needed; thus, nontrivial lower bounds are more readily shown for the former measu... |

81 |
CREW PRAMs and decision trees
- Nisan
- 1991
(Show Context)
Citation Context ...bject classifications. 68Q10, 68Q17, 68Q25, 68W01, 81P68. 1. Introduction. The query complexity of Boolean functions, also called blackbox or decision-tree complexity, has been well studied for years =-=[5, 7, 16]-=-. Counting how many queries are needed to evaluate a function is easier than counting how many computational steps are needed; thus, nontrivial lower bounds are more readily shown for the former measu... |

59 | Quantum lower bound for the collision problem
- Aaronson
- 2002
(Show Context)
Citation Context ..., nontrivial lower bounds are more readily shown for the former measure than for the latter. Also, query complexity has proved to be a powerful tool for studying the capabilities of quantum computers =-=[2, 3, 4, 7, 13]-=-. Numerous Boolean function properties relevant to query complexity have been defined, such as sensitivity, block sensitivity, randomized and quantum query complexity, and degree as a real polynomial.... |

56 |
Probabilistic Boolean decision trees and the complexity of evaluating game trees
- Saks, Wigderson
- 1986
(Show Context)
Citation Context ...san showed that R 0 (f) 2 # D(f) and R 2 (f) 3 = # (D(f)) [16]. On the other hand, the best known separation between deterministic and randomized query complexity is R 0 (f) = R 2 (f) = D(f) 0.753... =-=[19, 20]-=-, for f an AND/OR tree with two children per node. Whether better separations are possible is a long-standing open question, and one that might be fruitfully investigated with computer analysis 1 . Un... |

50 | Sampling algorithms: lower bounds and applications
- Bar-Yossef, Kumar, et al.
- 2001
(Show Context)
Citation Context ...bit negated.) Then let AD (f) be the maximum of ADS (f) over all S. Ambainis shows that Q 2 (f) =# (AD (f )). How e#cient an algorithm can we find for AD (f)? Second, Bar-Yossef, Kumar, and Sivakumar =-=[4] have defi-=-ned "approximate" versions of measures such as block sensitivity. Can we extend the algorithms given in this paper to compute those measures? 9. Acknowledgments. I thank Rob Pike and Lorenz ... |

32 | Circuit minimization problem
- Kabanets, Cai
- 2000
(Show Context)
Citation Context ...n Boolean function truth tables—both upper and lower bounds. Such problems have been considered since the 1950’s [19], but basic open questions remain, especially in the setting of circuit complexity =-=[11]-=-. This paper addresses the much simpler setting of query complexity. We do not know of a polynomial-time algorithm to find quantum query complexity; we raise this as an open problem. However, even fin... |

27 |
M.: Quantum decision trees and semidefinite programming
- Barnum, Saks, et al.
- 2003
(Show Context)
Citation Context ...uery complexity R0(f) LP This paper Block sensitivity bs(f) O(N 2.322 log N) This paper Quasisymmetry O(N) This paper Tree decomposition O(N 1.585 log N) This paper Quantum query complexity Q2(f) SDP =-=[5]-=- Randomized certificate complexity RC (f) LP Obvious There is also a complexity-theoretic rationale for studying algorithmic problems such as those considered in this paper. Much effort has been devot... |

26 | Quantum oracle interrogation: Getting all information for almost half the price - Dam - 1998 |

24 |
Lower bounds on learning decision lists and trees
- Hancock, Jiang, et al.
- 1996
(Show Context)
Citation Context ... set q[S] := 1 + max # m, max S(i)#{0,1} q # S S(i)=# # # ; } return max S#{0,1} n (n - q[S]); Again, if f is not given as a full truth table, then deciding whether C(f) # k for some k is NP-complete =-=[12]-=-. 3.3. Degree as a Polynomial. Let deg(f) be the minimum degree of an nvariate real multilinear polynomial p such that, for all X # {0, 1} n , p(X) = f(X). Degree was introduced to query complexity by... |

16 |
Sensitivity vs. block sensitivity of Boolean functions
- Rubinstein
- 1995
(Show Context)
Citation Context ...ction; truth table; query complexity; quantum computation. 1 Introduction The query complexity of Boolean functions, also called black-box or decision-tree complexity, has been well studied for years =-=[3,5,13,14,16,17]-=-. Numerous Boolean function properties relevant to query complexity have been defined, such as sensitivity, block sensitivity, randomized and quantum query complexity, and degree as a real polynomial.... |

15 |
On the Monte Carlo decision tree complexity of read-once formulae
- Santha
- 1991
(Show Context)
Citation Context ...san showed that R 0 (f) 2 # D(f) and R 2 (f) 3 = # (D(f)) [16]. On the other hand, the best known separation between deterministic and randomized query complexity is R 0 (f) = R 2 (f) = D(f) 0.753... =-=[19, 20]-=-, for f an AND/OR tree with two children per node. Whether better separations are possible is a long-standing open question, and one that might be fruitfully investigated with computer analysis 1 . Un... |

10 |
The complexity of minimizing disjunctive normal form formulas
- Czort
- 1999
(Show Context)
Citation Context ...eriel Command, USAF, under agreement number F30602-01-2-0524. 1 Query Property Complexity Source Deterministic query complexity D(f) O(N 1.585 log N) [10] Certificate complexity C(f) O(N 1.585 log N) =-=[8]-=- Degree as a real polynomial deg(f) O(N 1.585 log N) This paper Approximate degree # deg(f) About O # N 5 # (LP) Obvious Randomized query complexity R 0 (f) About O # N 7.925 # (LP) This paper Block s... |

9 | Exact learning when irrelevant variables are abound
- Guijarro, Lavín, et al.
- 1999
(Show Context)
Citation Context ...cy (DARPA) and Air Force Laboratory, Air Force Materiel Command, USAF, under agreement number F30602-01-2-0524. 1 Query Property Complexity Source Deterministic query complexity D(f) O(N 1.585 log N) =-=[10]-=- Certificate complexity C(f) O(N 1.585 log N) [8] Degree as a real polynomial deg(f) O(N 1.585 log N) This paper Approximate degree # deg(f) About O # N 5 # (LP) Obvious Randomized query complexity R ... |

7 | Minimization of decision trees is hard to approximate
- Sieling
- 2002
(Show Context)
Citation Context ...crucial: if f is non-total and only the inputs for which f is defined are given, then deciding whether D(f) ≤ k for some k is NPcomplete [16], and NP-hard even to approximate within a constant factor =-=[23]-=-. 3.2. Certificate Complexity. Given an input X to f, a certificate for X is a constant-valued restriction that agrees with X on the fixed variables. The size of the certificate is the number of fixed... |

6 | Average-case quantum query complexity
- Ambainis, Wolf
(Show Context)
Citation Context ..., nontrivial lower bounds are more readily shown for the former measure than for the latter. Also, query complexity has proved to be a powerful tool for studying the capabilities of quantum computers =-=[2, 3, 5, 11]-=-. Numerous Boolean function properties relevant to query complexity have been defined, such as sensitivity, block sensitivity, randomized and quantum query complexity, and degree as a real polynomial.... |

2 |
Trakhtenbrot, A survey of Russian approaches to Perebor (brute-force search) algorithms, Annals of the History of Computing 6
- A
- 1984
(Show Context)
Citation Context ...help this effort to have a better general understanding of the complexity of problems on Boolean function truth tables—both upper and lower bounds. Such problems have been considered since the 1950’s =-=[19]-=-, but basic open questions remain, especially in the setting of circuit complexity [11]. This paper addresses the much simpler setting of query complexity. We do not know of a polynomial-time algorith... |

1 |
Boolean Function Wizard 1.0 (software library
- Aaronson
- 2000
(Show Context)
Citation Context ...unded-error quantum query complexity if the memory of the quantum computer is restricted to O(log n) qubits. We have implemented most of the algorithms discussed in this paper in a linkable C library =-=[1]-=-, which is available for download. The paper is organized as follows. Section 2 gives preliminaries, and Section 3 reviews simple algorithms for deterministic query complexity, certificate complexity,... |

1 |
Lie groups and quantum circuits, talk at workshop on
- Solovay
- 2000
(Show Context)
Citation Context ...at the algorithm needs to attain. An alternative approach to approximating SQ 2,m (f) would be to represent each unitary matrix as a product of elementary gates. Kitaev [15] and independently Solovay =-=[21]-=- showed that a 2 m 2 m unitary matrix can be represented with arbitrary accuracy # > 0 by a product of 2 O(m)polylog(1/#) unitary gates. But this yields a 2 2 O(m) polylog(mn) algorithm, which is slow... |

1 | Quantum certificate complexity, to appear - Aaronson - 2003 |