### Citations

859 | Construction of Pseudorandom Generator from any One-Way Function
- Hastad, Impagliazzo, et al.
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Citation Context ...s remains a difficult open problem. What Wolfram has proposed is simply a candidate pseudorandom generator. There is no shortage of these (or equivalently, candidate one-way functions; H˚astad et al. =-=[8]-=- showed that either can be obtained from the other). Some such generators are based on NP-complete problems, but that is not considered evidence for their security, since breaking the generator might ... |

648 |
The complexity of computing the permanent
- Valiant
- 1979
(Show Context)
Citation Context ...am says that minimizing a DNF expression (p. 1096) and computing a permanent (p. 1146) are NP-complete; they are respectively ΣP 2-complete (as shown by Umans [5]) and #Pcomplete (as shown by Valiant =-=[6]-=-). Also, in Chapter 10, pseudorandom number generators based on cellular automata are proposed. Wolfram suggests that, since certain questions involving cellular automata are NP-complete, these genera... |

572 | Quantum complexity theory
- Bernstein, Vazirani
- 1993
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Citation Context ...han continuity in measurable quantities: the former, unlike the latter, does not enable an infinite amount of computation to be performed in a finite time. Also, as observed by Bernstein and Vazirani =-=[23]-=-, the linearity of quantum mechanics implies that small errors in amplitudes cannot be magnified during a quantum computation. 3.2. Bell’s theorem and causal invariance Our goal in this section is to ... |

384 |
The Character of Physical
- Feynman
- 1967
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Citation Context ...t R be a set of graph updating rules, which might be probabilistic. Then consider the following four assertions (which, though not mathematically precise, will be clarified by subsequent discussion). =-=(1)-=- R satisfies causal invariance. That is, given any initial graph (and choice of randomness if R is probabilistic), R yields a unique causal network. ∗ Indeed one can use almost any classical reversibl... |

245 | A public-key cryptosystem with worst-case/averagecase equivalence
- Ajtai, Dwork
- 1997
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Citation Context ...on average, under a simple distribution, so long as P ̸= NP. However, ‘worst-case/average-case equivalence’ has been shown for several cryptographic problems, including one studied by Ajtai and Dwork =-=[11]-=-. As for the cellular automaton conjecture, its validity depends on how it is formulated. Suppose we are given a one-dimensional, two-color cellular automaton on a lattice of bounded size n, and an en... |

217 |
Average Case Complete Problems
- Levin
- 1986
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Citation Context ...behavior. (p. 769) In computer science, the complexity of ‘typical’ instances of NP-complete problems has been investigated for decades. Highlights include Levin’s theory of average-case completeness =-=[9]-=- and studies of phase transitions in randomly generated combinatorial problems [10]. It remains open to show that some NP-complete problem is hard on average, under a simple distribution, so long as P... |

133 |
Bell’s theorem without inequalities
- Greenberger, Horn, et al.
- 1990
(Show Context)
Citation Context ...ly random’ or were the pseudorandom output of some updating rule. Our conclusion is also unaffected if we consider a ‘deterministic’ variant of Bell’s theorem due to Greenberger, Horne, and Zeilinger =-=[25]-=-, discussed by Wolfram on p. 1065. There three parties, Alice, Bob, and Charlie, are given input bits xA, xB, and xC respectively, satisfying the promise that xA ⊕ xB ⊕ xC = 0. The goal is to output b... |

126 | Finding hard instances of the satisfiability problem: A survey,” in Satisfiability Problem: Theory and Applications
- Cook, Mitchell
- 1997
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Citation Context ...P-complete problems has been investigated for decades. Highlights include Levin’s theory of average-case completeness [9] and studies of phase transitions in randomly generated combinatorial problems =-=[10]-=-. It remains open to show that some NP-complete problem is hard on average, under a simple distribution, so long as P ̸= NP. However, ‘worst-case/average-case equivalence’ has been shown for several c... |

66 | Deniable encryption
- Canetti, Dwork, et al.
- 1997
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Citation Context ...90). The most common cryptanalysis problems, such as inverting a one-way permutation, are in NP ∩ coNP, which means that they cannot be NP-complete unless NP = coNP. On the other hand, Canetti et al. =-=[7]-=- have proposed ‘deniable encryption’ schemes, in which a single ciphertext can correspond to many plaintexts. Such a scheme could conceivably be secure assuming only P ̸= NP, but finding a scheme for ... |

51 | The minimum equivalent DNF problem and shortest implicants
- Umans
- 2001
(Show Context)
Citation Context ...arding complexity. Wol-98 Book review fram says that minimizing a DNF expression (p. 1096) and computing a permanent (p. 1146) are NP-complete; they are respectively ΣP 2-complete (as shown by Umans =-=[5]-=-) and #Pcomplete (as shown by Valiant [6]). Also, in Chapter 10, pseudorandom number generators based on cellular automata are proposed. Wolfram suggests that, since certain questions involving cellul... |

44 | On one-dimensional quantum cellular automata
- Watrous
- 1995
(Show Context)
Citation Context ...thread hypothesis. This is an explicitly quantum cellular automaton or causal network, with a unitary transition rule. (For discussions of how to construct such automata, see van Dam [20] and Watrous =-=[21]-=-.) The reason seems to be that he does not want continuity anywhere in a model, not even in probabilities or amplitudes. In the notes, he describes an experiment with a quantum cellular automaton as f... |

43 |
Both Toffoli and controlled-NOT need little help to do universal quantum computation
- Shi
(Show Context)
Citation Context ...e effect that almost any nontrivial gate set is universal (that is, can approximate any unitary matrix to any desired precision, or any orthogonal matrix in case one is limited to reals). Indeed, Shi =-=[22]-=- has shown that a Toffoli gate∗plus any gate that does not preserve the computational basis, or a controlled-NOT gate plus any gate whose square does not preserve the computational basis, are both uni... |

26 | Attacking the busy beaver 5
- Marxen, Buntrock
- 1990
(Show Context)
Citation Context ...er of steps such a machine could make when started on a blank tape. (The best known lower bound for this number, called the ‘5th Busy Beaver shift number,’ is 47, 176, 870, due to Marxen and Buntrock =-=[4]-=-.) But if extremely short programs can produce ‘irreducibly complex’ behavior, then aren’t they already of interest? Indeed, in Chapter 11 Wolfram gives a remarkable construction, due to his employee ... |

25 |
Universality of tag systems with P = 2
- M
- 1964
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Citation Context ...is intuition fails for an interesting reason. Cook’s proof that Rule 110 is universal relies on simulating ‘cyclic tag systems,’ a variant of the tag systems studied in the 1960’s by Cocke and Minsky =-=[12]-=- among others (see also p. 670 of Wolfram). However, though Wolfram does not discuss this explicitly in the book, the known simulations of Turing machines by tag100 Book review systems require expone... |

14 | Quantum Cellular Automata
- Dam
- 1996
(Show Context)
Citation Context ...ibility than the thread hypothesis. This is an explicitly quantum cellular automaton or causal network, with a unitary transition rule. (For discussions of how to construct such automata, see van Dam =-=[20]-=- and Watrous [21].) The reason seems to be that he does not want continuity anywhere in a model, not even in probabilities or amplitudes. In the notes, he describes an experiment with a quantum cellul... |

13 |
2001a). “The Kochen-Specker Theorem,” Stanford Encyclopaedia of Philosophy (online). http://plato.stanford.edu/entries/Kochen-specker/ Held, Carsten (2001b). “The Interpretation of Spin in Geometric-Algebraic QM” http://www.uni-erfurt.de/wissenschaftsphil
- Held
- 1962
(Show Context)
Citation Context ... a Toffoli gate. On p. 1098, Wolfram reports that, out of 40, 320 such classical reversible gates, 38, 976 are universal. † A similar argument could be made on the basis of the Kochen-Specker theorem =-=[24]-=-. The reason we have not done so is that Wolfram never explicitly requires his model to be noncontextual.104 Book review (2) R satisfies the relativity postulate. That is, assuming the causal network... |

12 |
Quantum correlations with spacelike separated beamsplitters in motion: experimental test of multisimultaneity, Phys
- Stefanov, Zbinden, et al.
- 2002
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Citation Context ...all (i.e. contain too few edges) to transmit information in a way that violates causality. This is reminiscent of, for example, the ‘multisimultaneity’ model studied experimentally by Stefanov et al. =-=[19]-=-. There are several objections one could raise against this thread hypothesis. What we will show in Section 3.2 is that, if one accepts three of Wolfram’s own desiderata—determinism, relativity of ine... |

9 | Requardt(1998) Dimension theory on graphs and networks
- Nowotny, Manfred
(Show Context)
Citation Context ...13]; that particles could arise as ‘topological defects’ in such a network [16]; and that dimension and other geometric properties can be defined solely in terms of the network’s connectivity pattern =-=[17]-=-. The main difference we can discern between Wolfram’s model and earlier ones is that Wolfram’s is explicitly classical. Indeed, Wolfram requires the network evolution to be deterministic, by disallow... |

7 |
2002), A New
- Wolfram
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Citation Context ...ch remarks are obvious and are perfectly clear to anybody who is working on this problem. It does not do any good to point this out.” —Richard Feynman [1, p.161] 1. Introduction A New Kind of Science =-=[2]-=-, the 1280-page treatise by Mathematica creator Stephen Wolfram, has only a few things to say about quantum computing. Yet the book’s goal—to understand nature in computational terms—is one widely sha... |

7 |
A new approach to the geometrization of matter. gr-qc/0110060
- Crane
- 2001
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Citation Context ...usly by researchers in quantum gravity: in particular, that spacetime is a causal network arising from graph updating rules [13]; that particles could arise as ‘topological defects’ in such a network =-=[16]-=-; and that dimension and other geometric properties can be defined solely in terms of the network’s connectivity pattern [17]. The main difference we can discern between Wolfram’s model and earlier on... |

5 |
2000]: ‘A classical sequential growth dynamics for causal sets
- Rideout, Sorkin
(Show Context)
Citation Context ...njectures that spacetime is discrete at the Planck scale, of about 10 −33 centimeters or 10 −43 seconds. This conjecture is not new; it has long been considered in the context of loop quantum gravity =-=[13, 14]-=-, and has also received attention in connection with the holographic principle [15] from black hole thermodynamics. But are new ideas offered to substantiate the conjecture? For Wolfram, spacetime is ... |

3 |
t Hooft (1999): Quantum gravity as a dissipative deterministic system. Classical and Quantum Gravity 16(10
- unknown authors
(Show Context)
Citation Context ...at quantum mechanics is only an approximation to an underlying classical (and most likely deterministic) theory. Many physicists have sought such a theory, from Einstein to (in modern times) ’t Hooft =-=[18]-=-. But a series of results, beginning in the 1960’s, has made it clear that such a theory comes at a price. Although Wolfram discusses these results, in our view he has not understood what they entail.... |

3 |
2002) ‘The Man Who Cracked the Code to Everything’, Wired 10.06
- Levy
(Show Context)
Citation Context ... Bob’s location is made automatically. This is because the measurement involves E, an event which straddles the two locations (which would otherwise be spacelike separated). 4. Conclusion Steven Levy =-=[27]-=- opines in Wired that “probably the toughest criticism [of A New Kind of Science] will come from those who reject Wolfram’s ideas because the evidence for his contentions is based on observing systems... |

2 |
The universe in black and white, The Daily Telegraph
- Farmela
- 1999
(Show Context)
Citation Context ...utational terms—is one widely shared by the quantum computing community. Thus, many in the field will likely be curious: is this 2.5-kilogram tome worth reading? Notwithstanding newspaper comparisons =-=[3]-=- to Darwin’s Origin of Species, what is the book’s actual content? This review will not attempt a chapter-by-chapter evaluation, but will focus on two areas: computational complexity and fundamental p... |

2 |
Email correspondence
- Rowland
- 2002
(Show Context)
Citation Context ... two-party Bell inequality. Again, one needs to be able to talk about the state of G as a distribution or superposition over classical states. After Wolfram was sent a version of this review, Rowland =-=[26]-=-, an employee of Wolfram, wrote to us that the above argument fails for the following reason. We assumed that there exist two sequences of updating events, one in which Alice’s measurement precedes Bo... |

1 |
Causal evolution of spin networks, Nuclear Physics B508:409. gr-qc/9702025
- Markopoulou, Smolin
- 1997
(Show Context)
Citation Context ...njectures that spacetime is discrete at the Planck scale, of about 10 −33 centimeters or 10 −43 seconds. This conjecture is not new; it has long been considered in the context of loop quantum gravity =-=[13, 14]-=-, and has also received attention in connection with the holographic principle [15] from black hole thermodynamics. But are new ideas offered to substantiate the conjecture? For Wolfram, spacetime is ... |

1 |
The holographic principle (2002
- Bousso
(Show Context)
Citation Context ...r 10 −43 seconds. This conjecture is not new; it has long been considered in the context of loop quantum gravity [13, 14], and has also received attention in connection with the holographic principle =-=[15]-=- from black hole thermodynamics. But are new ideas offered to substantiate the conjecture? For Wolfram, spacetime is a causal network, in which events are vertices and edges specify the dependence rel... |