## grant GJ-43634x to M.I.T.Lectures on Network Complexity (1977)

### BibTeX

@MISC{77grantgj-43634x,

author = {},

title = {grant GJ-43634x to M.I.T.Lectures on Network Complexity},

year = {1977}

}

### OpenURL

### Abstract

These notes, often referred to as the “Frankfurt Lecture Notes”, are perhaps my most widely circulated unpublished work. Resulting from a series of lectures I gave at the University of Frankfurt in June of 1974, they summarize some early work on what is now known as circuit complexity. They circulated originally in the form of xeroxes of my hand-written notes. Later, in April of 1977, I revised the notes and had them typed up; copies of the typewritten version have also circulated widely. Now, with the availability of the world-wide web, I have decided to reissue them once again, this time in electronic form. In going over the notes again, I have tried to preserve the original style and to resist the temptation to make “improvements ” to either the content or its presentation. Nevertheless, I have fixed a few technical errors and have added a few missing assumptions here and there. I have also added a bibliography that was not present in the original. While I make no claims to its completeness, I have tried to add citations for the results referenced in the original notes, as well as giving references to a few related subsequent works. Even today, more that 20 years after the original lectures, readers may still find some material here of interest: • Section 1 presents counting arguments that establish upper and lower bounds on the maximum circuit complexity of any n-argument Boolean function over the full basis of 2-input gates. These and closely related results appear in [4, 12, 23, 25]. The particularly slick proof of Theorem 1.1 is due to Schnorr [20]. • Section 2 uses Turing time complexity T(n) to bound circuit complexity for families of Boolean functions. Savage [18] showed that the circuit complexity is at most O(T(n) 2). Here I present a result with Pippenger that reduces this bound to O(T(n)) for oblivious Turing machines and to O(T(n)log T(n)) for unrestricted Turing

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Citation Context ..., 26] note in passing that this bound can be improved to 2L(F) + O(n 2 log n) by using the4 INVERSION COMPLEXITY 20 sorting network of Ajtai, Komlós, and Szemerédi [1] instead of the Batcher network =-=[2]-=- in the proof of Theorem 4.5. The bound of that theorem then becomes O(n 2 log n). Still better bounds appear in [3, 26]. To prove Theorem 4.3 we need some additional concepts. Definition: Let F: K n ... |

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Citation Context ...ision: Beals, Nishino and Tanaka [3, 26] note in passing that this bound can be improved to 2L(F) + O(n 2 log n) by using the4 INVERSION COMPLEXITY 20 sorting network of Ajtai, Komlós, and Szemerédi =-=[1]-=- instead of the Batcher network [2] in the proof of Theorem 4.5. The bound of that theorem then becomes O(n 2 log n). Still better bounds appear in [3, 26]. To prove Theorem 4.3 we need some additiona... |

141 |
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Citation Context ...g arguments that establish upper and lower bounds on the maximum circuit complexity of any nargument Boolean function over the full basis of 2-input gates. These and closely related results appear in =-=[4, 12, 23, 25]-=-. The particularly slick proof of Theorem 1.1 is due to Schnorr [20]. • Section 2 uses Turing time complexity T(n) to bound circuit complexity for families of Boolean functions. Savage [18] showed tha... |

98 |
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Citation Context ...(n)log T(n)) for unrestricted Turing This research was supported in part by the National Science Foundation under research grant GJ-43634x to M.I.T.2 machines. These results subsequently appeared in =-=[17]-=- and were extended in [21]. • Section 3 illustrates the use of Turing machines to construct a small circuit for computing the transitive closure of a symmetric Boolean matrix. The Turing machine const... |

80 |
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Citation Context ... with minimal numbers of negations include [5, 10, 15]. General references with a wealth of information relating combinational complexity to other notions of finite function complexity include Savage =-=[19]-=- and more recently Wegener [27]. Both contain extensive bibliographies. Paterson [16] presents a nice collection of recent research papers on Boolean function complexity. Michael J. Fischer New Haven,... |

79 |
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Citation Context ...m for the other, and both can be done in time O(n log 2 7 · (log n) 1+ε ) for any ε > 0 using the fast matrix product method of Strassen [24] and the fast integer arithmetic of Schönhage and Strassen =-=[22]-=-. In case G is undirected, M becomes symmetric, and the above fact, restricted to symmetric matrices, is not known to hold. In fact, symmetric transitive closure is the same problem as finding the con... |

31 |
Boolean matrix multiplication and transitive closure
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Citation Context ... 1 iff i →G j (i.e., there is an edge from i to j in G). Then M ∗ describes the transitive closure G∗ of G, where i →G∗ j iff i = j or there is a path from i to j in G. ( ) ∨∧ Fact (Fischer and Meyer =-=[6]-=-, Munro [14]). matrix product and transitive closure can be done in essentially the same time, that is, a fast algorithm for one implies the existence of a fast algorithm for the other, and both can b... |

30 |
The Complexity of Boolean Functions. Wiley-Teubner series in computer science
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Citation Context ...ons include [5, 10, 15]. General references with a wealth of information relating combinational complexity to other notions of finite function complexity include Savage [19] and more recently Wegener =-=[27]-=-. Both contain extensive bibliographies. Paterson [16] presents a nice collection of recent research papers on Boolean function complexity. Michael J. Fischer New Haven, Connecticut April 19961 BOOLE... |

26 |
Efficient determination of the transitive closure of a directed graph
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Citation Context ... j (i.e., there is an edge from i to j in G). Then M ∗ describes the transitive closure G∗ of G, where i →G∗ j iff i = j or there is a path from i to j in G. ( ) ∨∧ Fact (Fischer and Meyer [6], Munro =-=[14]-=-). matrix product and transitive closure can be done in essentially the same time, that is, a fast algorithm for one implies the existence of a fast algorithm for the other, and both can be done in ti... |

23 | Computational work and time on finite machines
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Citation Context ...[4, 12, 23, 25]. The particularly slick proof of Theorem 1.1 is due to Schnorr [20]. • Section 2 uses Turing time complexity T(n) to bound circuit complexity for families of Boolean functions. Savage =-=[18]-=- showed that the circuit complexity is at most O(T(n) 2 ). Here I present a result with Pippenger that reduces this bound to O(T(n)) for oblivious Turing machines and to O(T(n)log T(n)) for unrestrict... |

21 |
A method of circuit synthesis
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Citation Context ...g arguments that establish upper and lower bounds on the maximum circuit complexity of any nargument Boolean function over the full basis of 2-input gates. These and closely related results appear in =-=[4, 12, 23, 25]-=-. The particularly slick proof of Theorem 1.1 is due to Schnorr [20]. • Section 2 uses Turing time complexity T(n) to bound circuit complexity for families of Boolean functions. Savage [18] showed tha... |

21 |
The network complexity and the Turing machine complexity of finite functions
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Citation Context ...ted Turing This research was supported in part by the National Science Foundation under research grant GJ-43634x to M.I.T.2 machines. These results subsequently appeared in [17] and were extended in =-=[21]-=-. • Section 3 illustrates the use of Turing machines to construct a small circuit for computing the transitive closure of a symmetric Boolean matrix. The Turing machine construction relies on the mail... |

15 |
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(Show Context)
Citation Context ...ority queue algorithm [7] but have never been published in their simple form. • Section 4 examines inversion complexity, the number of not-gates needed to compute a set F of Boolean functions. Markov =-=[13]-=- characterizes the exact minimal number of not-gates needed for arbitrary F in terms of the number of times F violates monotonicity along any monotone sequence of input vectors. Section 4 begins with ... |

12 |
Lattice theoretic properties of frontal switching functions
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(Show Context)
Citation Context ... been recently improved in [3, 26]. The question of how much the size increases as the number of not-gates is reduced further to the exact inversion complexity of F remains open. Related Work Gilbert =-=[9]-=- considered inversion complexity as early as 1954. He was motivated by relay contact networks in which devices for performing negation are more expensive or less reliable than those for performing the... |

11 | Fishspear: A priority queue algorithm
- Fischer, Paterson
- 1994
(Show Context)
Citation Context ... of a symmetric Boolean matrix. The Turing machine construction relies on the mail carrier algorithm that I developed with Paterson. These same ideas underlie the “Fishspear” priority queue algorithm =-=[7]-=- but have never been published in their simple form. • Section 4 examines inversion complexity, the number of not-gates needed to compute a set F of Boolean functions. Markov [13] characterizes the ex... |

6 |
More on the Complexity of Negation-Limited Circuits
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- 1995
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Citation Context ... be used to convert an arbitrary circuit of size C into an equivalent one of size 2C+O(n 2 log 2 n) that uses only ⌈log(n)+1⌉ not-gates. These results appear in [5] and have been recently improved in =-=[3, 26]-=-. The question of how much the size increases as the number of not-gates is reduced further to the exact inversion complexity of F remains open. Related Work Gilbert [9] considered inversion complexit... |

6 |
Practical decidability
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- 1972
(Show Context)
Citation Context ...g arguments that establish upper and lower bounds on the maximum circuit complexity of any nargument Boolean function over the full basis of 2-input gates. These and closely related results appear in =-=[4, 12, 23, 25]-=-. The particularly slick proof of Theorem 1.1 is due to Schnorr [20]. • Section 2 uses Turing time complexity T(n) to bound circuit complexity for families of Boolean functions. Savage [18] showed tha... |

6 |
The complexityof negation-limited networks—a brief survey
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- 1974
(Show Context)
Citation Context ...y ⌈log(n)+1⌉ not-gates. This result can be used to convert an arbitrary circuit of size C into an equivalent one of size 2C+O(n 2 log 2 n) that uses only ⌈log(n)+1⌉ not-gates. These results appear in =-=[5]-=- and have been recently improved in [3, 26]. The question of how much the size increases as the number of not-gates is reduced further to the exact inversion complexity of F remains open. Related Work... |

6 |
On the complexity of negationlimited Boolean networks (preliminary version
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- 1994
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Citation Context ... be used to convert an arbitrary circuit of size C into an equivalent one of size 2C+O(n 2 log 2 n) that uses only ⌈log(n)+1⌉ not-gates. These results appear in [5] and have been recently improved in =-=[3, 26]-=-. The question of how much the size increases as the number of not-gates is reduced further to the exact inversion complexity of F remains open. Related Work Gilbert [9] considered inversion complexit... |

5 |
Minimal Negative Gate Networks
- Nakamura, Tokura, et al.
- 1972
(Show Context)
Citation Context ...vices for performing negation are more expensive or less reliable than those for performing the other logical operations. Early papers to synthesize circuits with minimal numbers of negations include =-=[5, 10, 15]-=-. General references with a wealth of information relating combinational complexity to other notions of finite function complexity include Savage [19] and more recently Wegener [27]. Both contain exte... |

4 |
Synthesis of networks with a minimum number of negative gates
- Ibaraki, Muroga
- 1971
(Show Context)
Citation Context ...vices for performing negation are more expensive or less reliable than those for performing the other logical operations. Early papers to synthesize circuits with minimal numbers of negations include =-=[5, 10, 15]-=-. General references with a wealth of information relating combinational complexity to other notions of finite function complexity include Savage [19] and more recently Wegener [27]. Both contain exte... |

2 |
An efficient message-forwarding algorithm using sequential storage. Unpublished
- Fischer, Paterson, et al.
- 1982
(Show Context)
Citation Context ...be done in time O(n 2 ) by the algorithm below. Note added in revision: The algorithm below is known in the folklore. The remaining results in this section are due to Fischer, Paterson, and Pippenger =-=[8]-=- and have never been published. Let next(i) = { least j > i s.t. Mi,j = 1 0 if no such j To compute M ∗ : 1. for i ← 1 to n 2. [ j ← next(i); 3. if j ̸= 0 then Row j ← Row j ∨ Row i3 APPLICATION TO S... |

2 |
Boolean Function Complexity, volume 169
- Paterson, editor
- 1992
(Show Context)
Citation Context ...lth of information relating combinational complexity to other notions of finite function complexity include Savage [19] and more recently Wegener [27]. Both contain extensive bibliographies. Paterson =-=[16]-=- presents a nice collection of recent research papers on Boolean function complexity. Michael J. Fischer New Haven, Connecticut April 19961 BOOLEAN NETWORKS 3 1 Boolean Networks Let K = {0, 1}, f: K ... |

2 |
Berechungen in partiellen Algebren endlichen Typs
- Strassen
- 1973
(Show Context)
Citation Context |