## Circuit Minimization Problem (1999)

Venue: | In ACM Symposium on Theory of Computing (STOC |

Citations: | 32 - 1 self |

### BibTeX

@INPROCEEDINGS{Kabanets99circuitminimization,

author = {Valentine Kabanets and Jin-yi Cai},

title = {Circuit Minimization Problem},

booktitle = {In ACM Symposium on Theory of Computing (STOC},

year = {1999},

pages = {73--79},

publisher = {ACM Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study the complexity of the circuit minimization problem: given the truth table of a Boolean function f and a parameter s, decide whether f can be realized by a Boolean circuit of size at most s. We argue why this problem is unlikely to be in P (or even in P=poly) by giving a number of surprising consequences of such an assumption. We also argue that proving this problem to be NP-complete (if it is indeed true) would imply proving strong circuit lower bounds for the class E, which appears beyond the currently known techniques. Keywords: hard Boolean functions, derandomization, natural properties, NP-completeness. 1 Introduction An n-variable Boolean function f n : f0; 1g n ! f0; 1g can be given by either its truth table of size 2 n , or a Boolean circuit whose size may be significantly smaller than 2 n . It is well known that most Boolean functions on n variables have circuit complexity at least 2 n =n [Sha49], but so far no family of sufficiently hard functions has ...

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Citation Context ...rical parameters of R(I)) depends only on the size of I , and the sizes of I and R(I) are polynomially related. For example, the text-book reductions from SAT to 3-SAT, and from 3-SAT to Vertex Cover =-=[GJ79] are natur-=-al in the above sense. In fact, all "natural" NP-complete problems that we are aware of are complete under natural reductions; this includes the Minimum Size DNF Problem, for which a natural... |

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Citation Context ...ime poly(2 n ), then P 6= NP. So, if Yablonski succeeded in proving his intended claim, he would have found a negative solution to the P vs. NP problem even before that problem was formally stated in =-=[Coo71]-=-. Returning to our problem, we observe that MCSP is obviously in NP (just note that the input size is O(2 n ), and so we have enough time to check that a guessed circuit of appropriate size computes a... |

288 | Hardness vs. randomness
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Citation Context ...to use a hard Boolean function on O(log n) variables to derandomize BPP. We use their result to get the following algorithm in ZPP MCSP for every given BPP algorithm. (A similar argument was given in =-=[NW94]-=- to obtain another proof that BPP ` ZPP NP .) First, our algorithm guesses a truth table of a Boolean function on O(log n) variables of circuit complexity n \Omega\Gamma13 . This step is in ZPP MCSP s... |

185 |
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Citation Context ...uch that all of the output functions have circuit complexity greater than 2 m m (1 + (1 \Gamma fl) logm m ), for any fl ? 0. Our proof will use the following result that can be readily extracted from =-=[IW97]-=-. Theorem 5 (Impagliazzo-Wigderson) For every ffl ? 0, there exist c; d 2 N such that the truth table of a Boolean function f cn : f0; 1g cn ! f0; 1g of circuit complexity 2 fflcn can be transformed, ... |

176 | Natural proofs
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Citation Context ...Gamma Pr y2f0;1g 2k [C(y) = 1]j ? 1=s: The pseudorandom generator G k is called strong if it has hardness H(G k ) ? 2 k\Omega\Gamma50 . Let \Gamma be a complexity class. Following Razborov and Rudich =-=[RR97]-=-, we call a combinatorial property fC n g n?0 of n-variable Boolean functions f n \Gamma-natural with density ffi n if each C n contains a subset C n such that 1. the predicate f n ? 2 C n is computab... |

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Citation Context ...ed by linear-size circuits [26]. In particular, it follows that MCSP is not in P, with respect to the same oracle. 2.5 Two-Sided Error vs. Zero Error It is well-known that BPP _ ZPP NP [29] (see also =-=[23; 13; 19; 8]-=-). It is also obvious from the definitions that ZPP C_ RP C_ BPP. On the other hand, it is not known whether BPP C: RP or BPP C NP. We observe that if MCSP is easy, then any probabilistic algorithm wi... |

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Citation Context ...s truth table of size 2 n , or a Boolean circuit whose size may be significantly smaller than 2 n . It is well known that most Boolean functions on n variables have circuit complexity at least 2 n =n =-=[Sha49]-=-, but so far no family of sufficiently hard functions has been proven to exist in any relatively small uniform complexity class. As far as we know, every language in E = DTIME(2 O(n) ) may be decided ... |

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Citation Context ...k = `(log n) variables of circuit complexity 2 \Omega\Gamma k) for infinitely many k. \Xi 3.2 Implications for BPP We need the following two theorems on hardness-randomness trade-offs from [IW97] and =-=[BFNW93]-=-, respectively. Theorem 16 (Impagliazzo-Wigderson) If the class E contains a family of Boolean functions f n : f0; 1g n ! f0; 1g of circuit complexity at least 2 ffln , for some ffl ? 0, (i.o.), then ... |

73 | Circuit-size lower bounds and non-reducibility to sparse sets - Kannan - 1982 |

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Citation Context ...average-case algorithm for factoring that beats any known factoring algorithm; the best known (worst-case) deterministic factoring algorithm has the running time approximately 2 n14 on n-bit integers =-=[20; 24]-=-, while the best probabilistic algorithm runs in time approximately 2 x/'ff [14]. COROLLARY 3. If MCSP is in P, then, for any e > O, there is an algorithm running in time 2 n" that factors Blum intege... |

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Citation Context ... k exist in ZPP NP [KW98]. This is about the best possible one can get using relativizable techniques since there are oracles with respect to which all of P NP can be computed by linear-size circuits =-=[Wil85]-=-. In particular, it follows that MCSP is not in P, with respect to the same oracle. 2.5 Two-Sided Error vs. Zero Error It is well-known that BPP ` ZPP NP [ZH86] (see also [Sip83, Lau83, NW94, GZ97]). ... |

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Citation Context ...s 2 O(n) , given access to an NP-oracle. \Xi It was shown in [Kan82] that, for every k 2 N, \Sigma p 2 " \Pi p 2 contains a family of Boolean functions f n of circuit complexity greater than n k =-=; in [KW98], \Si-=-gma p 2 " \Pi p 2 was replaced by the class ZPP NP . By a padding argument, we easily get from Theorem 10 the following. Corollary 11 If MCSP is in P, then, for every k 2 N, there exists a langua... |

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Citation Context ...known (worst-case) deterministic factoring algorithm has the running time approximately 2 n=4 on n-bit integers [Pol74, Str76], while the best probabilistic algorithm runs in time approximately 2 p n =-=[LP92]-=-. Corollary 3 If MCSP is in P, then, for any ffl ? 0, there is an algorithm running in time 2 n ffl that factors Blum integers well on the average. The widely believed hardness of factoring may be tak... |

40 | Weak random sources, hitting sets, and BPP simulations
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Citation Context ...x2f0;1g n[C(x) = 1] = 0, and 1 if Pr x2f0;1g n[C(x) = 1] ? 1=2. As in the case of their global counterparts, the two local conditions stated above are also equivalent; the proof can be extracted from =-=[ACRT97]-=- (see also [BF99]). Now we show that, under the assumption that MSCP is easy, all of the global and local conditions stated above are equivalent. That is, if MCSP is in P, the following conditions are... |

38 | A survey of Russian approaches to Perebor (brute-force search) algorithms
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Citation Context ...to, Toronto, Canada. Email: kabanets@cs.toronto.edu. y Department of Computer Science, State University of New York at Buffalo, Buffalo, NY 14260. Email: cai@cs.buffalo.edu. 1 their truth tables (see =-=[Tra84]-=- for a more detailed discussion). It is not hard to see that if such a family of n-variable Boolean functions cannot be constructed in time poly(2 n ), then P 6= NP. So, if Yablonski succeeded in prov... |

37 |
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Citation Context ... bits, runs in time poly(n), and fools every circuit of size n on n inputs. It should be obvious that the existence of efficient discrepancy set generators implies BPP = P. Remarkably, Andreev et al. =-=[ACR98]-=- proved that the same conclusion can be achieved under the seemingly weaker assumption that efficient hitting set generators exist (see also [ACRT97, BF99, GW99] for simpler proofs). It turns out that... |

33 | Near-optimal conversion of hardness into pseudo-randomness - Impagliazzo, Shaltiel, et al. - 1999 |

28 | Another proof that BPP ⊆ PH (and more - Goldreich, Zuckerman - 1997 |

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Citation Context ...an be computed by linear-size circuits [Wil85]. In particular, it follows that MCSP is not in P, with respect to the same oracle. 2.5 Two-Sided Error vs. Zero Error It is well-known that BPP ` ZPP NP =-=[ZH86]-=- (see also [Sip83, Lau83, NW94, GZ97]). It is also obvious from the definitions that ZPP ` RP ` BPP. On the other hand, it is not known whether BPP ` RP or BPP ` NP. We observe that if MCSP is easy, t... |

21 |
A method of circuit synthesis
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(Show Context)
Citation Context ...into a pseudorandom generator Gd, : {0, 1} a'~ --> {0, 1} 2" running in time 2 °(") that has hardness H(Ga~) > 2 n. We also need a lower bound on the circuit complexity of most Boolean functions from =-=[15; 16]-=-. THEOREM 6 (LUPANOV). For any e > 0 and sufficiently large n, almost all n-variable Boolean functions need Boolean circuits of size greater than z" (1 + (1 - e) ] l°-~) n J° PROOF OF THEOREM 4. Let 7... |

17 | Einige Resultate über Berechnungskomplexität - Strassen - 1976 |

16 |
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Citation Context ... In fact, all "natural" NP-complete problems that we are aware of are complete under natural reductions; this includes the Minimum Size DNF Problem, for which a natural reduction from SAT is=-= given in [Mas79]. In -=-the next theorem, we use the notation SUBEXP = " ffl?0 DTIME(2 n ffl ). Theorem 15 If MCSP is NP-hard under a natural reduction from SAT, then 1. E contains a family of Boolean functions f n not ... |

16 | Super-Polynomial versus half-exponential circuit size in the exponential hierarchy - Miltersen, Vinodchandran, et al. - 1999 |

12 | Improved derandomization of BPP using a hitting set generator
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(Show Context)
Citation Context ... generators implies BPP = P. Remarkably, Andreev et al. [1] proved that the same conclusion can be achieved under the seemingly weaker assumption that efficient hitting set generators exist (see also =-=[2; 4; 7]-=- for simpler proofs). It turns out that these two assumptions are equivalent to the assumption that E contains a Boolean function of high circnit complexity. Namely, given an efficient hitting set gen... |

10 | One-sided versus two-sided error in probabilistic computation
- Buhrman, Fortnow
- 1999
(Show Context)
Citation Context ... = 0, and 1 if Pr x2f0;1g n[C(x) = 1] ? 1=2. As in the case of their global counterparts, the two local conditions stated above are also equivalent; the proof can be extracted from [ACRT97] (see also =-=[BF99]-=-). Now we show that, under the assumption that MSCP is easy, all of the global and local conditions stated above are equivalent. That is, if MCSP is in P, the following conditions are equivalent: 1. E... |

3 | BPP and the polynomial time hierarchy - Lautemann - 1983 |

3 |
The algorithmic difficulties of synthesizing minimal switching circuits
- Yablonski
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(Show Context)
Citation Context ...gon USA Copyright ACM 2000 1-58113-184-4/00/5...$5.00 We would fike to point out that the above problem was considered in the past; in fact, it was studied in the USSR already in the 50's (see, e.g., =-=[28; 27]-=-). Actually, Yablonski [28; 27] believed that he had shown the impossibility of eliminating the "brute-force search" when solving a related problem: "Compute a family {f,,}n>0 of n-variable Boolean fu... |

2 |
On the impossibility of eliminating PEREBOR in solving some problems of circuit theory
- Yablonski
- 1959
(Show Context)
Citation Context ...gon USA Copyright ACM 2000 1-58113-184-4/00/5...$5.00 We would fike to point out that the above problem was considered in the past; in fact, it was studied in the USSR already in the 50's (see, e.g., =-=[28; 27]-=-). Actually, Yablonski [28; 27] believed that he had shown the impossibility of eliminating the "brute-force search" when solving a related problem: "Compute a family {f,,}n>0 of n-variable Boolean fu... |

1 |
On the synthesis of certain classes of control systems
- Lupanov
- 1963
(Show Context)
Citation Context ...into a pseudorandom generator Gd, : {0, 1} a'~ --> {0, 1} 2" running in time 2 °(") that has hardness H(Ga~) > 2 n. We also need a lower bound on the circuit complexity of most Boolean functions from =-=[15; 16]-=-. THEOREM 6 (LUPANOV). For any e > 0 and sufficiently large n, almost all n-variable Boolean functions need Boolean circuits of size greater than z" (1 + (1 - e) ] l°-~) n J° PROOF OF THEOREM 4. Let 7... |

1 |
One-sided versus twosided error in probabifistic computation
- Buhrman, Fortnow
- 1999
(Show Context)
Citation Context ... generators implies BPP = P. Remarkably, Andreev et al. [1] proved that the same conclusion can be achieved under the seemingly weaker assumption that efficient hitting set generators exist (see also =-=[2; 4; 7]-=- for simpler proofs). It turns out that these two assumptions are equivalent to the assumption that E contains a Boolean function of high circnit complexity. Namely, given an efficient hitting set gen... |

1 |
Another proof that BPPCPH (and more
- Goldreich, Zuckerman
- 1997
(Show Context)
Citation Context ...ed by linear-size circuits [26]. In particular, it follows that MCSP is not in P, with respect to the same oracle. 2.5 Two-Sided Error vs. Zero Error It is well-known that BPP _ ZPP NP [29] (see also =-=[23; 13; 19; 8]-=-). It is also obvious from the definitions that ZPP C_ RP C_ BPP. On the other hand, it is not known whether BPP C: RP or BPP C NP. We observe that if MCSP is easy, then any probabilistic algorithm wi... |

1 |
Einige Resultate fiber Berechnungskomplexit£t. Jahresberichte der DMV
- Strassen
- 1976
(Show Context)
Citation Context ...average-case algorithm for factoring that beats any known factoring algorithm; the best known (worst-case) deterministic factoring algorithm has the running time approximately 2 n14 on n-bit integers =-=[20; 24]-=-, while the best probabilistic algorithm runs in time approximately 2 x/'ff [14]. COROLLARY 3. If MCSP is in P, then, for any e > O, there is an algorithm running in time 2 n" that factors Blum intege... |