## Bounded-width polynomial-size branching programs recognize exactly those languages (1986)

Venue: | in NC’, in “Proceedings, 18th ACM STOC |

Citations: | 213 - 14 self |

### BibTeX

@INPROCEEDINGS{Barrington86bounded-widthpolynomial-size,

author = {A. Barrington},

title = {Bounded-width polynomial-size branching programs recognize exactly those languages},

booktitle = {in NC’, in “Proceedings, 18th ACM STOC},

year = {1986},

pages = {1--5}

}

### Years of Citing Articles

### OpenURL

### Abstract

We show that any language recognized by an NC ’ circuit (fan-in 2, depth O(log n)) can be recognized by a width-5 polynomial-size branching program. As any bounded-width polynomial-size branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC’. Further, following

### Citations

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Citation Context ...riate definition of recognition of a language by a PBP. As these PBPs can be thought of as reoersible nonuniform log-space Turing machine computations, this suggests a comparison with work of Bennett =-=[9]-=-. The effect of nondeterminism on these classes must be examined as well, suggesting possible new attacks on the problem of whether L is equal to NL, the class of languages recognized by nondeterminis... |

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19 | Two lower bounds for branching programs
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Citation Context ...e defined by Lee [23] as an alternative to Boolean circuits in the description of switching problems-he called them “binary decision programs.” They were later studied in the Master’s thesis of Masek =-=[24]-=- under the name of “decision graphs.” Borodin et al. [lo] and Chandra et al. [ 1 l] raised the question of the power of bounded-width branching programs. Borodin et al. noted that the class BWBP conta... |

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6 |
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- 1986
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Citation Context ...perpolynomial lower bound for, say, the clique function would prove NC’ different from NP. The power of general poly-size permutation BPS (no restriction on width) was mentioned as an open problem in =-=[6]-=-. Cook and McKenzie [15] have just shown that the word problem for S, is complete for log space under NC’ reductions, even if the inputs and outputs are in pointwise notation (i.e., a permutation (r i... |

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5 |
personal communication
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2 |
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- 1983
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2 |
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Citation Context ... result is a nonuniform version of the well-known result that regular languages can be recognized by NC’ circuits. The proof in the uniform case essentially appears in [30] and is given explicitly in =-=[22]-=-. THEOREM 2. IfAs [Z] * is recognized by a w-BP B of length I, A is recognized by a fan-in 2 circuit of depth O(log I), where the constant depends on w. Proof: Recall our notion of recognition-we say ... |

1 |
Ci formulae on finite structures, Ann. Pure Applied Logic 24
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- 1983
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Citation Context ...in et al. noted that the class BWBP contains AC0 (languages recognized by unbounded fan-in, constant-depth, polynomialsize Boolean circuits) as well as the parity function (shown to be outside AC0 in =-=[17,2]-=-). They conjectured that the majority function was not in BWBP, in fact that for bounded width it requires exponential length. Subsequent results appeared which could be interpreted as progress toward... |

1 |
Bounded depth, polynomial-size circuits for symmetric functions, Theoret
- FAGIN, KLAWE, et al.
- 1985
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Citation Context ...degree structure (they had only just given the first proof that the structure of NC’ was nontrivial) and conjectured that majority was not reducible to parity. This study was taken up by Fagin et al. =-=[16]-=-, who found many new AC0 reducibilities among symmetric functions. Modulo the new parity lower bounds of Yao [35] and Hastad [19], they characterize those symmetric functions in AC’. They show that th... |

1 |
There are no p-complete families of symmetric Boolean functions, preprint
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Citation Context ...rge class of symmetric functions. Interestingly, no complete symmetric function exists in the projection-reducibility theory of Skyum and Valiant [32], by a recent result of Gereb-Graus and Szemeredi =-=[18]-=-. Chandra et al. [ 131 prove several natural functions AC0 equivalent to majority. In this section we show that solvability of a group is the key to the applicability of the methods used earlier for t... |

1 |
Rudiments of a branching program based complexity theory, preprint
- MEINEL
- 1986
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Citation Context ...tion of nondeterministic branching programs, that nondeterministic BWBP is also nonunifirm NC’. However, other definitions of nondeterministic branching programs are possible-see, for example, Meinel =-=[25]-=-. Our intuition must accept the fact that a Sstate NUDFA can count in polynomial time, as all symmetric functions are in NC’. It can also divide, as by [S] integer division is in nonuniform NC’. This ... |

1 |
A lower bound on complexity of branching programs
- AK
- 1984
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Citation Context ...BWBP, in fact that for bounded width it requires exponential length. Subsequent results appeared which could be interpreted as progress toward proving this conjecture. Chandra et al. [ 111 and Pudlak =-=[27]-=- showed superlinear length lower bounds for arbitrary constant width. In [lo] the idea was to work with width-2 and get exponential bounds. That paper did so for a restricted class of BPS, and Yao [34... |

1 |
Lower bounds for the size of circuits of bounded depth with basis {&, @}, preprint [Russian
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Citation Context ...hod of [17] does not seem to extend even to parity (mod 2) gates; as the restriction of a parity gate is still a parity gate. However, there has recently been dramatic progress in this area. Razborov =-=[28]-=-BOUNDED-WIDTH BRANCHING PROGRAMS 161 has proven the original conjecture of [17] by showing that any constant depth circuit of AND and mod 2 gates computing the majority function has exponential size.... |