## Computing with Restricted Nondeterminism: The Dependence of the OBDD Size on the Number of Nondeterministic Variables (1999)

Venue: | In Proceedings of the 19th Conference on Foundations of Software Technology and Theoretical Computer Science |

Citations: | 2 - 0 self |

### BibTeX

@INPROCEEDINGS{Sauerhoff99computingwith,

author = {Martin Sauerhoff},

title = {Computing with Restricted Nondeterminism: The Dependence of the OBDD Size on the Number of Nondeterministic Variables},

booktitle = {In Proceedings of the 19th Conference on Foundations of Software Technology and Theoretical Computer Science},

year = {1999},

pages = {342--355}

}

### OpenURL

### Abstract

. It is well-known that an arbitrary nondeterministic Turing machine can be simulated with polynomial overhead by a so-called guess-and-verify machine. It is an open question whether an analogous simulation exists in the context of space-bounded computation. In this paper, a negative answer to this question is given for nondeterministic OBDDs. If we require that all nondeterministic variables are tested at the top of the OBDD, i. e., at the beginning of the computation, this may blow-up the size exponentially. This is a consequence of the following main result of the paper. There is a sequence of Boolean functions fn : {0, 1} n # {0, 1} such that fn has nondeterministic OBDDs of polynomial size with O(n 1/3 log n) nondeterministic variables, but fn requires exponential size if only at most O(log n) nondeterministic variables may be used. 1 Introduction and Definitions So far, there are only few models of computation for which it has been possible to analyze the power of nondete...

### Citations

2939 | Graph-based algorithms for Boolean function manipulation
- Bryant
- 1986
(Show Context)
Citation Context ...ables, one follows a path starting at the source. At an interior node labeled by x i , the path continues with the edge labeled by a i . The output for a is the label of the sink reached in this way. =-=(2) A nondeterministic -=-branching program is syntactically a deterministic branching program on "usual" variables from X n and some "special" variables from Y n = {y 1 , . . . , y r(n) }, called nondeterm... |

621 | Communication complexity
- Kushilevitz, Nisan
- 1997
(Show Context)
Citation Context ... lemmas required for the proof of the main theorem of the paper. For a thorough introduction to communication complexity theory, we refer to the monographs of Hromkovi c [4] and Kushilevitz and Nisan =-=[10]-=-. A deterministic two-party communication protocol is an algorithm by which two players, called Alice and Bob, cooperatively evaluate a function f : X Y # {0, 1}, where X and Y are finite sets. Alice ... |

360 | The complexity of Boolean functions
- Wegener
- 1987
(Show Context)
Citation Context ...e of a deterministic (nondeterministic, resp.) branching program representing f n . For a history of results on branching programs, we have to refer to the literature, e. g., the monograph of Wegener =-=[15]-=- (and the forthcoming new monograph [16]). It is a fundamental open problem to prove superpolynomial lower bounds on the size of branching programs for explicitly defined Boolean functions even in the... |

160 |
Branching Programs and Binary Decision Diagrams
- WEGENER
- 2000
(Show Context)
Citation Context ...resp.) branching program representing f n . For a history of results on branching programs, we have to refer to the literature, e. g., the monograph of Wegener [15] (and the forthcoming new monograph =-=[16]-=-). It is a fundamental open problem to prove superpolynomial lower bounds on the size of branching programs for explicitly defined Boolean functions even in the deterministic case. The nondeterministi... |

91 |
bounds by probabilistic arguments
- Lower
- 1983
(Show Context)
Citation Context ... , d, let g i be the function computed by P i . Obviously, it holds that g i # IND n . By a simple counting argument (used in the same way by Hromkovi c and Schnitger in [6] and originally due to Yao =-=[17]-=-), we can conclude that there is an index i 0 # {1, . . . , d} such that |g -1 i 0 (1)| # w| IND -1 n (1)|/d. It is easy to see that | IND -1 n (1)| = n 2 n-1 , hence, |g -1 i 0 (1)| # nw 2 n-r-1 . It... |

89 | Private vs. common random bits in communication complexity
- Newman
- 1991
(Show Context)
Citation Context ...sources nondeterminism and randomness in more detail. The dependence of the OBDD size on the resource randomness has been dealt with to some extent in [13]. Analogous to a well-known result of Newman =-=[11]-=- for randomized communication complexity, it could be shown that O(log n) random bits (where n is the input length) are always sufficient to exploit the full power of randomness for OBDDs. On the othe... |

61 | On randomized one-round communication complexity
- Kremer, Nisan, et al.
- 1999
(Show Context)
Citation Context ...terature. We may interpret it as the description of direct storage access: The x-vector plays the role of the "memory contents," and the y-vector is an "address" in the memory. Kre=-=mer, Nisan, and Ron [9]-=- have shown that IND n has complexity# (n) for randomized one-way communication protocols with bounded error. It is easy to see that essentially log n bits of communication are sufficient and also nec... |

59 | Lower bounds for deterministic and nondeterministic branching programs
- Razborov
- 1991
(Show Context)
Citation Context ...ounds on the size of branching programs for explicitly defined Boolean functions even in the deterministic case. The nondeterministic case seems to be still harder (see, e. g., the survey of Razborov =-=[12]-=-). Nevertheless, several interesting restricted types of branching programs could be analyzed quite successfully, and for some of these models even exponential lower bounds could be proven. The goal i... |

18 | Lower bounds for depth-restricted branching programs - Krause - 1991 |

16 | Entropy of contact circuits and lower bounds on their complexity - Jukna - 1988 |

12 | On the size of randomized OBDDs and read-once branching programs for k-stable functions
- Sauerhoff
- 1999
(Show Context)
Citation Context ...cal lemma. In the following, the technique is called reduction technique for easier reference. 5 The reduction technique has appeared in various disguises in several papers. We use the formalism from =-=[14]-=- for our description. This makes use of the following standard reducibility concept from communication complexity theory. Definition 6 (Rectangular reduction). Let X f , Y f and X g , Y g be finite se... |

11 |
On the size of ordered binary decision diagrams representing threshold functions. Algorithms and Computation
- Hosaka, Takenaga, et al.
- 1994
(Show Context)
Citation Context ...gnment b to the nondeterministic variables such that the 1-sink is reached for the path belonging to the complete assignment consisting of a and b. Such a path to the 1-sink is called accepting path. =-=(3)-=- The size of a deterministic or nondeterministic branching program G, |G|, is the number of nodes in G. The deterministic (nondeterministic) branching program size of a function f n is the minimum siz... |

9 | Complexity Theoretical Results for Randomized Branching Programs
- Sauerhoff
- 1999
(Show Context)
Citation Context ...y to analyze the dependence of the size on the resources nondeterminism and randomness in more detail. The dependence of the OBDD size on the resource randomness has been dealt with to some extent in =-=[13]-=-. Analogous to a well-known result of Newman [11] for randomized communication complexity, it could be shown that O(log n) random bits (where n is the input length) are always sufficient to exploit th... |

8 |
Randomization and nondeterminism are incomparable for polynomial ordered binary decision diagrams
- Ablayev
(Show Context)
Citation Context ...ctions. Besides Boolean circuits and formulae, branching programs are one of the standard representations for Boolean functions. # This work has been supported by DFG grant We 1066/8-2. Definition 1. =-=(1)-=- A (deterministic) branching program (BP) G on the set of input variables X n = {x 1 , . . . , x n } is a directed acyclic graph with one source and two sinks, the latter labeled by the constants 0 an... |

4 |
c. Communication Complexity and Parallel Computing
- Hromkovi
- 1997
(Show Context)
Citation Context ...cation protocols and state two lemmas required for the proof of the main theorem of the paper. For a thorough introduction to communication complexity theory, we refer to the monographs of Hromkovi c =-=[4]-=- and Kushilevitz and Nisan [10]. A deterministic two-party communication protocol is an algorithm by which two players, called Alice and Bob, cooperatively evaluate a function f : X Y # {0, 1}, where ... |

3 |
Nondeterministic communication with a limited number of advice bits
- c, Schnitger
- 1996
(Show Context)
Citation Context ...sary to have randomized OBDDs of polynomial size (see [13] for details). What do we know about the resource nondeterminism? In the context of communication complexity theory, Hromkovi c and Schnitger =-=[6]-=- have proven that, contrary to the randomized case, the number of nondeterministic advice bits cannot be bounded by O(log n) without restricting the computational power of the model. An asymptotically... |

1 |
c and M. Sauerhoff. Communication with restricted nondeterminism and applications to branching program complexity
- Hromkovi
- 1999
(Show Context)
Citation Context ... restricting the computational power of the model. An asymptotically exact tradeoff between one-way communication complexity and the number of advice bits has been proven by Hromkovi c and the author =-=[5]-=-. These results lead to the conjecture that there should be sequences of functions which have nondeterministic OBDDs of polynomial size, but which require exponential size if the amount of nondetermin... |