### Abstract

In this paper, we consider a linear decision tree such that a linear threshold function at each internal node has a bounded weight: the sum of the absolute values of its integer weights is at most w. We prove that if a Boolean function f is computable by such a linear decision tree of size (i.e., the number of leaves) s and rank r, then f is also computable by a depth-2 threshold circuit containing at most s(2w+1)^{r} threshold gates with weight at most (2w+1)^{r+1} in the bottom level. Combining a known lower bound on the size of depth-2 threshold circuits, we obtain a 2^{Ω(n/ logw)} lower bound on the size of linear decision trees computing the Inner-Product function modulo 2, which improves on the previous bound 2^{√n} if w = 2^{o(√n}).

Original language | English |
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Title of host publication | SOFSEM 2015 |

Subtitle of host publication | Theory and Practice of Computer Science - 41st International Conference on Current Trends in Theory and Practice of Computer Science, |

Editors | Tiziana Margaria-Steffen, Giuseppe F. Italiano, Jean-Jacques Quisquater, Roger Wattenhofer, Jaroslav Pokorný |

Publisher | Springer Verlag |

Pages | 412-422 |

Number of pages | 11 |

ISBN (Electronic) | 9783662460771 |

Publication status | Published - Jan 1 2015 |

Event | 41st International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2015 - Pec pod Sněžkou, Czech Republic Duration: Jan 24 2015 → Jan 29 2015 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8939 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 41st International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2015 |
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Country | Czech Republic |

City | Pec pod Sněžkou |

Period | 1/24/15 → 1/29/15 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*SOFSEM 2015: Theory and Practice of Computer Science - 41st International Conference on Current Trends in Theory and Practice of Computer Science,*(pp. 412-422). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8939). Springer Verlag.

**Lower bounds for linear decision trees with bounded weights.** / Uchizawa, Kei; Takimoto, Eiji.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*SOFSEM 2015: Theory and Practice of Computer Science - 41st International Conference on Current Trends in Theory and Practice of Computer Science,.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8939, Springer Verlag, pp. 412-422, 41st International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2015, Pec pod Sněžkou, Czech Republic, 1/24/15.

}

TY - GEN

T1 - Lower bounds for linear decision trees with bounded weights

AU - Uchizawa, Kei

AU - Takimoto, Eiji

PY - 2015/1/1

Y1 - 2015/1/1

N2 - In this paper, we consider a linear decision tree such that a linear threshold function at each internal node has a bounded weight: the sum of the absolute values of its integer weights is at most w. We prove that if a Boolean function f is computable by such a linear decision tree of size (i.e., the number of leaves) s and rank r, then f is also computable by a depth-2 threshold circuit containing at most s(2w+1)r threshold gates with weight at most (2w+1)r+1 in the bottom level. Combining a known lower bound on the size of depth-2 threshold circuits, we obtain a 2Ω(n/ logw) lower bound on the size of linear decision trees computing the Inner-Product function modulo 2, which improves on the previous bound 2√n if w = 2o(√n).

AB - In this paper, we consider a linear decision tree such that a linear threshold function at each internal node has a bounded weight: the sum of the absolute values of its integer weights is at most w. We prove that if a Boolean function f is computable by such a linear decision tree of size (i.e., the number of leaves) s and rank r, then f is also computable by a depth-2 threshold circuit containing at most s(2w+1)r threshold gates with weight at most (2w+1)r+1 in the bottom level. Combining a known lower bound on the size of depth-2 threshold circuits, we obtain a 2Ω(n/ logw) lower bound on the size of linear decision trees computing the Inner-Product function modulo 2, which improves on the previous bound 2√n if w = 2o(√n).

UR - http://www.scopus.com/inward/record.url?scp=84922032283&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84922032283&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84922032283

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 412

EP - 422

BT - SOFSEM 2015

A2 - Margaria-Steffen, Tiziana

A2 - Italiano, Giuseppe F.

A2 - Quisquater, Jean-Jacques

A2 - Wattenhofer, Roger

A2 - Pokorný, Jaroslav

PB - Springer Verlag

ER -