Table 6:

Description
. | DPP Format
. | Parameters
. |
---|---|---|

TupleILP | ||

Sub graph must have ≤ w_{1} active tuples | $\u2211i\u2208FYii\u2264w1+1$ (16) | – |

Active hypothesis term must have ≤ w_{2} edges | $H\theta [:,:,i]\u2299Y\u2264w2\u2200i\u2208Ht$ (17) | H_{θ} is populated by hypothesis term matrix H with dimension ((n + k) × (n + k) × l) and the values are given by: $Hijk=1,\u2200k\u2208Ht,i\u2208H,j\u2208F,tk\u2208trm(hi),tk\u2208trm(fj)1,\u2200k\u2208Ht,i\u2208F,j\u2208H,tk\u2208trm(hj),tk\u2208trm(fi)0,otherwise$ (18) |

Active tuple must have active subject | $Y\u2299T\theta S>=E\u2299A\theta $ (19) | A_{θ} populated by adjacency matrix A, $T\theta S$ by subject tuple matrix T^{S} with dimension ((n + k) × (n + k)) and the values are given by: $TijS=1,i\u2208H,j\u2208F,|trm(hi)\u2229trm(fjS)|>01,i\u2208F,j\u2208H,|trm(hj)\u2229trm(fiS)|>00,otherwise$ (20) |

Active tuple must have ≥ w_{3} active fields | $Y\u2299T\theta S+Y\u2299T\theta P+Y\u2299T\theta O\u2265w3(Y\u2299A\theta )$ (21) | A_{θ} populated by adjacency matrix A and $T\theta S$, $T\theta P$, $T\theta O$ populated by subject, predicate and object matrices T^{S}, T^{P}, T^{O} respectively. Predicate and object tuples are converted into T^{P},T^{O} matrices similar to T^{S} |

Active tuple must have an edge to some hypothesis term | Implemented during graph construction by only considering tuples that have lexical overlap with a hypothesis | – |

ExplanationLP | ||

Limits the total number of abstract facts to w_{4} | $diag(Y)\xb7F\theta AB\u2264w4$ (22) | $F\theta AB$ is populated by Abstract fact matrix F^{AB}, where: $FijAB=1,i\u2208H,j\u2208FA0,otherwise$ (23) |

Description
. | DPP Format
. | Parameters
. |
---|---|---|

TupleILP | ||

Sub graph must have ≤ w_{1} active tuples | $\u2211i\u2208FYii\u2264w1+1$ (16) | – |

Active hypothesis term must have ≤ w_{2} edges | $H\theta [:,:,i]\u2299Y\u2264w2\u2200i\u2208Ht$ (17) | H_{θ} is populated by hypothesis term matrix H with dimension ((n + k) × (n + k) × l) and the values are given by: $Hijk=1,\u2200k\u2208Ht,i\u2208H,j\u2208F,tk\u2208trm(hi),tk\u2208trm(fj)1,\u2200k\u2208Ht,i\u2208F,j\u2208H,tk\u2208trm(hj),tk\u2208trm(fi)0,otherwise$ (18) |

Active tuple must have active subject | $Y\u2299T\theta S>=E\u2299A\theta $ (19) | A_{θ} populated by adjacency matrix A, $T\theta S$ by subject tuple matrix T^{S} with dimension ((n + k) × (n + k)) and the values are given by: $TijS=1,i\u2208H,j\u2208F,|trm(hi)\u2229trm(fjS)|>01,i\u2208F,j\u2208H,|trm(hj)\u2229trm(fiS)|>00,otherwise$ (20) |

Active tuple must have ≥ w_{3} active fields | $Y\u2299T\theta S+Y\u2299T\theta P+Y\u2299T\theta O\u2265w3(Y\u2299A\theta )$ (21) | A_{θ} populated by adjacency matrix A and $T\theta S$, $T\theta P$, $T\theta O$ populated by subject, predicate and object matrices T^{S}, T^{P}, T^{O} respectively. Predicate and object tuples are converted into T^{P},T^{O} matrices similar to T^{S} |

Active tuple must have an edge to some hypothesis term | Implemented during graph construction by only considering tuples that have lexical overlap with a hypothesis | – |

ExplanationLP | ||

Limits the total number of abstract facts to w_{4} | $diag(Y)\xb7F\theta AB\u2264w4$ (22) | $F\theta AB$ is populated by Abstract fact matrix F^{AB}, where: $FijAB=1,i\u2208H,j\u2208FA0,otherwise$ (23) |

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