B+E and d4 Emmanuel Lonca CRIL, Universit dArtois and CNRS KOCOON - - PowerPoint PPT Presentation

B+E and d4

Emmanuel Lonca

CRIL, Université d’Artois and CNRS

KOCOON workshop, 18 dec 2019

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Outline

One talk, two tools !

◮ B+E : a preprocessor ◮ d4 : a compiler

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Outline

One talk, many tools !

◮ B+E : a preprocessor ◮ d4 : a compiler and a reasoner ◮ pddl2cnf, cnf2eadt, ...

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B+E

◮ B+E : preprocessor initialy designed for model counting ◮ based on variable definability (gates) detection

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Definability

Exploiting definability for model counting, the old (pmc) way :

◮ replacing a variable by its definition ◮ gate replacement exploits explicit definitions

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Definability

Exploiting definability for model counting, the old (pmc) way :

◮ replacing a variable by its definition ◮ gate replacement exploits explicit definitions

◮ for k variables, 22k possible gates 4/15

Definability

Exploiting definability for model counting, the old (pmc) way :

◮ replacing a variable by its definition ◮ gate replacement exploits explicit definitions

◮ for k variables, 22k possible gates ◮ restricted to some gates (AND, XOR, ...) 4/15

Implicit vs. explicit definability

One does not need to identify the gates themselves but it can be enough to determine that such gates exist ! Let Σ ∈ L, X ⊆ P and y ∈ P :

◮ explicit definability : ∃Φ ∈ PROPX s.t. Σ |

= ΦX ↔ y

◮ implicit definability : ∀γX over X, γX ∧ Σ |

= y or γX ∧ Σ | = ¬y

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Implicit vs. explicit definability

One does not need to identify the gates themselves but it can be enough to determine that such gates exist ! Let Σ ∈ L, X ⊆ P and y ∈ P :

◮ explicit definability : ∃Φ ∈ PROPX s.t. Σ |

= ΦX ↔ y

◮ implicit definability : ∀γX over X, γX ∧ Σ |

= y or γX ∧ Σ | = ¬y Beth, 1953 : implicit and explicit definability coincide

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Implicit vs. explicit definability

One does not need to identify the gates themselves but it can be enough to determine that such gates exist ! Let Σ ∈ L, X ⊆ P and y ∈ P :

◮ explicit definability : ∃Φ ∈ PROPX s.t. Σ |

= ΦX ↔ y

◮ implicit definability : ∀γX over X, γX ∧ Σ |

= y or γX ∧ Σ | = ¬y Padoa, 1903 : Σ defines y in terms of X iff Σ ∧ Σ′

X ∧ y ∧ ¬y′ |

= ⊥

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B+E

B+E = Bipartition + Eliminate

Algorithm 1: B input : a CNF formula Σ

• utput: a set O of output variables, i.e., variables defined in Σ in

terms of I = Var(Σ) \ O

1 Σ, O← backbone(Σ); 2 V ← sort(Var(Σ)); 3 I←∅; 4 foreach x ∈ V do 5

if defined?(x, Σ, I ∪ succ(x, V ), max#C) then

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O←O ∪ {x};

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else

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I←I ∪ {x};

9 return O 6/15

B+E

B+E = Bipartition + Eliminate

Algorithm 2: E input : a CNF formula Σ and a set of output variables O ⊆ Var(Σ)

• utput: a CNF formula Φ such that Φ ≡ ∃E.Σ for some E ⊆ O

1 Φ←Σ ; iterate←true ; P←O; 2 while iterate do 3

E←P ; P←∅ ; iterate←false;

4

while E = ∅ do

5

x←select(E, Φ) ; E←E \ {x};

6

if #(Φx) × #(Φ¬x) > max#Res then

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P←P ∪ {x}

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else

9

R←removeSub(Res(x, Φ), Φ);

10

if #((Φ \ Φx,¬x) ∪ R) ≤ #(Φ) then

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Φ←(Φ \ Φx,¬x) ∪ R;

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iterate←true;

13

else P←P ∪ {x};

14 return Φ 6/15

Variable reduction

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Literal reduction

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Solved (d4)

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B+E and compilation

◮ KOCOON → MC KC ◮ B+E(Σ) ≡ Σ

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B+E and compilation

◮ KOCOON → MC KC ◮ B+E(Σ) ≡ Σ but B+E(Σ) can be queried if only variables of I

are involved

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B+E and compilation

◮ KOCOON → MC KC ◮ B+E(Σ) ≡ Σ but B+E(Σ) can be queried if only variables of I

are involved

◮ some positive results for planning

◮ PDDL → CNF ◮ B+E with I containing all action, initial state and goal variables 10/15

B+E and compilation

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B+E and compilation

◮ B+E very efficient for MC, and may help for KC (planning) ◮ https ://www.cril.univ-artois.fr/KC/bpe2.html

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B+E and compilation

◮ B+E very efficient for MC, and may help for KC (planning) ◮ https ://www.cril.univ-artois.fr/KC/bpe2.html ◮ send your benchmarks to compile@cril.fr

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Compile ! project

The purpose of the Compile ! project is to make available on the Web some ressources, especially pieces of software developed at CRIL, which are somehow relevant to knowledge compilation purposes.

◮ translators : bn2cnf, cn2cnf, pddl2cnf, whatever2cnf, tt2bm ◮ preprocessors : pmc, B+E ◮ counters and compilers : several tools including dmc and d4

See the Compile ! project homepage

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d4

◮ d4 can now execute queries on d-DNNF (interactive shell)

◮ conditioning ◮ consistency check (under assumptions) ◮ model counting (under assumptions) ◮ linear optimization (query + transform)

◮ and load/save them

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B+E and d4

Emmanuel Lonca

CRIL, Université d’Artois and CNRS

KOCOON workshop, 18 dec 2019

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