Us Using Knowled edge e Compilati tion to to PP PP PP -Complet - PowerPoint PPT Presentation

Us Using Knowled edge e Compilati tion to to PP PP PP -Complet Solve e PP ete e Pro roblem ems YooJung Choi UCLA Dagstuhl 2017

Agenda • Beyond NP • Sentential Decision Diagrams (SDDs) • Solving MAJ-MAJ-SAT using SDDs • Application: Same-Decision Probability

Pro robabilisti tic c Infer eren ence Input: Bayesian Network SDP PP PP MAP NP PP PP Marginals NP MPE Complete problems: hardest in their class A. Darwiche

Pro roto toty typical Pro roblem ems MAJ-MAJ-SAT PP PP NP PP E-MAJ-SAT MAJ-SAT PP NP SAT Boolean expressions: (A or (not B) or C), ((not A) or D or (not E)), …. A. Darwiche

Reducti tions Oztok et al, KR 2016 Prototypical problems SDP MAJ-MAJ-SAT PP PP Huang et al, AAAI 2006 MAP NP PP E-MAJ-SAT Darwiche, KR 2002 Marginals MAJ-SAT PP Park, AAAI 2002 NP MPE SAT Boolean expressions: (A or (not B) or C), ((not A) or D or (not E)), …. A. Darwiche

Solvi ving by y Knowled edge e Compilati tion Prototypical problems SDP MAJ-MAJ-SAT PP PP Systematic MAP NP PP E-MAJ-SAT Approach (Compile to Marginals PP MAJ-SAT Boolean Circuits) NP MPE SAT Boolean expressions: (A or (not B) or C), ((not A) or D or (not E)), …. A. Darwiche

MAJ-MAJ-SAT A B C Boolean expression: T T T (A or B) and (not C) T T F Split variables X ={C}, Y ={A,B} T F T T F F F T T F T F F F T F F F A. Darwiche

MAJ-MAJ-SAT A B C Boolean expression: T T T (A or B) and (not C) T T F Split variables X ={C}, Y ={A,B} T F T T F F MAJ-MAJ-SAT: Is there a majority of X -instantiation under which the majority of Y -instantiations satisfying? F T T F T F F F T F F F A. Darwiche

MAJ-MAJ-SAT A B C Boolean expression: T T T (A or B) and (not C) T T F Split variables X ={C}, Y ={A,B} T F T MAJ-MAJ-SAT: Is there a majority of X -instantiation T F F under which the majority of Y -instantiations satisfying? F T T No F T F F F T F F F A. Darwiche

Knowled edge Compilati tion encoding NNF Circuit or and and (A and (not B)) or(C and (not D)) or or or or or ((not C) and D) Compiler and and and and and and and and …  A  B  D  C B A C D MAJ-MAJ-SAT Answer in E-MAJ-SAT MAJ-SAT Linear Time SAT A. Darwiche

NNF NNF Circu cuits ts  L K L   P   L   P  A  L  K L   P  P A L P P K  K A  A A  A A. Darwiche

De Deco composability ty (DNN DNNF) F) Darwiche, JACM 2001  L K L   P   L   P  A  L  K L   P  P A L P P K  K A  A A  A A. Darwiche

Deco De composability ty (DNN DNNF) F) Darwiche, JACM 2001 SAT in linear time MAJ-MAJ-SAT PP PP NP PP E-MAJ-SAT MAJ-SAT PP  L K L   P   L   P  A  L  K L   P  P A L P P NP SAT K  K A  A A  A A. Darwiche

De Dete terminism (d-DN DNNF) F) Darwiche, JANCL 2000 L  L   L K  P   L   P  A  L  K  P  P A L P P Input: L, K, P , A K  K A  A A  A A. Darwiche

De Dete terminism (d-DN DNNF) F) Darwiche, JANCL 2000 MAJ-SAT in linear time MAJ-MAJ-SAT PP PP NP PP E-MAJ-SAT MAJ-SAT PP L  L   L K  P   L   P  A  L  K  P  P A L P P NP SAT Input: L, K, P , A K  K A  A A  A A. Darwiche

Structu tured De Deco composability ty Pipatsrisawat & Darwiche, AAAI 2008  L K L   P   L   P  A  L  K L   P  P A L P P K  K A  A A  A A. Darwiche

vtree • Structu tured De Deco composability ty • • Pipatsrisawat & Darwiche, AAAI 2008 L K P A  L K L   P   L   P  A  L  K L   P  P A L P P K  K A  A A  A A. Darwiche

vtree • Structu tured De Deco composability ty • • Pipatsrisawat & Darwiche, AAAI 2008 L K P A  L K L   P   L   P  A  L  K L   P  P A L P P K  K A  A A  A A. Darwiche

vtree • Structu tured De Deco composability ty • • Pipatsrisawat & Darwiche, AAAI 2008 L K P A  L K L   P   L   P  A  L  K L   P  P A L P P K  K A  A A  A A. Darwiche

Parti titi tioned ed De Determ erminism (SDDs DDs) Darwiche, IJCAI 2011 A. Darwiche

Parti titi tioned ed De Determ erminism (SDDs DDs) Darwiche, IJCAI 2011 L  L   L K  P   L   P  A  L  K  P  P A L P P Input: L, K, P , A K  K A  A A  A A. Darwiche

Parti titi tioned ed De Determ erminism (SDDs DDs) Darwiche, IJCAI 2011 L  L   L K  P   L   P  A  L  K  P  P A L P P Input: L, K, P , A K  K A  A A  A A. Darwiche

Parti titi tioned ed De Determ erminism (SDDs DDs) Darwiche, IJCAI 2011 L  L   L K  P   L   P  A  L  K  P  P A L P P Input: L, K, P , A K  K A  A A  A A. Darwiche

Parti titi tioned ed De Determ erminism (SDDs DDs) Darwiche, IJCAI 2011 MAJ-MAJ-SAT PP PP NP PP E-MAJ-SAT MAJ-SAT PP NP SAT L  L   L K  P   L   P  A  L  K  P  P A L P P Input: L, K, P , A K  K A  A A  A A. Darwiche

Parti titi tioned ed De Determ erminism (SDDs DDs) Not yet… Darwiche, IJCAI 2011 MAJ-MAJ-SAT PP PP NP PP E-MAJ-SAT MAJ-SAT PP NP SAT L  L   L K  P   L   P  A  L  K  P  P A L P P Input: L, K, P , A K  K A  A A  A MAJ-MAJ-SAT in linear time using appropriate vtree Oztok & Darwiche, KR 2016 A. Darwiche

Solvi ving MAJ-MAJ-SAT using SDDs DDs Will solve a functional variant: How many x instantiations are there, that lead to more than T satisfying assignments?

Constr trained SDDs DDs X -constrained vtree contains a node v x • on the right-most path • s.t. X is precisely the set of variables outside of v x

Constr trained SDDs DDs X -constrained vtree contains a node v x • on the right-most path • s.t. X is precisely the set of variables outside of v x 1 X -constrained node 2 3 L K P A X ={L,K}

Constr trained SDDs DDs X -constrained vtree contains a node v x • on the right-most path • s.t. X is precisely the set of variables outside of v x 1 2 L X -constrained node X ={L,K} 3 K A P

1 Constr trained SDDs DDs 2 3 f L K P A  L K L   P   L   P  A  L  K L   P  P A L P P K  K A  A A  A

1 Constr trained SDDs DDs 2 3 f L K P A  L K L   P   L   P  A  L  K L   P  P A L P P K  K A  A A  A

1 Constr trained SDDs DDs 2 3 f L K P A  L K L   P   L   P  A  L  K L   P  P A L P P K  K A  A A  A SDD v2 library will include support for constrained SDDs

MAJ-MAJ-SAT Algori rith thm ෍ MC 𝑔 𝐲 > 𝑈] Oztok & Darwiche, KR 2016 𝐲 𝐘 = {L, K} 𝑈 = 2  L K L   P   L   P  A  L  K L   P  P A L P P K  K A  A A  A

MAJ-MAJ-SAT Algori rith thm ෍ MC 𝑔 𝐲 > 𝑈] Oztok & Darwiche, KR 2016 𝐲 𝐘 = {L, K} 𝑈 = 2  L K L   P   L   P  A  L  K L   P  P A L P P 1 1 1 0 1 1 1 0 1 0 1 1 1 1 1 0 1 1 0 1 1 K  K A  A A  A 1 1 1 1 1 1

MAJ-MAJ-SAT Algori rith thm ෍ MC 𝑔 𝐲 > 𝑈] Oztok & Darwiche, KR 2016 𝐲 𝐘 = {L, K} 𝑈 = 2 1 1 2 2 1 0 0 0 0 0 1 2  L K L   P   L   P  A  L  K L   P  P A L P P 1 1 1 0 1 1 1 0 1 0 1 1 1 1 1 0 1 1 0 1 1 K  K A  A A  A 1 1 1 1 1 1

MAJ-MAJ-SAT Algori rith thm ෍ MC 𝑔 𝐲 > 𝑈] Oztok & Darwiche, KR 2016 𝐲 𝐘 = {L, K} 𝑈 = 2 1 [2>2]=0 2 [3>2]=1 [1>2]=0 1 1 1 2 2 1 0 0 0 0 0 1 2  L K L   P   L   P  A  L  K L   P  P A L P P 1 1 1 0 1 1 1 0 1 0 1 1 1 1 1 0 1 1 0 1 1 K  K A  A A  A 1 1 1 1 1 1

MAJ-MAJ-SAT Algori rith thm ෍ MC 𝑔 𝐲 > 𝑈] Oztok & Darwiche, KR 2016 𝐲 𝐘 = {L, K} 𝑈 = 2 0 0 2 1 [2>2]=0 2 [3>2]=1 [1>2]=0 1 1 1 2 2 1 0 0 0 0 0 1 2  L K L   P   L   P  A  L  K L   P  P A L P P 1 1 1 0 1 1 1 0 1 0 1 1 1 1 1 0 1 1 0 1 1 K  K A  A A  A 1 1 1 1 1 1

MAJ-MAJ-SAT Algori rith thm ෍ MC 𝑔 𝐲 > 𝑈] Oztok & Darwiche, KR 2016 2 𝐲 𝐘 = {L, K} 𝑈 = 2 0 0 2 1 [2>2]=0 2 [3>2]=1 [1>2]=0 1 1 1 2 2 1 0 0 0 0 0 1 2  L K L   P   L   P  A  L  K L   P  P A L P P 1 1 1 0 1 1 1 0 1 0 1 1 1 1 1 0 1 1 0 1 1 K  K A  A A  A 1 1 1 1 1 1

Ap Appl plications Is it even worth solving PP PP problems?

Same-Deci ecision Pro robability ty Threshold for Pregnancy is 90% A. Darwiche