ࡱ> g`abcdefy<(0 g/ 0DTimes New Roman(0(z[ 0 DTahomaew Roman(0(z[ 0 " DWingdingsRoman(0(z[ 0 0DArialngsRoman(0(z[ 0 @DCourier Newman(0(z[ 0 1PDSymbol Newman(0(z[ 0  C0.  @n?" dd@  @@`` ` X  Up=<'IO&'4**J01,q6"8!.2N  7:D<_AC F DkUj> L> 3C;B =,G H'%N.9?(&/  35?(E 4!.!.!)*f`qn<< J  ; ; ?;!),1<W<7BS.!]-</58<=>U 0AA @8 +̵ ʚ;2Qa;ʚ;g4CdCd@z[ 08ppp@ <4ddddl 0L<4!d!dl 0L<4BdBdl 0L80___PPT10 ?  %)PSearching while Keeping a Trace The Evolution from SAT to Knowledge Compilation4Q($00Adnan Darwiche Computer Science Department UCLA 1$"4OSearching while Keeping a Trace The evolution from SAT to Knowledge Compilation,P(0jSatisfiability (SAT) Knowledge compilation The connection: The trace of search Implications Open questions]'Satisfiability (SAT)wIs a set of Boolean constraints satisfiable? Input to SAT is typically a CNF SAT is mostly solved by DPLL search P.ZZFZZ-Kb M/Satisfiability (SAT)SAT Solvers: Significant growth in last decade; many solvers publicly available (source code); millions of variables & constraints not uncommon. Applications: Verification, planning, diagnosis, CAD, non-propositional reasoning (e.g., SMTs), & ^ZZ  )`5Knowledge Compilation Map What s the space of possible target compilation languages? Can it be synthesized in a semantically systematic way? How do the languages compare? Succinctness (relative size) Operations they support in polytimeT;Z8ZZAZ;8A Applications Knowledge Compilation MapFor a given application: identify needed operations Choose most succinct language that supports desired operations Compile knowledge base into chosen languageZNegation Normal FormNegation Normal FormDecomposability Determinism 6 1 Inference by Compiling to d-DNNFDeterministic Conformant Planning Blai Bonet and Hector Geffner (KR 2006) Probabilistic Conformant Planning Jinbo Huang (AIPS 2006) Model-based diagnosis Paul Elliott and Brian Williams (AAAI 2006) Anthony Barrett (IJCAI 2005) Databases (query re-write) Yolife Arvelo, Blai Bonet and Maria Esther Vidal (AAAI 2006) Inference in Bayesian Networks (2006 competition) Mark Chavira, Adnan Darwiche (IJCAI 2005) Inference in Probabilistic Relational Models Mark Chavira, Adnan Darwiche and Manfred Jaeger (IJAR 2006) *"P(P"PPPIPP=P2P+P-P@P"("I  =  2+-?" -[Q4ngSAT by DPLL Search Recent Trend: Exhaustive DPLLCount number of models: Model counters, e.g., relsat, cachet Generate all/subset of models: Image computation in model checking SMTs (non-propositional reasoning) Variations on DPLL SearchZ%Z ZHZZ%H,.M:piThe Language of Search81 Trace of DPLL 92Exhaustive DPLL@9Trace of DPLL:]WTrace of DPLL: a Formula;4Dealing with RedundancyC<Dealing with Redundancy<5Dealing with RedundancyibThis is an OBDD!D=This is an OBDD!^XA Non-traditional OBDD Compiler G@FBDDb\ FBDD vs OBDDFBDD more succinct than OBDD (dynamic var ordering in sat) Top-down vs bottom-up algorithms OBDD: equivalence test (canonical) FBDD: probabilistic equivalence test Both allow model counting >@a,&{TODealing with RedundancyRLevel One: Unique nodes (done) Level Two: Avoid redundant compilation (searches),3S$jcRedundant CompilationKCFormula Caching{Majercik and Litmman, 1998 Darwiche, 2002 Bacchus et al, 2003, 2004 Huang, 2004 Sang, Kautz, Beam, 2004, 2005 Thurley, 2006HBa[Beyond BDDs&  LD#Decomposition (Component Analysis)$$(PK  SNMEFBDD vs d-DNNF:d-DNNF more succinct than FBDD (effectiveness of decomposition) Deterministic equivalence test open Probabilistic equivalence test apply Other queries same& 8 ]d^The Language of Search Relation to AND/OR Search (CP)AND/OR graphs are deterministic and decomposable AND/OR search algorithms are doing enough work to compile networks into (multi-valued equivalent of) d-DNNF Capable of more than answering a single query (model counting, belief revision, etc)Ze_ ImplicationsSAT techniques harnessed for knowledge compilation c2d compiler based on Rsat Solver (SAT-race 06) Language properties (succinctness/tractability) help characterize power and limitations of search *JzUnderstanding DPLL Take any program X that runs exhaustive DPLL-style search Examine traces, if traces L, then X can answer all queries tractable for L X is hopeless on any input having no polynomial-size representation in LX;ZZ%ZtZ    &E4. Power of DPLL Traces of several model counters (Relsat, Cachet, e.g.) are in d-DNNF Are doing enough work to compile formulas into d-DNNF solve tasks beyond model counting (e.g., minimum cardinality, probabilistic equivalent testing) d}!>#UP'Limitation of DPLL: General determinismha8Beyond DPLL: Decomposability (D) without determinism (d)9 (, SummarySOverview of recent results in knowledge compilation Overview of recent trends in exhaustive DPLL search A connection between SAT and knowledge compilation (search trace): SAT techniques harnessed for compilation into various languages Language properties (succinctness, tractability) characterize power and limitations of search algorithms.PP/= ` 33PP` 13` 3333` Q_{` 333fpKNāvI` j@v۩ῑ΂H>?" dd@,?n<d@ `7 `2@`7``2 n?" dd@   @@``PR    @ ` `p>> ] U 0  D (  D D <hu" B   D Td" B   D <"U_ B   D Td">& B   D ND"P B   D <"p B   D C x,?d?"bUv B    D <  #" `   T Click to edit Master title style! !$  D 0@ "   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  D 6 "]}  ^*   D 6X  "   C A. Darwiche   B D s *޽h ? 333380___PPT10.(г%  Blends     @ H(  HT + H "+bb P@ H# "Dwoh H s *"PP H Bd" P@bb P 0  H# "Nyh H s *"P   H Bd"P 0 z  H <" a*h  H s *"  H  f?d?"+)  H BI ?#" ` p  T Click to edit Master title style! !  H 0 L " `    W#Click to edit Master subtitle style$ $ H 6dP "`p   b*  H 6T "   G A. Darwiche   B H s *޽h ? 333380___PPT10.(г% 0 zrp (  p p 0 m s$   P*   p 0v  /$  R*  d p c $ ?NH   p 0q  x!  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S p 6(w s   P*   p 6}  /  R*  H p 0jB ? 3380___PPT10.}&T P,(  , , 0Pf s$  f X*  , 0Vf  /$ f Z*  , 6Zf s  f X*  , 6G  / f Z* H , 0jB ? 3380___PPT10.9+B#style.visibility<*h%(D' =-s6Bwipe(left)*<3<*hD' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h+%(D' =-s6Bwipe(left)*<3<*h+D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h+O%(D' =-s6Bwipe(left)*<3<*h+OD' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*hO\%(D' =-s6Bwipe(left)*<3<*hO\D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h\k%(D' =-s6Bwipe(left)*<3<*h\k+8+0+h + K0 ,(  , , s *$ D `<$D  0  x , c $ D    * , Z@??"6?@`NNN?NLp  bx A & okX => B A & okX => B B & okY => C B & okY => Cv= P   H , 0޽h ? 33___PPT10u..wNwM+D=' = @B + K0 PPb(  P P s *@ D0`<$D  0  x P c $ D    H P 0޽h ? 33___PPT10u..wNwM+D=' = @B +4   0D (  F 0 `  @   <` L bx A & okX => B A & okX => B B & okY => C B & okY => Cv= P     No?#" `0 `z      @ ,$D  0  Z??"6?@`NNN?N   HCompiled Structure   Zd?G~6?"6?@`NNN?N  `  >Compiler   z @      ,$D  0dT  @   #  @ <   Z??"6?@`NNN?N  lEvaluator (Polytime)  2   N??"6?@`NNN?N@      Z??"6?@`NNN?N @  =Queries   Z?G27H?"6?@`NNN?N   < "`  )Knowledge CompilationH  0޽h ? ̙33( ___PPT10+iD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* +!  H @ P (  F 0 `  @   <$ L bx A & okX => B A & okX => B B & okY => C B & okY => Cv= P     No?#" `0 `F      @   Z??"6?@`NNN?N   D ?&   Z?G~6?"6?@`NNN?N  `  >Compiler   F @      dT  @   #  @ <   Z??"6?@`NNN?N  lEvaluator (Polytime)  2   N??"6?@`NNN?N@      ZT??"6?@`NNN?N @  =Queries   Z?G27H?"6?@`NNN?N   < "`  )Knowledge CompilationH  0޽h ? ̙33y___PPT10Y+D=' = @B +    p  (  F 0 `  @   < L bx A & okX => B A & okX => B B & okY => C B & okY => Cv= P     No?#" `0 `:  Zx ??"6?@`NNN?N @  r< ..... Prime Implicates OBDD &    Z?G~6?"6?@`NNN?NP  >Compiler   F @     dT  @   #  @ <   Z??"6?@`NNN?N  lEvaluator (Polytime)  2   N??"6?@`NNN?N@      ZT??"6?@`NNN?N @  =Queries    Z?G27H?"6?@`NNN?N   < "`  )Knowledge CompilationH  0޽h ? ̙33y___PPT10Y+D=' = @B +}  l$(  lr l S < D    r l S  D   H l 0޽h ? 3333___PPT10i.|+D=' = @B +` K0 og(  A  ZA ? yDiagnosis Is this a normal behavior? What are the possible faults? Planning Can this goal be achieved? Generate a set of plans Probabilistic reasoning What is the probability of X given Y Non-monotonic reasoning (penalty logics) Does X follow preferentially from Y Formal verification / CAD: Is it possible that the design will exhibit behavior X? Are two designs equivalent?   9 3%)$T 9 3%)  $  T\~  s *dJ D    H  0޽h ? ̙33y___PPT10Y+D=' = @B +  <(  ~  s *N D    ~  s *h4 D@`  H  0޽h ? ̙33y___PPT10Y+D=' = @B +9 h5`5DI 4(  z  p  p ,$D  04N  p    p 2  N??"6?@`NNN?N0p P   ZW??"6?@`NNN?N   B Succinctness    N ` p  ` p2  T??"6?@`NNN?N` @ N  p    p3   Zd\??"6?@`NNN?N p cPolytime Operations      Z`??"6?@`NNN?N P  Consistency (CO) Validity (VA) Clausal entailment (CE) Sentential entailment (SE) Implicant testing (IP) Equivalence testing (EQ) Model Counting (CT) Model enumeration (ME) R RL   Zf??"6?@`NNN?N  |HProjection (exist. quantification) Conditioning Conjoin, Disjoin, NegateI I   `k??"6?@`NNN?Np0   D ( = E ZPo??"6?@`NNN?NiE0 U  uADecomposability Determinism Smoothness Flatness Decision OrderingBB  F ZTt??"6?@`NNN?N{= u JNegation Normal Form    Z\x??"6?@`NNN?N~   BA"  Z|??"6?@`NNN?N~   3B   Z??"6?@`NNN?N~ i  D B"  Z~??"6?@`NNN?N~ b  3A  Z??"6?@`NNN?N~ `  3C   Z??"6?@`NNN?N~   D D"  Zȏ??"6?@`NNN?N~ x  3D   Z??"6?@`NNN?N uR  D C"   Zȗ??"6?@`NNN?N   Aand"  Z??"6?@`NNN?N   5and   Z$??"6?@`NNN?N c  Aand"  Z??"6?@`NNN?N M  5and  Z??"6?@`NNN?N I  5and   Zȩ??"6?@`NNN?N   Aand"  Z<??"6?@`NNN?N   5and   Z??"6?@`NNN?N ~N  Aand"   Z8??"6?@`NNN?NEk 4or ! Z??"6?@`NNN?Nk @or" " Z8??"6?@`NNN?Nck @or" # Z4??"6?@`NNN?N&k 4or $ Z??"6?@`NNN?N4i 5and % Z??"6?@`NNN?NqAi 5and & ZD??"6?@`NNN?NK  @or"|B ' TD?<$|B (@ TD?2B ) ZD1?N;B *@ ZD1?N|B + TD?Nk|B ,@ TD?N&B -@ ZD1?L@ B . ZD1?PC B / ZD1?PShC B 0 ZD1?PV%C |B 1@ TD?TD @ |B 2@ TD?T C |B 3 TD?L~C |B 4@ TD?L~C B 5 ZD1?  B 6 ZD1?|  |B 7 TD?|  |B 8 TD?|  B 9@ ZDg ?| = B : ZDg ?| = B ;@ ZDg ?| ? B < ZDg ?|  B = ZD1? B B >@ ZD1?  |B ? TD?| t |B @@ TD? % B A ZD1?| % B B@ ZDjJ?| t% B C ZD1?| j' B D@ ZD1?| j  I <p "`  0A Knowledge Compilation MAPH  0޽h ? ̙335-___PPT10 +D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*+K&  r%j%9:%(  B#F uR  y$  T ?D FA&  T ? 5B  T ? H B&  T, ?  5A  TH ?   5C   T ? H D&   T ?* 5D   TD ?_R H C&   T ?u XA  Eand&   T ?) A  7and  TT ? A  Eand&  T8 ? n A  7and  TX ?9 A  7and  Tt ? A  Eand&  T ? 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" Dor&~B  ND?r p~B B ND?M 7B  TDjJ?\qB B TDjJ??\q~B  ND?c~B  B ND?c}B !B TDjJ?b B " TDjJ?&p ( B # TDjJ?&( B $ TDjJ?&( ~B %B ND? 2x ~B &B ND?2x( ~B ' ND?K( ~B (B ND? ( B ) TDjJ?b b[B * TDjJ?y y .~B + ND?" " }~B , ND?T T}B -B TDg ?b [B . TDg ? [B /B TDg ? gB 0 TDg ? y B 1 TDjJ?k [B 2B TDjJ? y [~B 3 ND?" g~B 4B ND? TgB 5 TDjJ? T[B 6B TDjJ? gB 7 TDjJ? [B 8B TDjJ?" g 9 T ?B  sADecomposability Determinism Smoothness Flatness Decision OrderingBBr : S  D    H  0޽h ? ̙33y___PPT10Y+D=' = @B +'  &&;>T&(    ZT ?u+_ NA.  Z ?~_ 9B  ZT ?_ P B.  Z ? _ 9A  Z ? _ 9C  Z ?_ P D.   Z ?_ 9D   Z ?J8_ P C.   Z ? `>  Mand.   Z ?   7and   Zp ?   Eand&  Z ? vT  ;and  Z ? $   7and  Z ?   Mand.  Z ? t  ;and  Z ? Q/  Eand&  Z ?9  :or  Z ?I  Lor.  Z( ?   Dor&  ZH ?b  6or  Z ?cAB 7and  Z ?kIB 7and  Z\  ?  Dor&|B  TD?[ |B @ TD?~6 iB  ZDԔ?EB @ ZDԔ?(E|B  TD?L{|B @ TD?L B  @ ZDԔ? K B ! ZDԔ? Y B " ZDԔ?  B # ZDԔ? v |B $@ TD? a |B %@ TD? a |B & TD? 4 |B '@ TD?  B ( ZDԔ? KKNB ) ZDԔ? b b *|B * TD? i|B + TD? ==iB ,@ ZDg ? 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