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Counting: v(n) is number of solutions in T(n) Optimization: v(n) is the optimal solution in T(n) Belief updating: v(n), probability of evidence in T(n). Partition function: v(n) is the total probability in T(n). Goal: compute the value of the root node recursively using dfs search of the AND/OR tree. AND/OR search tree and algorithms are ([Freuder & Quinn85], [Collin, Dechter & Katz91], [Bayardo & Miranker95],[Darwiche 2001], [Bacchus et. Al, 2003]) Space: O(n) Time: O(exp(m)), where m is the depth of the pseudo-tree Time: O(exp(w* log n)) BFS is time and space O(exp(w* log n) (2PPPsPP2 1) ()(Z%o )^   A 5   From AND/OR Tree   To an AND/OR Graph   dAND/OR Search Graphs$cAny two nodes that root identical subtrees/subgraphs (are unifiable) can be merged Minimal AND/OR search graph: of R relative to tree T is the closure under merge of the AND/OR search tree of R relative to T, where inconsistent subtrees are pruned. Canonicity: The minimal AND/OR graph relative to T is unique for all constraints equivalent to R. _ZZMk [># v=Context based cachingCaching is possible when context is the same context = parent-separator set in induced pseudo-graph = current variable + ancestors connected to subtree below -y /D>Caching ?Caching !<All Four Search Spacesv#Properties of minimal AND/OR graphs$$Complexity: Minimal AND/OR R relative to pseudo-tree T is O(exp(w*)) where w* is the tree-width of R along T. Minimal OR search graph is O(exp(pw*)) where pw* is path-width w* d" pw*, pw* d" w*log n Canonicity: The minimal AND/OR search graph is unique (canonical) for all equivalent formulas (Boolean or Constraints), consistent with its pseudo tree. b PdPPZP PPPP \1  #X,;JGSearching AND/OR GraphswAO(j): searches depth-first, cache i-context j = the max size of a cache table (i.e. number of variables in a context)d.J(Gt1Context-Based Caching OutlineBackground in Graphical models AND/OR search trees and Graphs Minimal AND/OR graphs From AND/OR search graphs to AOMDDs Compilation of AOMDDs AOMDDs and earlier BDDs ZZZZ=$.Pq OR Search Graphs vs OBDDs,AND/OR Search Graphs; AOBDDs(0AND/OR Search Graphs; AOBDDs(AOBDD ConventionsAOBDD ConventionsHCombining AOBDD (apply)IExample:JExample (continued)KAOBDD vs. OBDD Complexity Complexity of apply: Complexity of apply is bounded quadratically by the product of input AOBDDs restricted to each branch in the output pseudo-tree. Complexity of VE-AOBDD: is exponential in the tree-width along the pseudo-tree.8Z:,4 @AOMDDs and tree-BDDs$ 2Tree-BDDs (McMillan1994) are : AND/OR BDDS are *! Related workRelated work in Search Backjumping + learning (Freuder 1988,Dechter 1990,Bayardo and Mirankar 1996) Recursive-conditioning (Darwiche 2001) Value elimination (Bacchus et. Al. 2003) Search over tree-decompositions (Trrioux et. Al. 2002) Related to compilation schemes: Minimal AND/OR  related to tree-OBDDs (McMillan 94), d-DNNF (Darwiche et. Al. 2002) Case-factor diagrams (Mcallester, Collins, Pereira, 2005) Tutorial references: Dechter, Constraint processing, Morgan Kauffman, 2003 Dechter, Tractable Structures of Constraint Satisfaction problems, In Handbook of constraint solving, forthcoming. PP!PPPzP1PPP!    QR% 10p+ ConclusionAND/OR search should always be used. AND/OR BDDs are superior to OBDDs. A search algorithm with good and no-good learning generates an OBDD or an AND/OR Bdds. Dynamic variable ordering can be incorporated With caching, AND/OR search is similar to inference (variable-elimination) The real tradeoff should be rephrased: Time vs space rather than search vs inference, or search vs model-checking. 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O,$@ 0B H @ TD1?".,$@ 0 I Z 1?" \{ ,$ 0 9A  J Z1?"L 3 ,$ 0 9B  K N1?" j4 ,$ 0 9E  L Z1?" ,$ 0 9G  M N1?"  ,$ 0 9D  N Zh1?"  ,$ 0 9F  O Z!1?",$ 0 9C  P N%1?" ,$ 0 HConstraint graph m Q Z(1?"& ,$ 0 =Semantics: set of all solutions Primary task: find a solutionH>   H  0޽h ? ̙33+3#3___PPT103+rD0' = @B D0' = @BA?%,( < +O%,( < +D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bwipe(right)*<3<* D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D'' =%(Dd'' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8 %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*9 %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*: %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*; %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*< %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*= %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*> %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*? %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@ %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*A %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*B %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*C %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*E %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*F %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*G %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*I %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*J %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*K %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*M %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*N %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*O %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Q %(++0+I  ++0+J  ++0+K  ++0+L  ++0+M  ++0+N  ++0+O  ++0+P  ++0+Q  +   $f (     6g $hj = {(C), (A v B v C), (A v B v E), (B v C v D)}.50 cd  <A ?`f~  s *h ak   H  0޽h ? 33___PPT10i.˱+D=' = @B + r-K0 ^h(    s *8   <$ 0  ~  s *  ak    z     [ {g ,$D 0 2 TA wl? ?   wl   S 7jBCYDEF8c1@1bw%Y@  $2  6"`H| =A 0 !2  6"`|   =D 0 !2   6L"`   =B 0 !2   6"` 1  =C 0 !Z  B s *P Z   s *P Z   s * -| 2  6"` |   =E 0 !Z B s *- | 2  6T"`m  =F 0 !Z  s *T B c $|-| T B c $  T  c $| | T  c $ |   0  `,$ 0 All these tasks are NP-hard identify special cases approximate PDP&d  T\ ??/ T $,$ 0 dExamples: Constraint networks Belief networks Cost networks Markov random fields Influence diagrams,  Z eH  0޽h ?             3333@8___PPT10..e@Ѩ+D8' = @B D ' = @BA?%,( < +O%,( < +D' =%(D@' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4L%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L~%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(++0+ ++0+ ++0+ +" -K0 ! 0j--!(  !~ ! s * ak    ! 0<Ѥ?7 PBelief updating (sum-prod) 2 ! 0TҤ N DMPE (max-prod) 2 ! 0֤~ |FCSP  consistency (projection-join)$ 2$ ! 0' F E#CSP (sum-prod) 2F  0  ! u 2 ! 6٤"` DP(X)$0 # 2  ! 6ݤ"`@(`  FP(Y|X)$0 # 2  ! 6D"` @ `  XP(Z|X)60 # + T  !B c $@T  ! c $ @2  ! 6"` 0  FP(T|Y)$0 # 2 ! 6 "`@ `0  FP(R|Y)$0 # T !B c $`6 2 T ! c $6  2 ! 6"`N n 0  XP(L|Z)60 # + 2 ! 6, "`  0  FP(M|Z)$0 # T !B c $6  T ! c $| 6 N   ! s *A l??"`V  ux ld ! s *A l??"`T % x ldf ! s *A l??"`C  x ldf ! s *A l??"`x  x ldf ! s *A! l??"`C k x ld ! s *A" l??"`v k x ld$ ! 6A# l? ?"6?@`NNN?Nix ld ! s *A% l??"`0bx ldf ! s *A& l??"` Sk x ld ! s *A' l??"` N x ldf ! s *A( l??"` /k x ld  ! s *A) l??"` 1 x ldfB !! ZDjJ?"'B "!B ZDjJ?" B #!B ZDjJ?"  B $!B ZDjJ?"@  B %! ZDjJ?"  B &! ZDjJ?" l B '! ZDjJ?"l  B (! ZDjJ?" l d  B )! ZDjJ?" l  B *! ZDjJ?"Lh  B +! ZDjJ?" LB ,! ZDjJ?"L -!  ` ?? i-Trees are processed in linear time and memory . .H ! 0޽h ?o! ! !! ! ! ! !! !!! !!! !!! ̙33___PPT10i.?+D=' = @B + -K0 Pj!6(  !~ ! s *`& ak   x ! c $4'    H ! 0޽h ? 3333___PPT10i.%`)+D=' = @B +r  F?>?pj88!;(  ! !  0e0e    BC,DEF&  8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||Q,?8-Q@`"<  Q ,$D 0 !  0e0e    BCJDEF&  8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||`Jl$#`J@`"<   ,$D  0 !  0e0e    BCADEF&f  8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||,Abl@`"< x ,$D  0 !  0e0e    B&C8DEF&  8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||0c8& 0@`"< T,$D 0 !  0e0e    B1CDE$F.  8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| ]1b(!\3*@`"< ` <U ,$D 0r !  0e0e    BCDEF  8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||ll @`"< E%,$D 0~ ! s *7 ak   2  ! 6H9S"`? F  AE$0 2  ! 6<S"`?Y p   AK$0 2  ! 6@S"`?4 AF$0 2  ! 6\CS"`? 7 <  AL$0 f  ! S C"? sp f !@ S C"?q f ! S C"?r f ! S C"? 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AK$0 n " S C"?&72 " 6S"`?0B AG$0 n " S C"?)Nn " S C"?4n " S C"?9n " S C"?!$Tn " S C"?Tn " S C"?j2 " 68 S"`?j8  AN$0 n " S C"?Nn "B S C"?k%n "B S C"?NH " 0޽h ??`"" """ """ """ """ """""""""" """ """ """ """ """""""" """!""""""$"#"""%"""&"""'"0"*"-","<".","*"/")"0"1"3"2"4"2")"5"6"2"7"9"8":"<"0"="+">"?" +";"@"!3")"A""8"C"B"#<"+"D"$9"*"E"%+")"F"&N"["O"'N"L"P"(R"Q"S")Q"K"T"*U"Q"V"+X"W"Y",M"\"]"-M"Z"^".R"K"_"/W"a"`"0["M"b"1X"L"c"2M"K"d"3k"w"l"4k"i"m"5o"n"p"6q"n"r"7t"s"u"8j"x"y"9j"v"z":s"|"{";w"j"}"<t"i"~"=""">"""?"""@"""A"""B"""C"""D"""E"""F"""G"""H"""I"""J"""K"""L"""M"""N"""O"""P"""Q"""R"""S"""T"""U"""V"""W"""X"""Y"""Z"""["""\"""]"""^"""_"""`"""a"""b"""c"""d"""e"""f"""g"""h"""i"""j"""k"""l"""m"""n"""o"""p"""q"""r"""s"""t"""u"""v"""w"""x"""y"""z"""{"""|"""}"""~"""""""""""""""""" 3333!z!___PPT10Z!.U+D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =4@BBBB%())))?D' =1:Bhidden*o3>+B#style.visibility<*"%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =%(D' =%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*G"%(D' =-s6Bwipe(left)*<3<*G"D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*("%(D' =%(Df' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*("%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*e"%(D' =-o6Bwipe(up)*<3<*e"D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*J"%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*J"%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-u6Bwipe(right)*<3<*"D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h"%(DW' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*"%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(+8+0+" +X  iFaFjpp "=(   "~  " s *  ak   z / H   " y/ ,$D 0  "  `!??"`H  DA=yellow 0  2  " Z?? / @  "B T??9 z ( 9PH   " ( P ,$D 0  "  `D'33??"`vPH  CA=green 0   " T??( 9cr  "  `* ??4 u,$ 0 V Inference may require too much memory Condition (guessing) on some of the variables<W1)sz J pp  "  p,$D 02  " <1"` B   AC$0 # 2  " <5"`*   AB$0 # 2  " <4"`0 .  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AC$0 # 2 ~" < x"` Z AK$0 # 2 " <,͡"`&( AG$0 # Z " s *{  2 " <$ա"`0   AL$0 # 2 " <١"`\ >  AD$0 # Z " s *>  Z " s *q(2 " <D"``6 AF$0 # Z " s *q1(2 " <"`1; AH$0 # Z " s *6 { 2 " <"`F 6  AM$0 # 2 " <( "`0 AJ$0 # Z "B s *,1F2 " <"` 6  AE$0 # Z " s *  > Z " s *6  Z " s *  Z " s *q(,FZ "B s *1 "B T??Q " T?? F  z 6  "  P,$D  02 " <"`6 .  AC$0 # 2 " <"` Z AK$0 # 2 " <4"`&( AG$0 # Z " s *{  2 " <"`0   AL$0 # 2 " <!"`\ >  AD$0 # Z " s *>  Z " s *q(2 " <"``6 AF$0 # Z " s *q1(2 " <"`1; AH$0 # Z " s *6 { 2 " <"`F 6  AM$0 # 2 " <$)"`0 AJ$0 # Z "B s *,1F2 " <4"` 6  AE$0 # Z " s *  > Z " s *6  Z " s *  Z " s *q(,FZ "B s *1 "B T??Q " T?? F 3 "  `$* ??4 u K Inference may require too much memory Condition on some of the variables<L1) F   "  T < "#  2 " <D."`b. 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