The State Explosion Problem. Symbolic Encoding using Decision Diagrams. CiteSeer Database. Overview. Boolean Functions.

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1 The Stte Eplosion Prolem Smoli Enoding using Deision Digrms 6.42J/6.834J ognitive Rootis Mrtin Shenher (using mteril from Rndl rnt, ln Mishhenko, nd Geert Jnssen) Mn prolems suffer from stte spe eplosion: the numer of sttes is eponentil in the numer of vriles in the sstem. Deision digrms (DDs): grph-sed, nonil representtion of funtions tht voids spe eplosion in mn prtil ses. iteseer Dtse Overview inr Deision Digrms (DDs) Etensions: DDs, MDDs, ZDDs, SDDs Deision Digrm Pkges omining Smoli Enoding nd Serh oolen Funtions inr Vriles,, n Funtion f:,, n {,} Sum of Produts (DNF) f = ( ) ( ) ( ) Produt of Sums (NF) f = ( ) Truth Tle f(,,) Shnnon Epnsion onsider oolen funtion f (, 2,, n ). Then: f = ( f = ) ( f = ) oftor = oftor =

2 inr Deision Trees Reursive Shnnon epnsion Vrile Ordering Impose ritrr totl ordering on vriles Vriles must oe ordering long ll pths Propert: No onfliting ssignments long pth Ordered inr Deision Tree Order < < voiding low-up Deision tree no more spe-effiient thn truth tle Still epliitl enumertes ll the 2 n possile vlutions Ide: Trnsform to direted li grph (DG) Promotes shring of su-epressions Ordered inr Deision Tree Rule : ollpse Lef Nodes No longer tree. 2

3 Rule 2: Remove Redundnt Tests Rule 2: Remove Redundnt Tests Rule 3: Isomorphi Sugrphs Rule 3: Isomorphi Sugrphs Rule m eome pplile gin Finl Representtion Redued, Ordered inr Deision Digrm (RODD) 3

4 ODT to RODD Summr n RODD n e otined from n ODT repetedl ppling the following redution rules (until none of the them n e pplied nmore): Remove duplite terminl (lef) nodes Remove duplite non-terminl (internl) nodes Remove nodes with redundnt tests noniit of RODDs Is there unique DD for eh oolen funtion? Theorem: DD nonil, iff redued nd ordered. Redued: None of the redutions pplile Ordered: There is totl vrile ordering Enles heking equivlene of oolen funtions heking equivlene of orresponding RODDs. Equivlene heking Emple Do two iruits ompute n identil funtion? si tsk in forml hrdwre verifition ompre new design to known good design Solution omintoril Serh Prove ll ssignments fil to hek equivlene Tpill, must eplore signifint frtion of inputs Eponentil time ompleit iruit iruit 2 O O2 O O2 Diff Solution using RODDs Emple RODDs Funtions equl iff RODDs identil Never enumerte epliit funtion vlues Eploit struture nd regulrit of funtions O O2 4 it dder 3 nodes 64 it dder 57 nodes out S Liner growth S 3 S 2 S S out

5 Influene of Vrile Ordering Funtion f = ( ) (2 2) (3 3) Vrile Ordering RODD sie depends strongl on vrile ordering Finding vrile ordering tht produes miniml RODD sie is intrtle ut, heuristis eist for finding good vrile orderings Funtions eist whose RODD hs eponentil sie for n vrile ordering ut, suh funtions re rrel enountered 3 3 Good ordering: Liner growth d ordering: Eponentil growth Mnipulting RODDs Proedure ppl(f,g) for implementing ll the 2 4 = 6 two-rgument logil opertions on oolen funtions ppl Opertor Three possile ses Root f < Root g ppl(f,g) = ( ppl(f =, g = )) ( ppl(f =, g = )) ppl (f =, g) ppl (f =, g) Reursive desent into sutrees = Reursive desent into sutrees = ppl ppl (f =, g = ) (f =, g = ) Root f = Root g Root f > Root g ppl (f, g = ) ppl (f, g = ) ppl Opertor Pseudoode ppl Opertor Emple Funtion ppl( F, G ) if ( lredomputed( F, G ) ) return the result elsif ( F {,} nd G {,} ) return oper( F, G ) elsif ( Vr( F ) = Vr( G ) ) u retenode( Vr(F), ppl(f,g ), ppl(f,g)); elsif ( Vr( F ) < Vr( G ) ) u retenode( Vr(F), ppl(f,g ), ppl(f,g )); else /* ( Vr( F ) > Vr( G ) ) */ u retenode( Vr(G), ppl(f,g ), ppl(f,g )); Insertomputed( F,G,u ); return u; ompleit: O( F G ) d 2 d , 2, 2 6, 2 6, 5 3, 2 5, 2 3, 4 4, 3 5, 4 5

6 ppl Opertor Emple Generting RODDs Inrementll, 2, 2 6, 2 6, 5 T T2 O new_vr("") new_vr(" ) new_vr("") T nd(, ) T2 nd(, ) O Or(T, T2) Network of omponents s RODD 3, 2 5, 2 3, 4 d d Deision Digrms T T2 O 4, 3 5, 4 RODDs Summr Eploit struture nd regulrities of funtions Eponentil numer of disrete sttes n e ptured in polnomil-sie RODD Stisfiilit, Tutolog, omplement onstnt ppl (nd, or, et.) polnomil in RODD sie Vrile ordering importnt, ut diffiult to find Overview inr Deision Digrms (DDs) Etensions: DDs, MDDs, ZDDs, SDDs Deision Digrm Pkges omining Smoli Enoding nd Serh lgeri DDs (DDs)) [hr[ 93] Etension to funtions with non-inr vlues noniit of representtion (s for DDs) pplitions in omintoril optimition Multi-Vlued DDs (MDDs)) [Km[ 9] Etension to funtions with non-inr vriles noniit of representtion (s for RODDs) However, inr enoding of domins often etter f(,,) 3 2 DD 2 3 MDD 2 6

7 Zero-suppressed DDs (ZDDs)) [Minto 93] dpted to sprse funtions (mn s in on-set) Modified rule 2: Remove node if -edge points to Shred DDs (SDDs)) [Minto 9] Glol tle storing unique nodes Equivlene hek for funtions eomes onstnt ZDD: Defult f g h g f h DD Templtes [Goel[ Hsteer rnt 3] Shre not onl funtions, ut lso funtion tpes: f: g: d f: g: d Overview inr Deision Digrms (DDs) Etensions: DDs, MDDs, ZDDs, SDDs Deision Digrm Pkges omining Smoli Enoding nd Serh d f g {\,\d} Deision Digrm Pkges Pkge hrteristis ode Nme uthor(s) ffilition D.3 rmin iere ETH Zurih UD udd.9 Jørn Lind-Nielsen ITU, Denmrk L L Rjeev Rnjn U erkele MU MU Dvid Long MU/TT UD UDD 2.3. Fio Someni oulder, O EST EST ed Roert Meoli Mrior, Sloveni IM IM Geert Jnssen IM Wtson MON MON nders Møller TT/RIS PDT usinto/orno Politenio di Torino ST Sttidd. Stefn Edelkmp Freiurg, Germn TGR TiGeR 3.? oudert/mdre/touti ull/de/xori TUD TUDD.8.3 Stefn Höreth Drmstdt, Germn ode Nme of the pkge Lng Progrmming lnguge T Tpe (Pointers/Indies) R Trversl (Depth/redth) M Supports Mngers Nm Mimum # of nodes Si Sie of node in tes 64 Sie inreses on 64-it Soure: [Jnssen 22] Vm P G T DV Et Yer Pro Mimum # of vriles Unique tle per vrile Ref. ount or Mrk-Sweep omputed tle shred Dnmi Vrile ordering # of funtions in PI First er when ville Qulit of ode 7

8 ompring Pkges ode Lng T R M Nm Si 64 Vm P G T DV Et Yer Pro I D Y 2^25 8-2^ N MS S N UD /++ I D N 2^32 2 N 2^2 N MS M Y L P Y 2^28 6 Y 2^6 Y R 8 S Y MU P D Y 2^29 6 Y 2^6 Y R 8 S Y UD /++ P D Y 2^28 6 Y 2^6 N R6 S Y EST P D N 2^32 36 Y 2^32 N R32 M N 72 2 IM I D Y 2^3 6 N 2^24 N MS S Y MON I D Y 2^24 6 N 2^6 N MS M N D PDT I D N 2^3 24 N 2^6 N MS M N ST ++ I D N 2^23 8 N 2^ 9 N MS S N TGR P D Y 2^3 6 Y 2^6 Y R6 M Y 9 994? TUD I D Y 2^3 2 N 2^6 Y R6 S Y Overview inr Deision Digrms (DDs) Etensions: DDs, MDDs, ZDDs, SDDs Deision Digrm Pkges omining Smoli Enoding nd Serh Soure: [Jnssen 22] rnh-nd nd-ound Serh Smoli rnh-nd nd-ound Eh serh node is soft onstrint suprolem Lower ound (l): Optimisti estimte of est solution in sutree Upper ound (u): est solution found so fr Prune, if l u. Enode Smolill Eh serh node is set of soft onstrint suprolems Lower ound Funtion (f l ): Optimisti estimtes of est solutions in sutree Upper ound (u): est solution found so fr Prune, if f l u. Domin Splitting Generlie to serh over sets: Prtition domins into sets hoose suset for unssigned vrile omine ll ompletel ssigned onstrints Emple: 4-Queens4 Vriles: Rows Domins: olumns onstrints: Q Q Q Q 8

9 Emple Serh Tree onstrints involved Emple Domin splitting with prtitions Solution Solution Emple Domin splitting with prtitions Sinking Opertion is new onstrint where ll vlues of tuples hve een repled Generlies the test to funtions onstrint onstrint sink( e2,.5) Solution Smoli rnh-nd nd-ound Funtion ( : ssignments, : vlue): vlue if Distne lower ound: l = identit. then if then return let e n unssigned vrile for eh do return return Enode funtions f t, f l s deision digrms. 9