Transactions on Information and Communications Technologies vol 16, 1996 WIT Press, ISSN

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1 Trasactos o Iformato ad Commucatos Techologes vol 6, 996 WIT Press, wwwwtpresscom, ISSN Itellget decso mag based o the caocal optmzato model wth a dsuctve costrat: theory ad applcatos V Dosoy Departmet of Mathematcs, Smferopol State Uversty 4, Yaltsaya Street, Smferopol, Urae Emal: dosoy@pdssusltcrmeaua Abstract Itellget decso mag must be accompaed wth mportat abltes such as geeralzato of emprcal data, deductve ferece usg Kowledge Base ad a optmal choce from a set of alteratves The caocal optmzato model wth a dsuctve costrat cosdered ths paper allows the coecto of the optmzato based approach wth owledge based ad case based approaches for IDSS developmet Some applcatos Epert Systems ad ecologcal IDSS are preseted Itroducto The combato of Artfcal Itellgece ad Operatos Research methods has bee sgfcat recet years Decso Support ad Epert Systems developmet stpulate the mportace of Icomplete Iformato based optmzato methods ad models vestgatos Both Case-based ad Kowledge-based approaches use Icomplete Ital Iformato ad they are eeded wth specal optmzato procedures to realze decso mag Let us cosder a problem etr () where B = { } f, Ω B, f PS () 2 (), s the set of vertces of the ut -dmesoed cube, df ( K ) 2( ) { } =,,, PS = f: B R s the class of pseudo-boolea fuctos, Ω s a set of admssble solutos Model () descrbes a choce process from the admssble alteratves set Ω wth a goal fucto f whch estmates ay alteratve We cosder decso mag methods whe compoets f ad/or Ω () are wealy defed, for eample, we have two subsets W ad W such that W Ω, W I Ω= These sets W, W are complete formato, whch partally specfes a set of admssble solutos Ω I aother case we have a set of productos cotaed the Kowledge Base to

2 Trasactos o Iformato ad Commucatos Techologes vol 6, 996 WIT Press, wwwwtpresscom, ISSN deduce the goal predcate " s admssble soluto" Ths producto set, geerally speag, may be complete {, f,, l } If the goal fucto f s partally defed by a set ( ) = or f s represeted by the partal bary relato the we have complete formato for the goal fucto f Thus, we have some complete formato I (f, Ω) to choose soluto accordg to model () The correct model compoets f ad Ω est but they are uow ad represeted wth I (f, Ω) Ths paper descrbes some elemets of the sythetc approach (SA) for soluto of the wealy defed pseudo-boolea scalar optmzato problem ad ts applcatos AI SA s based o the recostructo of the problem () the caocal form wth a dsuctve costrat: m etr () K = (2) where () K = σ & & m (DNF) K () σ= = f, () = σ r r L s elemetary coucto Dsuctve ormal form s eual to ff Ω As usually, σ = f σ= ad Followg results are very mportat for further cosderato σ = f Theorem [] Ay pseudo-boolea fucto optmzato problem wth costrats whch defe a o-empty set of admssble solutos Ω of the problem () ca be represeted a euvalet form (2) wth the dsuctve costrat Theorem 2 [] If the step's umber eeded to reduce the problem of codtoal optmzato of a lear pseudo-boolea fucto f form () to a euvalet form (2) wth dsuctve costrat s bouded by a polyomal of the dmeso of the problem, t s solvable polyomal tme 2 Case-based ad owledge-based approaches for the costrat recostructo Let tal formato about admssble set Ω be gve a form { } W, W : W Ω, W I Ω=, card W U W = L << Ay 2 admssble vector of varables other words, f, K, W the Ω ad f W W s a ad ay β W s ot a admssble vector I B \ Ω Ths tal the { } formato s true, but complete descrpto of the set Ω s uow

3 Trasactos o Iformato ad Commucatos Techologes vol 6, 996 WIT Press, wwwwtpresscom, ISSN We use the Patter Recogto learg techue based o emprcal geeralzato of the partal data represeted wth a Boolea learg table T L (Fg) 2 D () 2 m m m 2 m β β β 2 β β m β β m m 2 β m = " Ω" m m Fg : A Boolea learg table T l We use a bary decso tree based learg algorthm to realze sythess of the decso rule { } D : B, such as If D () D () s the true decso rule, { ΩI T }, L =, { B \ Ω} I TL}, TL D (), Ω =, Ω, the, geerally speag, t s possble, such vectors est that D( ) D ( ) The followg theorem [2] s a specal case of Chebyshev eualty We deote µ a umber of leaves of the bary decso tree (BDT) Ths BDT s bult wth a table T L ad correspods to =, Let P(E) be probablty of BDT decso error, or P(E) s the rule D () D ( T L ) probablty of the evet () () D D, B ( ) BDT s called admssble ff T D () = D() L Theorem 3 [2] For ay admssble BDT wth µ leaves whch s bult wth a true learg table T L PPE ( ) for ay ε ( 75µ / L) ( ε) < ep ( ε 2 L/ µ ) Corollary Amog all admssble BDT the most statstcally relable are BDT havg least leaves umber µ Costructg optmal ( µ m ) BDT s NP-complete problem [6] Ths problem s well ow There s o eed to eumerate all appromate methods of optmal BDT sythess

4 Trasactos o Iformato ad Commucatos Techologes vol 6, 996 WIT Press, wwwwtpresscom, ISSN It s especally sgfcat followg two facts ) A euvalet BDT (geerally speag, ouue) ests for ay Boolea fucto f Ths fact follows from the Shao epaso ) BDT defes DNF for the correspodg fucto f whch ca be foud O(µL) steps Eample Let gve the problem ma( ) wth the partal formato gve o Fg " Ω" Fg2 : The partal formato about set Ω Fg 3: The admssble BDT Oe admssble BDT s show o Fg 3 Ths BDT defes the DNF D = () whch couctos correspod to the braches termatg wth leaves mared by BDT defes the orthogoal DNF, but t s evdet that = The recostructed problem ( ) ma , 2 23 =,,,,, f = = 3 has soluto ( 2 3) = ( ) ad ( ) fucto f s accurately defed 2 3 I ths eample Now we cosder the owledge-based approach to the dsuctve costrats fdg We suppose that the Logcal Producto System [4] (LPS) s gve to deduce the goal fact = Ω I ths case we fd DNF D () such as D () = whe the goal g () " " g () has bee deduced Ths DNF D () s called the Logcal Descrpto of the Deducve Rego Deducblty (LDD) [4] The procedure of fdg LDD D-algorthm ad DSalgorthm s suggested [4] Eample 2 Let the LPS {, β, γ β γ } S = g g s gve The correspodg AND/OR graph s show o Fg 3 By usg ths graph we ca fd LDD D = () 2 2 3

5 Trasactos o Iformato ad Commucatos Techologes vol 6, 996 WIT Press, wwwwtpresscom, ISSN Remar Both D- ad DS- algorthms use dept-frst search wth bactracg The complety of the LDD fdg caot be estmated as polyomal Eamples ad 2 show the DUALITY of these two owledge represetato forms wth emprcal data ad declaratve owledge (productos) A dualty-based decso mag method s suggested [5] 3 The problem wth the partally defed lear goal fucto ad the eactly defed dsuctve costrat g β γ The caocal form (2) wth a dsuctve costrat plays a mportat role for the codtoal pseudo-boolea optmzato problem As show above, cyberetcs models of owledge represetato ca be used to buld the model (2) We suppose ow dsuctve costrat s eactly obtaed but tal formato about the goal fucto f s gve partally Let f be lear () = = (, ) = f c C ad we have the partal formato the form of pars, f,, f are gve {( ( ) ) = } where the values ( ) eactly Wth ths tal formato we have the euato system ( ) c + c + L + c = f, =,, 2 2 = K s uow where vector C ( c,, c ) 2 3 Fg4: AND/OR graph correspodg to the LPS The partal tal formato allows to fd the system of geeratrces C = ( C, K, C ) of the cove polyhedral coe KC ( ) cotag the uow true vector C of the goal fucto f coeffcets Methods for the geeratrces fdg was developed by Cherov [7] As stated ths paper the approach ca be used wth tal formato the form f f for of partally defed bary relato ( ) ( p), {,, p AA= K s} A Ω { Ω \ A} Let us suppose that for the problem ( C ) ma,, m σ = the partal formato about f() = ( C ) C = ( C,, C ) C K( C) σ r r & L & (3) s gve as the set of geeratrces s true wth addtoal formato about

6 Trasactos o Iformato ad Commucatos Techologes vol 6, 996 WIT Press, wwwwtpresscom, ISSN sgs of the coeffcets of the vector C More strctly, t s ow as ether c c = or c > for all c,, = Sce C K( C) < or, we ca ote C = µ C+ L + µ C, µ p, p =, Let us deote µ = µ + L + µ C N = C µ It s evdet that CN cov( C) ( ) ( ) ( ) ( ) { } G cov C G C cov C, = : sg c = sg c, =,, sg() z It s evdet that CN G cov( C) Theorem 4 If C K( C) ad Let G be a subset, z > =, z =, z <, C ((, ) ma (, ) the Ω ( C, ) ma ( C, ) C G C = C The proof of the theorem 4 s very smple ad omtted where, CN G cov( C) = Ω We wll use geeratrces fd codtos of the possblty of the etract soluto possblty Sce dsuctve costrat s gve the form m ad C accordg to theorem 4 to resolve the formulated problem ad to K () =, K () σ r & L & = we ca cosder m regos Ω σ r = r correspodg to couctos () such as, K, Ω K m Ω K = Ω UL U Ω = m Ω ( ) ( () ) Solutos of the followg m problems (4), as wll be show below, allow the formulated problem wth complete tal formato to be resolved ( λc+ L + λ C ) ma, () K = λc+ L + λ C = C G λ+ L + λ = λ, =, We'll fd Boolea vectors Ω, =, m, such as C G Ω C, C, e ( ) ( )) C G ( C, ) = ma ( C, ) Ω Coordates of the vectors = (,, ) (4) K are defed by formulas

7 Trasactos o Iformato ad Commucatos Techologes vol 6, 996 WIT Press, wwwwtpresscom, ISSN Where s arbtrary value from { }, {, K, r } { K r} { K r} { K r} σ, (,, ) ( c > ) =, (,, ) ( c < ) ( ) ( c = ),,,,,, = Problems (4) ca be rewrtte the form ( c + L + c ) ma λ λ = λ+ L + λ = λ, =, C = c, K, c, K, c, C C ) ( We deote ( ) Let us assocate every problem (5) wth a problem ( c + L + c ) m λ λ = λ+ L + λ = λ, =, Etreme soluto of the problems (5), (6) are attaable o the smple' vertces, where (-)- mesoal smple s s We fd where It s evdet that = : =, S λ λ λ λ = B = ma b, A = m b, b = = C G A = ma C, C, B c ( ) ( ) Ω The codto of the eact soluto of the formulated problem follows from these eualtes ad theorem 4: If ( A B ) for ay, m, ad we have eact soluto (5) (6) (, ma, ) the C G ( C ) = ( C ) Ω (7) = spte of the fact that vector C s uow If codto (7) s ot fulflled the we caot fd the eact soluto but t s possble to apply a game-theoretc approach to resolve the problem wth ucertaty Really, the ucerta stuato eteror medum (or Nature), whch ca be cosdered as a adversary, chooses oe, from the set strateges { C, K, C} = C to mmze the goal fucto f() = ( C )

8 Trasactos o Iformato ad Commucatos Techologes vol 6, 996 WIT Press, wwwwtpresscom, ISSN The choce of C vertce e ) C s eual to the choce of the ut vector e S λ ( ths smple's The possble strateges, =, m, whch are chose to mamze f() ad adversary's strateges C, K, C, K, C are used to form the two-perso zero-sum ocooperatve game wth a pay matr m Ths game Γ= { } { C} = m H = h = c,, H may be solved wth usual methods 4 Applcato usg the caocal model: the ecologcal-ecoomc decso support system TAVRIDA The ma problem of the regoal ecologcal-ecoomc cotrol s a choce of the actos pacage from the set of admssble actos to trasfer the rego s state from the crtcal stuato IDSS TAVRIDA s maly teded for the soluto to ths problem Let { $, $ } K be a set of the actos whch were selected by eperts We deote the varables, =, : =,, f the acto $ wll be accomplshed otherwse Let us deote C,K C the epedtures reured for the actos $, K $ realzato ad Ω the set of bary vectors (,, ) = K such that ay Ω defes the actos pacage whch guaratees that the rego s state wll be trasfered from the crtcal stuato (the rs class) f the actos $ (such that = ) wll be accomplshed Ecologcal processes are very complcated, therefore the set may be eactly defed oly theoretcally We assume that the set Ω s partally defed by two oempty subsets W ad W such that W W { B Ω Ω}, \ These subsets gve by eperts are represeted the learg table T l, where l = W U W Thus we have the model wth complete formato: m c {, } =, = W Ω W { B \ Ω} Ths model may be trasformed to the caocal form wth a dsuctve costrat by usg above preseted case-based approach for the costrat recostructo

9 Trasactos o Iformato ad Commucatos Techologes vol 6, 996 WIT Press, wwwwtpresscom, ISSN Whe the Kowedge-based approach s used, the predcate g Ω, Ω =, Ω () s the goal oe I ths case Kowledge Base (KB) s used for DNF D () (LDDR) of the fucto g Ω sythess Productos cotaed the KB descrbe the complcated ecologcal processes IDSS TAVRIDA cotas the Data Base of ecologcal factors, the Data Base of ecologcal actos, the Data Base of ecologcal factors geeratg sources, the Data Base of eperts ad epert estmatos ad the Kowledge Base The causes of the crtcal ecologcal stuatos arse ad results of ecologcal actos eecuto are descrbed by the set of productos I ths paper emphass s placed o the optmzato models IDSS, therefore descrpto of other possbltes of the TAVRIDA are omtted 5 The dual epert system DUELS [5] The Dual s called Epert System usg otly owledge-based ad case-based decso mag by meas of sythess of the DNF s for both methods as show above the secto 2 Let the DNF D () have bee obtaed by meas of emprcal data I geeralzato (emprcal ducto) The learg table for the D () I sythess s etracted from the emprcal Case Data Base cluded DUELS The DNF s called the Logcal Descrpto of the Iductve Rego Deducblty D I () (LDI) Above, the DNF D () s called Logcal Descrpto of the Deductve Rego I Deducblty (LDD) LDD DD() s obtaed by meas of the deductve ferece usg the set productos cotaed the Kowledge Base whch s cluded DUELS We suppose that both D () I ad DD() DNFs defe the rules ri ad r D for the same goal predcate g () such that r I : g () s true ff D () I =, r D : g () s true ff D () D = Coucto D () & D () D () I D = ID s called the Logcal Descrpto of the Combed Rego Deducblty (LDID) DUELS provdes a computato of the LDID D () whch defes the rule rid : ID r ID : g( ) s true ff D () ID = g Ω

10 Trasactos o Iformato ad Commucatos Techologes vol 6, 996 WIT Press, wwwwtpresscom, ISSN Let Pr ( I ) ad ( ) Pr ( I ) ad Pr ( D ) error probablty of the rule r ID s eual to Pr ( ID ) Pr ( I ) Pr ( D ) tha both Pr ( I ) ad Pr ( D ) Pr D be the error probabltes of the rules r I ad r D If ad these rules are statstcally depedet, the the = ad less DUELS may be used for costructo of the dsuctve costrats The Dual approach has great possbltes for applcatos dagoss, data ad owledge verfcato (by meas of comparso DNFs DI() ad D ( D ), owledge acusto from the emprcal data for egeerg ad ecoomc systems The ma sgfcace of the DUELS s the possblty of the optmzato model s sythess usg complete formato Refereces Dosoy VI Pseudo-Boolea optmzato wth a dsuctve costrat, JComp Maths Math Phys, Pergamo, 994, Vol34, 2, Dosoy VI Decso Trees Learg algorthms J Comp Maths Math Phys, Naua, Moscow, 982, Vol22, 4, Dosoy VI Wealy defed problems of Boolea lear programmg wth a partally specfed set of admssble solutos J Comp Maths Math Phys, Naua, Moscow, 988, Vol28, 9, Dosoy VI The Logcal Producto Systems: aalyss ad sythess, Cyberetcs ad Syst Aal, Kev, 994, 4, Dosoy VI The Dual Epert Systems, Tech Cyber, Moscow, 993, 5, -9 6 Hyafl L, Rvest RL Costractg optmal bary decso trees s NP-complete, If Proc Letters, 976, Vol5,, Cherov SN A lear eualtes, Naua, Moscow, 968