Theoretical Foundations of the UML Gao Complexity Lecture 09: - PowerPoint PPT Presentation

Theoretical Foundations of the UML Gao Complexity Lecture 09: Realisability → Joost-Pieter Katoen Lehrstuhl für Informatik 2 Software Modeling and Verification Group moves.rwth-aachen.de/teaching/ss-20/fuml/ May 18, 2020 Bag petty Joost-Pieter Katoen Theoretical Foundations of the UML 1/35

Realise bitity problem - - { , Mn } finite of set HSCs input M a : , , . . . { My Mn ) weak A that realises output CFM a : , . . , / I = { , Mn ) LCA ) Mi . - , . . acceptance no sync condition message IT Fp F= PEP of Main Lecture theorem g ( by CFM ) * realise Act ble Finite L E weak is a if if and only L closed F is under inference relation ↳ { , Mn ) of linearis atoms M Recall . . , - , : / - 't * ① formed formed L E well Act Act well is E w - - . , ( Vp =vrp ) Tff EP Wrp FUEL L F w . . L ) ( Iff ⑦ L L K closed implies KW is under we

of of Topic hard today how it checking is : - - what the is complexity ? Act L f whether E closed under is complete Result this NP problem is : co - . finite Giroary of checking whether set MSG : a able by complete real week CFM CONP is is is a - . Explanation - ? ① what is NP completeness co TDP ) ( ② Josh dependency problem ③ of onto the Polynomial reduction TDP { realisation :b the problem C ismdle.CI?edg 's ability The lies reals problem CONP in .

pleteness Iom PSPACE y . replete NP \ Npn NP co PSPACE E EXPTIME tie D= COP cop HEP Decision problem then → , NP ) ( P =/ =/ believed His that NP comp , Examples . - # ? P number number prime is given a a - formula problem Boolean SAT Np - - , , complete ( ) ) ⇐ nxz Ing eh V x n . , formula determine whether Boolean is a a - ? tautology complete comp -

of characterisation NP Simple co : of effty which for the class problems proofs of exist verifiable counterexamples - Aivey of decision all the class NP is co : IT H that problems NP such E . ② that show problem decision To our i ? Ct ) decision 't Act L E closed under F is . fy complete comp is we a - , and that GNP hard provide problem is a - - € (f) to reduction polynomial . ( ODP ) Problem Join Dependency

IDP example : I c ) , b inputs U= universe a : n , . k=4 KEIN cardinality e.g z . . , Uk records in R relation E a 3 e.g # . , . database a ) } { ( ,a ) , b ) a ) , ( ( a. a ,b , b , ( b Be a a. a. a ,a a a. , , , , k ] [ set Ind index 4 7 over e.g . . . , . tables sub } } { - { { 1.2.33 { z.s.gg I r.gg - , , - Brig In ' b a a a b a a a a b a a → b a a a ( RPI , RFI ) " ( HEE VI ERI JDP implies U E : . . ? ER I R from the database reconstruct the tables sub can we ? ,RrIm Rrf - . . . ,

TI ) ( for Cle ) HI TIER ER example a- a- → our : - . ( b. b. b ) , b ) ) a) Mass a- ( b. b. b TI a- a- e.g = = = , . ( b. b. b) ¢ RTI but obligation so no , for HR a- to be R in Indeed a- . . a ) ( b) a- EIR a. a. a. = . not a a- RF bn ) ✓ a. a) ( RRI e a. = , Job a ) bz ) ✓ ( a. r Rr Iz a- Iz a. e = dependency ✓ ( 9 a) R RE b 3) Ertz a. e = RTI svbtebtes RFI the ,RrIz Intuition by combining : , , , a ) ¢ ( ER but ( R that a. a) would imply a. a. ga a. , ,

problem ) Definition ( dependency Join - f- of ) finite Let be elements set U universe a I cardinality ) KE IN Uk =/ ate ) R I database E records U a C- a . , , n . . , , , k ] Im } - { In sets E index Ind E . - . . , . . , . ) . ) - ( Ertj I , airy Igi in , ing ai , - . . . , . . . . , Um Ij { rIj=b } b- . ) Rr FIER a- E . . k ) Ind of fi least i Constraint at E every appears : . Ij in once some there JDP exist join does dependency : a , E Uk for it holds all a- , RRIJ ) ER ( a- Eertj implies tf Ind Ij c- e. . . database ) ( R Intuition relation be reconstructed : can = - by subset tables having multiple each joining a of shored R attributes the the by records is .

E ngos ] Yannakokis Theorem Maier Sagd , , , - complete DDP is NP co -

finite " theorem problem The decision is given a ( by CFM ) sable of set reali week MSG a A complete NP co - Proof - ① CONP This decision problem lies in . ⑦ decision problem hard GNP This is - . ① Lemma problem realise by The decision bility a - weak CONP CFM in is . sketch ) Boff ( of that the complement the show problem reliability lies NP in . { , Mn ) check not that M realise ble To is , . , - . follows NP is in as : we pursue for nondeterministic Guess ally every a process . , Mn ) { PEP let MSC Mp Ms E an . , . . . of Mprp actions be the Wp is sequence occurring at Mp in process p .

( for b the Check that projections Up . . , EP ) consistent i. e. every process are pi , - formed combination their well complete is a - MSC M . Mn } ¢ { whether Ms Check M c . , . . - , . nonrealisebility check XD Ergo NP we in can . . . bility NP ② Lemmy realise problem hard The co is i - . from Pref reduction polynomial JDP provide the a : 's ability onto real problem the : set of MSCS Ind ) [ Uk k 1- RE U { Mp , ) > , Mir , , , , - . . - d of instance TDP # tuples R in - of the instance realisability problem Ind ) that ( k JDP such R - u E . , , , { , } ( by iff sable real M ship is . a , , , . . . ) weak CFM , MIR , ) L Mi iff f- under closed is . . , .

Polynomial reduction : three contain several index mnllnple Ind sets as may - iv. Log . k ] to i G belongs that E assume we every . . T least sets at two If Ij this Ij Ind E is . , . - ) not the just duplicate Ind Ij in case , Im } { { pm } D= Ind to In p , - = , , . . . . , . . , - - ability port SDP real , 's for set index each i. one process e . Uk , In ) { AT AT R with e . - = , . . . . { Me , Main } MSG ↳ , - . . , structure has Maj the MSC i.e same every . , . , the only exchanges the some message . message content differs . R for have So record database MSC in 7 every we , , .

defined follows These MSG as are . { Is } example Inde Ii By E. : , . } 7,34 ) { { } { Iz= I Ig 7,33 3,4 2 = = , In } for of { AT structure MSC My I C- : , , . . . E . %) p , Be 's - Cx , i x * x : Xz x +3 . As ,ak ) R For a- Cap ME tuple obtain C- we = . . , finite of by by a- I set replacing The MSCS . ER } time I a- Me the be polynomial Clearly this done th reduction can ,

My Pp ( Remotes contains either sends or : ) if if only j Ip and Xj receives message E addition alton Linearis In My has MSC unique a , , ( Mz ) set Lin singleton i.e is a . . ind ) dependency ( , he R Claims U join is a , , - if if and only { , ) real ble My ,M,r isa is a - . - - , CFM ) ( by week a . " Pino " contraposition that ⇐ By Assume : . ,M,p , ) Ind ) { Mp real able CU , k , R and is is . , . , . riot exists dependency Then there join is a - C- Uk sak ) ( ay that e- such = - - , TI for a- R all RI I Ind E C- ¢* ) If R but . a- rIj c- RRIJ there Take Since Ind is Ij E . JE b- JER b- R rIj that I such a = .

ME Consider MSC By the construction . , a- rj " since & depends a- r Heng M only Ij on . rIj=5JrIj ME rj follows Mbf It rj a- = - , to all applies Ij Ind M thus This Mgj E E = , . , ) { Mp to j Mir This applies any so . . - . , , M Mfa belong to Mfm all . . . , - { Mi , MIR , ) sable Kali ME EM is Since - - . , , . - , ) closed K { under MIR is M , , , - - - I to ¢ R Contradiction . . " " Ind ) k ( ⇒ along similar lines let U R be goes : , . . , , { , Mgr , } contraposition dependency By assume M join a , . . , . . , - M sable riot if of M reali But M t M is . , JER rig b- " { Mi " off , } from tuples read Mr we can . . . , , E Uk k tuple each j that there for such I is such a - b- J a- ¢ rIj Contradiction that for j but a- R each = . , DX .