Embedded Systems 5. Midterm, Thursday December 18, 2008, Final, Thursday February 12, 2009, 16-19

Transcription

1 Embedded Sysems Exam Daes / egisraion Miderm, Thursday December 8, 8, 6-8 Final, Thursday February, 9, 6-9 egisraion hrough HISPOS oen in arox. week If HISPOS no alicable Non-CS, Erasmus, ec send o finkbeiner@cs.uni-sb.de No searae course sign-u BUT: Please indicae uorial, mar nr, name, on homework submissions. - -

2 Peri Nes EVIEW Def.: N=C,E,F is called a Peri ne, iff he following holds. C and E are disjoin ses. F C E E C; is binary relaion, flow relaion Def.: Le N be a ne and le x C E. x := {y y F x} is called he se of recondiions. x := {y x F y} is called he se of oscondiions. Comeing Trains Examle: x x x Boolean marking and comuing changes of markings EVIEW A Boolean marking is a maing M: C {,}. Firing evens x generae new markings on each of he condiions c according o he following rules: a ransiion a x can be fired, iff x, i.e. all recondiions of x are marked and x is no marked, afer firing x is unmarked and x is marked M M, iff M resuls from M by firing exacly one ransiion - 4 -

3 Comeing Trains Examle is C/E ne: EVIEW M8 M M M3 M4 M5 M6 M7 Ms M3s M4s M5s M6s M7s M8s Condiion/even nes C/E nes EVIEW Def.: A Peri ne N=C,E,F ogeher wih a se of markings M is called condiion/even ne C/E ne, iff N is simle and has no isolaed elemens M is closed w.r.. firing and inverse firing wo markings in M can be ransformed ino each oher by firing and inverse firing for each even e E, here exiss a marking in M, ha allows firing a e - 6-3

4 Examle Thalys rains: more comlex EVIEW Thalys rains beween Cologne, Amserdam, Brussels and Paris. Synchronizaion a a Brussels and Paris s Place/ransiion nes EVIEW Def.: P, T, F, K, W, M is called a lace/ransiion ne P/T ne iff. N=P,T,F is a ne wih laces P and ransiions T. K: P N {ω} \{} denoes he caaciy of laces ω symbolizes infinie caaciy 3. W: F N \{} denoes he weigh of grah edges 4. M : P N {ω} reresens he iniial marking of laces W Segmen of some ne M defauls: K = ω W = In he following: assume iniial marking is finie, caaciy ω

5 Examle: Job rocessing sysem free rocessors job done job waiing begin execuion finish execuion rocessing Comuing changes of markings EVIEW Firing ransiions generae new markings on each of he laces according o he following rules: - - 5

6 Acivaed ransiions Transiion is acivaed iff EVIEW Acivaed ransiions can ake lace or or fire, bu don have o. o. The order in in which acivaed ransiions fire is is no fixed i i is is non-deerminisic. - - Shorhand for changes of markings EVIEW Firing ransiion: Le W, if \ + W, if \ = W, + W, if P: M = M+ M = M+ +: vecor add - - 6

7 eachabiliy relaion M [> M` iff M = M+ M [ε> M` iff M=M` M [q> M` iff M``. M [q> M`` and M`` [> M` M [*> M iff q. M [q> M` eachabiliy se M = {M M [*> M } eachabiliy grah GM: nodes M, edges { M,,M` M [> M } Unbounded Peri ne

8 Boundedness A lace is safe if i conains a mos one oken in all reachable markings. A ne is safe if all laces are safe. A lace is k-bounded if i conains a mos k oken in all reachable markings. A lace is bounded if i is k-bounded for some k. A ne is bounded if each lace is bounded Boundedness Theorem : A P/T ne wih finie iniial marking is bounded iff is reachabiliy se is finie

9 Boundedness Theorem : A P/T ne is unbounded iff here exis wo reachable markings M, M, such ha M[*>M and M > M Proof of Theorem Lemma: Every infinie sequence of markings M i conains a weakly monoonically growing infinie subsequence M`i, i.e., for j<k, M`j M`k

10 Proof of Theorem Algorihm for deciding boundedness Exlore GM deh-firs: If here exiss a marking M on he sack such ha M <M, so wih resul UNBOUNDED; If enire grah exlored, reurn BOUNDED. - -

11 Weak Peri ne comuers A P/T ne wih r disinguished inu laces in i, a finie number of inernal laces s i, one exra ouu lace ou, one exra sar lace on, and one exra so lace off is called a weak Peri ne comuer for he funcion f: N r N iff here exiss for each xn r an iniial marking M x such ha M x on= and M x in i =x i for x i r; M x ou=m x off; M x s i =; For all reachable markings M M x, Mon= and Moff and Mou fx; For all reachable markings M M x, if Moff= hen M is dead; For all k fx, here exiss a reachable marking M such ha Mou=k and Moff=. - - Addiion Source: Mahias Janzen, Comlexiy of Place/Transiion Nes

12 Mulilicaion Source: Mahias Janzen, Comlexiy of Place/Transiion Nes GM is no rimiive recursive Proofidea: Le An:N N be defined by An=A n A m=m+ A n+ = A n+ m+=a n A n+ m A=5; A=7; A=5; A3=65535; An majorizes he rimiive recursive funcions. We can imlemen An as a P/T ne ha grows slowly wih n. Source: Mahias Janzen, Comlexiy of Place/Transiion Nes

13 A m=m gm+:=fgm, g=f - 6-3

14 Deadlock A dead marking deadlock is a marking where no ransiion can fire. A Peri ne is deadlock-free if no dead marking is reachable Liveness A ransiion is dead a M if no marking M is reachable from M such ha can fire in M. A ransiion is live a M if here is no marking M reachable from M where is dead. A marking is live if all ransiions are live. A P/T ne is live if he iniial marking is live. Observaions: A live ne is deadlock-free. No ransiion is live if he ne is no deadlock-free

15 Examle Liveness Examle Liveness

16 eversibliy A P/T ne is reversible iff for every MM, M [*> M Theorem 3: In a reversible ne, a ransiion is live iff is no dead in M

17 Comuaion of Invarians We are ineresed in subses of laces whose number of labels remain invarian under ransiions, e.g. he number of rains commuing beween Amserdam and Paris Cologne and Paris remains consan Imoran for correcness roofs, e.g. he roof of liveness Comeing Trains Examle: Place Invarian,,

18 Shorhand for changes of markings + + = if,, \ if, \ if, W W W W Le P: M = M+ Firing ransiion: +: vecor add M = M+ EVIEW Marix N describing all changes of markings Def.: Marix N of ne N is a maing N: P T Z inegers such ha T: N,= Comonen in column and row indicaes he change of he marking of lace if ransiion akes lace. + + = if,, \ if, \ if, W W W W EVIEW

19 Examle: N = s EVIEW Characerisic Vecor = c if if Le: = = = c c P j j j = j Scalar roduc

20 Condiion for lace invarians Accumulaed marking consan for all ransiions if = = n c c Equivalen o N T c = where N T is he ransosed of N = = = c c P j j j More deailed view of comuaions Sysem of linear equaions. Soluion vecors mus consis of zeros and ones. = n n m m n n c c c

21 Comeing Trains Examle