From LTL to Symbolically Represented Deterministic Automata

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1 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy From LTL to Symoliclly Represented Deterministic Automt Andres Morgenstern Klus Schneider Sven Lmerti Mnuel Gesell Octoer 29th 2013 Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 1

2 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Tle of Contents 1 Motivtion nd Prolem Setting 2 Determinizing Non-Confluent Automt 3 Symolic Deterministion vi the Automt Hierrchy Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 2

3 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Verifiction Modern (emedded) systems re more nd more complex: testing is often not sufficient (Forml) verifiction is wy to gurntee correctness Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 3

4 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy This tlk is out the Temporl Logic LTL used to specify infinite ehvior of systems usge: Specifictions for Verifiction / Model Checking utomt cn e used to reson out LTL non-deterministic utomt: Model-Checking of Kripke Structures LTL stisfiility deterministic utomt: Model Checking of Proilistic Systems LTL Gme Solving / Controller Synthesis CTL* Stisfiility How to efficiently generte deterministic utomt from LTL? Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 4

5 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy LTL LTL LTL descries properties of infinite computtions σ 0, σ 1,... Xα: σ 1 = α Gα : i. σ i = α Fα: i. σ i = α [α U β]: δ. σ δ = β 0 < δ. σ i = α [α U β] = [α U β] Gα Exmples G : holds everywhere F : holds somewhere GF: holds infinitely often FG: fter n initil phse, holds lwys Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 5

6 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Review on Finite Automt on Finite Words Nondeterministic Automt, s 1 s 0 s 2 utomt red finite words lphet Σ = {, } Initil stte s 0 s 3 s 4 Finl sttes s 3, s 4 utomton ccepts, if one of the runs end in finl stte,, Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 6

7 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Determiniztion: Suset Construction Deterministic Automt Nondeterministic A., {s 0 } {s 0, s 1 } {s 0, s 2 } s 1 s 0 s 2 {s 0, s 1, s 3 } {s 0, s 2, s 4 } s 3 s 4 {s 0, s 2, s 3 } {s 0, s 1, s 4 },, {s 0, s 2, s 3, s 4 } {s 0, s 1, s 3, s 4 } Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 7

8 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Finite Automt on Infinite Words - ω-automt Büchi utomton utomt red infinite words, lphet Σ = {, } Initil stte s 0, s 0 s 1 L(A) = ( + ) ω = FG Accepting stte s 1 ccepts, if ccepting sttes visited infinitely often Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 8

9 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Finite Automt on Infinite Words - ω-automt Nondeterministic Automton,, q 1 q 2 Suset construction,, {q 1 } {q 1, q 2 } L(A) = ( + ) ω = FG L(A) = ( + ) ω Suset construction gives wrong result! Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 9

10 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Determiniztion of ω-automt Sfrs construction (1989) genertes deterministic Automt from nondeterministic Büchi Automt Ech stte of the generted Automton is n ordered tree in which ech node is lelled y suset of the sttes of the nondeterministic utomton such tht ech node... Complicted: First Implementtion 2006 y J. Klein : ltl2dstr Doesn t llow direct symolic implementtion Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 10

11 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Our Contriutions Non-Confluent Automt Ide: exploit the specil structure of those utomt tht stem from LTL formuls Result: Deterministic Automton with tuple of stte sets insted of Tree structure Exploiting the Temporl Logic Hierrchy LTL nondeterministic Büchi LTL nondeterministic Co-Büchi- or Sfety- Automt esier to determinize = symolic implementtion with BDDs Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 11

12 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy 1 Motivtion nd Prolem Setting LTL Finite Automt on Finite Words Finite Automt on Infinite Words - ω-automt 2 Determinizing Non-Confluent Automt Non-Confluent Automt The Determiniztion Procedure 3 Symolic Deterministion vi the Automt Hierrchy The Automt Hierrchy The Temporllogic-Hierrchy The Brekpoint Construction Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 12

13 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Bsis: Bckwrd-Determinism Determinism Bckwrd-Determinism q 0 q 1 q 2 q 1 q 2 q 3 Bckwrd-determinism = incoming edges must e lelled differently every Büchi Automton from LTL is ckwrd-deterministic! Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 12

14 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Runs re unique Runs of ckwrd-deterministic utomt re non-confluent c q q q q c q c q never c c q q q q c q c q every Büchi utomton from LTL is ckwrd-deterministic! different runs cn not merge = non-confluence every run uniquely determined y lst visited stte Ide: prllel suset-construktions set encode whetherf stte hs een visited! Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 13

15 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Bsic Ide S 0 S 1 S 2 S 3 n suset constructions S i Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 14

16 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Bsic Ide S 0 S 1 S 2 S 3 n suset constructions S i {r 0 } {r 0, r 1 } runs my split Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 14

17 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Bsic Ide S 0 S 1 S 2 S 3 n suset constructions S i {r 0 } {r 0, r 1 } {r 0, r 1 } {r 1 } runs my split move ccepting to right Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 14

18 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Bsic Ide S 0 S 1 S 2 S 3 n suset constructions S i {r 0 } {r 0, r 1 } {r 0, r 1 } {r 1 } runs my split move ccepting to right {r 0, r 1 } {r 1 } Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 14

19 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Bsic Ide S 0 S 1 S 2 S 3 n suset constructions S i {r 0 } {r 0, r 1 } {r 0, r 1 } {r 1 } runs my split move ccepting to right {r 0, r 1 } {r 1 } {r 0, r 1, r 2 } {r 1, r 2 } runs my split Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 14

20 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Bsic Ide S 0 S 1 S 2 S 3 n suset constructions S i {r 0 } {r 0, r 1 } {r 0, r 1 } {r 1 } runs my split move ccepting to right {r 0, r 1 } {r 1 } {r 0, r 1, r 2 } {r 1, r 2 } {r 0, r 1, r 2 } {r 1, r 2 } {r 2 } runs my split move ccepting to right Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 14

21 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Bsic Ide S 0 S 1 S 2 S 3 n suset constructions S i {r 0 } {r 0, r 1 } {r 0, r 1 } {r 1 } runs my split move ccepting to right {r 0, r 1 } {r 1 } {r 0, r 1, r 2 } {r 1, r 2 } {r 0, r 1, r 2 } {r 1, r 2 } {r 2 } {r 0, r 1, r 2 } {r 1, r 2 } {r 2 } {r 0, r 1, r 2 } {r 1, r 2 } {r 2 } {r 1 } runs my split move ccepting to right move ccepting to right Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 14

22 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Bsic Ide S 0 S 1 S 2 S 3 n suset constructions S i {r 0 } {r 0, r 1 } {r 0, r 1 } {r 1 } runs my split move ccepting to right {r 0, r 1 } {r 1 } {r 0, r 1, r 2 } {r 1, r 2 } {r 0, r 1, r 2 } {r 1, r 2 } {r 2 } {r 0, r 1, r 2 } {r 1, r 2 } {r 2 } {r 0, r 1, r 2 } {r 1, r 2 } {r 2 } {r 1 } runs my split move ccepting to right move ccepting to right check if S i j i S j Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 14

23 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy A smll exmple NDet Deterministic utomton 1 {1},, 2 3 c prllel suset-constructions in sets S 0... S n 1 Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 15

24 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy A smll exmple NDet Deterministic utomton 1 2 {1},, {1, 2},, 3 c dedends red mrking Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 15

25 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy A smll exmple NDet Deterministic utomton {1},, 1 {1, 2},, 2 {1, 2}, {2}, 3 c F sttes new suset-construction right Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 15

26 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy A smll exmple NDet Deterministic utomton 1 2 {1},, {1, 2},, {1, 2}, {2}, {1, 2},, 3 c dedends red mrkingg Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 15

27 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy A smll exmple NDet Deterministic utomton 1 2 {1},, {1, 2},, {1, 2}, {2}, {1, 2},, 3 c {1, 2}, {2}, Split F sttes sme Zustnd! Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 15

28 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy A smll exmple NDet Deterministic utomton {1},, 1 {1, 2},, 2 {1, 2}, {2}, 3 c sme stte! Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 15

29 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy A smll exmple NDet Deterministic utomton {1},, 1 {1, 2},, 2 {1, 2}, {2}, {1, 2, 3}, {1, 2, 3}, 3 c prllel suset-construction! Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 15

30 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy A smll exmple NDet Deterministic utomton {1},, 1 {1, 2},, 2 {1, 2}, {2}, {1, 2, 3}, {1, 2, 3}, 3 c {1, 2, 3},, runs non-confluent lue mrking nd removl right Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 15

31 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy A smll exmple NDet Deterministic utomton c c {1},, c {1, 2}, {2}, {1, 2, 3},,,,,, c, {3},, c c {1, 2, 3},, cept, if set ( often red) nd ( lue)! Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 15

32 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Det. non-confluent utomt: Conclusion Bsics LTL formul led to non-confluent utomt runs re unique n prllel suset-constructions Advntge symolic implementtion is possile Drwck limit : 20 sttes for the non-deterministic utomton BDDs explode for igger exmples due to ig numer of BDD vriles Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 16

33 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Overview 1 Motivtion nd Prolem Setting LTL Finite Automt on Finite Words Finite Automt on Infinite Words - ω-automt 2 Determinizing Non-Confluent Automt Non-Confluent Automt The Determiniztion Procedure 3 Symolic Deterministion vi the Automt Hierrchy The Automt Hierrchy The Temporllogic-Hierrchy The Brekpoint Construction Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 17

34 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy ω-automt,,c,,c,,c ω-automt ω-automt red infinite Words. Different cceptnce conditions: Büchi : visit F sttes infinitely often : GF FGc c Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 18

35 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy ω-automt,,c,,c,,c ω-automt ω-automt red infinite Words. Different cceptnce conditions: Büchi : visit F sttes infinitely often : GF FGc Co-Büchi: visit F sttes finitely often : FG FGc c Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 18

36 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy ω-automt,,c,,c,,c ω-automt ω-automt red infinite Words. Different cceptnce conditions: Büchi : visit F sttes infinitely often : GF FGc Co-Büchi: visit F sttes finitely often : FG FGc Streett: oolen comintion of (co)-büchi (in Normlform) c Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 18

37 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy ω-automt,,c,,c,,c ω-automt ω-automt red infinite Words. Different cceptnce conditions: Büchi : visit F sttes infinitely often : GF FGc Co-Büchi: visit F sttes finitely often : FG FGc Streett: oolen comintion of (co)-büchi (in Normlform) Sfety : visit only F sttes Liveness : visit F sttes t lest once Prefix : oolen comintion of Sfety und Liveness (in Normlform) c Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 18

38 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy The Automt Hierrchy (Wgner, 1979) (N)Det Sfety ool. com. Det Büchi ool. com. NDet Prefix Det Prefix NDet Büchi (N)Det Streett NDet totl Liveness Det Liveness ool. com (N)Det Co-Büchi ool. com C 1 C 2 := utomton from C 1 cn e trnslted to one from C 2 Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 19

39 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy The Temporllogic Hierrchy (Mnn&Pnueli, 1987) (N)Det Sfety TL Sfety NDet totl Liveness Det Liveness TL Liveness ool. com. NDet Prefix Det Prefix TL Prefix ool. com Det Büchi TL Büchi (N)Det Co-Büchi TL Co-Büchi ool. com. NDet Büchi (N)Det Streett TL Streett ool. com C 1 C 2 := utomton from C 1 cn e trnslted to one from C 2 Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 19

40 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Symolic Deterministion vi Automt Hierrchy (N)Det Sfety Det Büchi Sfr TL Sfety Suset NDet totl Liveness Det Liveness TL Büchi NDet Prefix Det Prefix TL Prefix (N)Det Co-Büchi NDet Büchi (N)Det Streett TL Streett TL Liveness TL Co-Büchi Brekpoint BDD-represented Automt for TL Sfety nd TL Co-Büchi Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 20

41 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Symolic Deterministion vi Automt Hierrchy (N)Det Sfety Det Büchi TL Sfety ool. com. TL Büchi ool. com. Dul NDet Prefix Det Prefix Dul NDet Büchi (N)Det Streett TL Prefix TL Streett NDet totl Liveness Det Liveness TL Liveness ool. com (N)Det Co-Büchi TL Co-Büchi ool. com BDD-represented Automt for TL Liveness, TL Prefix nd TL Streett Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 20

42 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Temporllogic-Hierrchy TL Sfety P G ::= V Σ,, P F XP G [P G U P G ] GP G TL Prefix P Prefix ::= P G, P F,, TL Büchi P GF ::= P Prefix,, P FG XP GF [P GF U P GF ] GP GF TL Streett P Prefix ::= P GF, P FG,, TL Liveness P F ::= TL co-büchi P FG ::= V Σ,, P G XP F [P F U P F ] FP F P Prefix,, P GF XP G [P F U P F ] FP FG Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 21

43 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Temporllogic-Hierrchy TL Sfety P G ::= V Σ,, P F XP G [P G U P G ] GP G TL Prefix P Prefix ::= P G, P F,, TL Büchi P GF ::= P Prefix,, P FG XP GF [P GF U P GF ] GP GF TL Streett P Prefix ::= P GF, P FG,, TL Liveness P F ::= TL co-büchi P FG ::= V Σ,, P G XP F [P F U P F ] FP F P Prefix,, P GF XP G [P F U P F ] FP FG Exmple formul: TL Sfety G, [ U ] TL Liveness F, [ U ] TL Büchi GF, G ( F) FG, FG ( [ U ]) TL co-büchi Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 21

44 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Temporllogic-Hierrchy TL Sfety P G ::= V Σ,, P F XP G [P G U P G ] GP G TL Prefix P Prefix ::= P G, P F,, TL Büchi P GF ::= P Prefix,, P FG XP GF [P GF U P GF ] GP GF TL Streett P Prefix ::= P GF, P FG,, TL Liveness P F ::= TL co-büchi P FG ::= V Σ,, P G XP F [P F U P F ] FP F P Prefix,, P GF XP G [P F U P F ] FP FG Every formul from TL X cn e trnslted to NDet X -utomton Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 21

45 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Symolic Deterministion vi Automt Hierrchy (N)Det Sfety Det Büchi Sfr TL Sfety Suset NDet totl Liveness Det Liveness TL Büchi NDet Prefix Det Prefix TL Prefix (N)Det Co-Büchi NDet Büchi (N)Det Streett TL Streett TL Liveness TL Co-Büchi Brekpoint BDD-represented Automt for TL Liveness, TL Prefix nd TL Streett Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 22

46 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Symolic Deterministion vi Automt Hierrchy (N)Det Sfety Det Büchi TL Sfety ool. com. TL Büchi ool. com. Dul NDet Prefix Det Prefix Dul NDet Büchi (N)Det Streett TL Prefix TL Streett NDet totl Liveness Det Liveness TL Liveness ool. com (N)Det Co-Büchi TL Co-Büchi ool. com BDD-represented Automt for TL Liveness, TL Prefix nd TL Streett Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 22

47 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy The Brekpoint Construction co-büchi utomt co-büchi cceptnce : FGF fter initil phse, never leve F Bsic Ide let suc (Q) e the successors of set Q under input we keep two Sets Q, Q F Q: ordinry suset-construction Q F : suset-construction, reduced to F suc { (Q F ) = suc (Q) F if Q F = = Brekpoint suc (Q F ) F else if Q F fter initil phse forever, ccept Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 23

48 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Brekpoint Construction: n exmple Co-Büchi: FGF,,c 1,,c 2,,c c 3 Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 24

49 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Brekpoint Construction: n exmple Co-Büchi: FGF,,c 1 2,,c Brekpoint Construction: {1} {1, 2} {2},, c, c c 3,,c {1, 2, 3} {2, 3},, c, c {1, 2, 3} {3}, c {1, 2, 3} Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 24

50 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Experiments Sources K. Etessmi nd G. Holzmnn: Optimizing Büchi utomt F. Somenzi nd R. Bloem: Efficient Büchi utomt from LTL formule M. Dwyer, G. Avrunin, nd J. Corett: Property specifiction ptterns for finite-stte verifiction overll : 94 formul 90 elong to TL Streett Rther smll formul (mx 20 temporl opertors) trnsltion done in seconds, mx. Mem : 30 m Comprle results with ltl2dstr Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 25

51 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Experiments: Rel Word Exmples AMBA Cse Study Specifiction of n riter for the AMBA Bus from industry: cpus, cches,... per client 10 Sfety/ 1 Büchi property totl.40 Temporl opertots / client Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 26

52 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy The AMBA specifiction A1: G ((hmstlock hurst == incr) XF usreq) A2: GFhredy N A3: G (hlock[i] husreq[i]) i=0 N A4: ( husreq[i] hlock[i] hredy) i=0 N G0: G (hmster==i (usreq husreq[i])) i=0 G1: G ( hredy strt) G2: G ((hmstlock hurst==incr strt) X [ husreq B strt]) hmstlock [ hurst == BURST4 strt] G3: G hredy X strt U [3] hredy strt [ ] X strt U [4] hredy strt G4: N (hredy (hgrnt[i] X(hmster == i)) i=0 G5: G (hredy (locked X(hmstlock))) N ( ( ( ))) X(hmster == i) (hmster == i) G6: G X strt X(hmstlock) hmstlock i=0 N G7: G ((decide X(hgrnt[i])) (hlock[i] X(locked))) i=0 N ( ( )) X(hgrnt[i]) hgrnt[i] G8: G ( decide) X(locked) locked i=0 N G9: GF ( husreq[i] hgrnt[i]) Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 27

53 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy AMBA:Experiments AMBA Cse Study Tools Specifiction of n riter for the AMBA Bus from industry per client 10 Sfety/ 1 Büchi property totl.40 Temporl opertots / client ltl2dstr: not le to generte (explicit) deterministic utomton for 2 clients, 3.5 GB our tool: 16 clients in seconds, mount: < 100 m More Experiments hve een done, similr results Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 28

54 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Determiniztion vi Automt Hierrchy: Conclusion Min Ide Locte formul syntcticlly in Hierrchy Suset (Brekpoint) construction symoliclly oolen comintion of Formuls / Automt Advntges Deterministic utomt never explicitely represented Efficient: due to oolen comintion suutomt very smll (less thn < 20 ndet sttes) Nerly ll formul elong to TL Streett Disdvntges Not every formul is in TL Streett! Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 29

55 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Future Work: Determiniztion vi Automt Hierrchy Previous Work Fct: Every LTL formul is equivlent to formul from TL Streett! (Mnn, Pnueli) = Exmple : FG [ U ] = FG [ U ] GF Prolem : Proof goes vi deterministic utomt = useless? Future Work Find trnsltion from LTL to TL Streett! Currently: Blow-up in the size t lest exponentil Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 30

56 Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy Thnks for your ttention! Andres Morgenstern, Klus Schneider, Sven Lmerti, Mnuel Gesell From LTL to Deterministic Automt 31

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