LTL Translation Improvements in Spot

Transcription

1 LTL Trnsltion Improvements in Spot Alexndre Duret-Lutz VECoS'11 16 September 2011 Alexndre Duret-Lutz LTL Trnsltion Improvements 1 / 19

2 Context High-level model M Does the model M veries the property ϕ? LTL property ϕ M = ϕ or counterexmple Alexndre Duret-Lutz LTL Trnsltion Improvements 2 / 19

3 Automt-Theoretic LTL Model Checking High-level model M Stte-spce genertion Stte-spce utomton A M Synchronized product L (A M A ϕ ) = L (A M ) L (A ϕ ) Product Automton A M A ϕ Emptiness check L (A M A ϕ )? = LTL property ϕ LTL trnsltion Negted property utomton A ϕ M = ϕ or counterexmple Alexndre Duret-Lutz LTL Trnsltion Improvements 2 / 19

4 Automt-Theoretic LTL Model Checking High-level model M Stte-spce genertion Stte-spce utomton A M Synchronized product L (A M A ϕ ) = L (A M ) L (A ϕ ) Product Automton A M A ϕ Emptiness check L (A M A ϕ )? = LTL property ϕ LTL trnsltion Negted property utomton A ϕ M = ϕ or counterexmple Alexndre Duret-Lutz LTL Trnsltion Improvements 2 / 19

5 Automt-Theoretic LTL Model Checking High-level model M On-the-y genertion of stte-spce utomton A M On-the-y synchronized product L (A M A ϕ ) = L (A M ) L (A ϕ ) Emptiness check L (A M A ϕ )? = LTL property ϕ LTL trnsltion Negted property utomton A ϕ M = ϕ or counterexmple Alexndre Duret-Lutz LTL Trnsltion Improvements 2 / 19

6 Automt-Theoretic LTL Model Checking High-level model M Custom Model Checker On-the-y genertion of stte-spce utomton A M SPOT On-the-y synchronized product L (A M A ϕ ) = L (A M ) L (A ϕ ) Emptiness check L (A M A ϕ )? = LTL property ϕ LTL trnsltion Negted property utomton A ϕ M = ϕ or counterexmple Alexndre Duret-Lutz LTL Trnsltion Improvements 2 / 19

7 Let's Tlk bout the LTL Trnsltion We trnslte LTL to generlized Büchi utomt with trnsition-bsed cceptnce. Most people prefer non-generlized Büchi utomt with stte cceptnce. Our trnsltion is competitive, thnks to: A simple tbleu construction tht use BDD for simplictions Two techniques to improve determinism Severl LTL rewriting rules tht ese the trnsltion (not shown) Alexndre Duret-Lutz LTL Trnsltion Improvements 3 / 19

8 Trnsition-bsed Generlized Acceptnce Conditions 1 Introduction 2 Trnsition-bsed Generlized Acceptnce Conditions 3 Trnslting LTL into (TG)BA eciently 4 Conclusion Alexndre Duret-Lutz LTL Trnsltion Improvements 4 / 19

9 Dierent Kinds of Büchi Automt (G F G F b) Trnsition-bsed Generlized Generlized Büchi Automton Büchi Automton Büchi Automton ā āb ā b 3 1 b āb 1 b b 2 b b ā b ā b 2 3 b b b b ā b āb Sme expressive power. Converting BA to GBA, or GBA to TGBA, is trivil. The opposite direction requires degenerliztion. (T)GBA occur nturlly when trnslting LTL. Alexndre Duret-Lutz LTL Trnsltion Improvements 5 / 19

10 Trnslting LTL into (TG)BA eciently 1 Introduction 2 Trnsition-bsed Generlized Acceptnce Conditions 3 Trnslting LTL into (TG)BA eciently Gols: Size nd Speed Core Trnsltion: Tbleu using BDDs Improving Determinism with BDDs nd WDBA Results 4 Conclusion Alexndre Duret-Lutz LTL Trnsltion Improvements 6 / 19

11 Trnsltion of Litterture Formuls Cumulted sizes of utomt for 188 formuls from the litterture Products with rndom stte-spce of 200 sttes Σ A ϕ Σ A M A ϕ st. tr. st. tr. BA Spin (A 9) LTL2BA Modell ( 1) Spot Spot det Spot WDBA TGBA Spot Spot det Spot WDBA A = 15min timeout = bogus trnsltion Produce more deterministic ut. WDBA minimiztion when pplicble Alexndre Duret-Lutz LTL Trnsltion Improvements 7 / 19

12 Rozier & Vrdi's Sclbility Experiment (1/2) An LTL formul C n tht cn be encoded by n2 n -stte utomton. K. Y. Rozier nd M. Y. Vrdi. LTL stisbility checking. In Proc. of SPIN'07, vol of LNCS, pp Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 8 / 19

13 Rozier & Vrdi's Sclbility Experiment (1/2) An LTL formul C n tht cn be encoded by n2 n -stte utomton. E.g. C 3 = (( (G( (X( X( X )))))) (( b) X( b X b)) (G(( b) (X((X X b) ((( ) (b X X X b) (( b) (X X X b))) U ))))) (G(( b) (X((X X b) ((b ( ) X X X b) U( (( ) ( b) (X((X X b) ((( ) (b X X X b) (( b) X X X b)) U ))))))))))) K. Y. Rozier nd M. Y. Vrdi. LTL stisbility checking. In Proc. of SPIN'07, vol of LNCS, pp Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 8 / 19

14 Rozier & Vrdi's Sclbility Experiment (1/2) An LTL formul C n tht cn be encoded by n2 n -stte utomton. E.g. C 3 = (( (G( (X( X( X )))))) (( b) X( b X b)) (G(( b) (X((X X b) ((( ) (b X X X b) (( b) (X X X b))) U ))))) (G(( b) (X((X X b) ((b ( ) X X X b) U( (( ) ( b) (X((X X b) ((( ) (b X X X b) (( b) X X X b)) U ))))))))))) b ā b ā b b ā b ā b b āb ā b b āb āb āb b āb āb b āb ā b b āb ā b b ā b K. Y. Rozier nd M. Y. Vrdi. LTL stisbility checking. In Proc. of SPIN'07, vol of LNCS, pp Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 8 / 19

15 Rozier & Vrdi's Sclbility Experiment (2/2) 100s 80s 60s 40s 20s 0s spot-0.4 spot-0.5 ltl2b-1.1 C 6 C 8 C 10 C 12 C 14 Time to trnslte C n into BA Other explicit trnsltors re o the chrt: Modell took nerly 6 minutes to compute C 4 nd rn out of memory on C 5. Spin took more thn 11 hours to trnslte C 1 into 33-stte utomton with 447 trnsitions (insted of 2 sttes nd 2 trnsitions). Mny trnsitions hving unstisble gurds such s ((!b) && () && (b)). All experiments done on n Intel Core2 with 8GB of RAM. Alexndre Duret-Lutz LTL Trnsltion Improvements 9 / 19

16 Tbleu Construction for LTL (X ) (b U ) 1 Lbel the initil stte by the formul to trnslte J.-M. Couvreur. On-the-y veriction of liner temporl logic. In Proc. of FM'99, vol of LNCS, pp Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 10 / 19

17 Tbleu Construction for LTL (X ) (b U ) 1 Lbel the initil stte by the formul to trnslte 2 Rewrite ech stte lbel ϕ s β i X ψ i i Boolen formul LTL formul Since f U g = g (f X(f U g)) we hve: (X ) (b U ) = ( X ) (b (X ) X(b U )) J.-M. Couvreur. On-the-y veriction of liner temporl logic. In Proc. of FM'99, vol of LNCS, pp Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 10 / 19

18 Tbleu Construction for LTL (X ) (b U ) 1 Lbel the initil stte by the formul to trnslte 2 Rewrite ech stte lbel ϕ s β i X ψ i i Since f U g = g (f X(f U g)) we hve: (X ) (b U ) = ( X ) (b X( (b U ))) J.-M. Couvreur. On-the-y veriction of liner temporl logic. In Proc. of FM'99, vol of LNCS, pp Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 10 / 19

19 Tbleu Construction for LTL (X ) (b U ) b (b U ) 1 Lbel the initil stte by the formul to trnslte 2 Rewrite ech stte lbel ϕ s β i X ψ i i Then connect ϕ to ech ψ i using β i s lbel Since f U g = g (f X(f U g)) we hve: (X ) (b U ) = ( X ) (b X( (b U ))) J.-M. Couvreur. On-the-y veriction of liner temporl logic. In Proc. of FM'99, vol of LNCS, pp Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 10 / 19

20 Tbleu Construction for LTL (X ) (b U ) b (b U ) 1 Lbel the initil stte by the formul to trnslte 2 Rewrite ech stte lbel ϕ s β i X ψ i i Then connect ϕ to ech ψ i using β i s lbel Since f U g = g (f X(f U g)) we hve: (X ) (b U ) = ( X ) (b X( (b U ))) = X J.-M. Couvreur. On-the-y veriction of liner temporl logic. In Proc. of FM'99, vol of LNCS, pp Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 10 / 19

21 Tbleu Construction for LTL (X ) (b U ) b (b U ) 1 Lbel the initil stte by the formul to trnslte 2 Rewrite ech stte lbel ϕ s β i X ψ i i Then connect ϕ to ech ψ i using β i s lbel Since f U g = g (f X(f U g)) we hve: (X ) (b U ) = ( X ) (b X( (b U ))) = X ; = X J.-M. Couvreur. On-the-y veriction of liner temporl logic. In Proc. of FM'99, vol of LNCS, pp Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 10 / 19

22 Tbleu Construction for LTL (X ) (b U ) b (b U ) 1 Lbel the initil stte by the formul to trnslte 2 Rewrite ech stte lbel ϕ s β i X ψ i i Then connect ϕ to ech ψ i using β i s lbel Since f U g = g (f X(f U g)) we hve: (X ) (b U ) = ( X ) (b X( (b U ))) = X ; = X ; (b U ) = ( (b X(b U ))) J.-M. Couvreur. On-the-y veriction of liner temporl logic. In Proc. of FM'99, vol of LNCS, pp Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 10 / 19

23 Tbleu Construction for LTL (X ) (b U ) b (b U ) b b U 1 Lbel the initil stte by the formul to trnslte 2 Rewrite ech stte lbel ϕ s β i X ψ i i Then connect ϕ to ech ψ i using β i s lbel Since f U g = g (f X(f U g)) we hve: (X ) (b U ) = ( X ) (b X( (b U ))) = X ; = X ; (b U ) = b X (b U ) J.-M. Couvreur. On-the-y veriction of liner temporl logic. In Proc. of FM'99, vol of LNCS, pp Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 10 / 19

24 Tbleu Construction for LTL (X ) (b U ) b (b U ) b b U b 1 Lbel the initil stte by the formul to trnslte 2 Rewrite ech stte lbel ϕ s β i X ψ i i Then connect ϕ to ech ψ i using β i s lbel Since f U g = g (f X(f U g)) we hve: (X ) (b U ) = ( X ) (b X( (b U ))) = X ; = X ; (b U ) = b X (b U ) b U = ( X ) (b X (b U )) J.-M. Couvreur. On-the-y veriction of liner temporl logic. In Proc. of FM'99, vol of LNCS, pp Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 10 / 19

25 Tbleu Construction for LTL (X ) (b U ) b;p (b U ) b U b;p b;p 1 Lbel the initil stte by the formul to trnslte 2 Rewrite ( ech stte lbel ϕ s β i X ψ i ) P γ ij i j Then connect ϕ to ech ψ i using β i s lbel nd {P γ ij } j s promises Since f U g = g (f X(f U g) P g ) we hve: (X ) (b U ) = ( X ) (b X( (b U )) P ) = X ; = X ; (b U ) = b X (b U ) P b U = ( X ) (b X (b U ) P ) J.-M. Couvreur. On-the-y veriction of liner temporl logic. In Proc. of FM'99, vol of LNCS, pp Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 10 / 19

26 Tbleu Construction for LTL (X ) (b U ) 1 Lbel the initil stte by the formul to trnslte b 2 Rewrite ( ech stte lbel ϕ s (b U ) β i X ψ i ) P γ ij b i j Then connect ϕ to ech ψ i using b U β i s lbel nd {P γ ij } j s promises b 3 Crete Büchi cceptnce sets complementing ech promise Since f U g = g (f X(f U g) P g ) we hve: (X ) (b U ) = ( X ) (b X( (b U )) P ) = X ; = X ; (b U ) = b X (b U ) P b U = ( X ) (b X (b U ) P ) J.-M. Couvreur. On-the-y veriction of liner temporl logic. In Proc. of FM'99, vol of LNCS, pp Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 10 / 19

27 Implementing the Trnsltion with BDDs (1/2) r( ) = r( ) = r(p) = Vr[p] r( p) = Vr[p] r(f g) = r(f ) r(g) r(f g) = r(f ) r(g) r( (f g)) = r( f ) r( g) r( (f g)) = r( f ) r( g) r(x f ) = Next[f ] r( X f ) = Next[ f ] r(f U g) = r(g) (r(f ) Next[f U g] P[g]) r( (f U g)) = r( g) (r( f ) Next[ (f U g)]) Alexndre Duret-Lutz LTL Trnsltion Improvements 11 / 19

28 Implementing the Trnsltion using BDDs (2/2) r((x ) (b U )) = r(x ) r(b U ) = Next[] (r( ) (P[ ] r(b) Next[b U ])) = Next[] ( Vr[] (P[ ] Vr[b] Next[b U ])) Alexndre Duret-Lutz LTL Trnsltion Improvements 12 / 19

29 Implementing the Trnsltion using BDDs (2/2) r((x ) (b U )) = r(x ) r(b U ) = Next[] (r( ) (P[ ] r(b) Next[b U ])) = Next[] ( Vr[] (P[ ] Vr[b] Next[b U ])) This BDD is then mssged into n irredundnt sum of products: = ( Vr[] Next[]) (Vr[b] Next[] Next[b U ] P[ ]) S. Minto. Fst genertion of irredundnt sum-of-products forms from binry decision digrms. In Proc. of SASIMI'92, pp Alexndre Duret-Lutz LTL Trnsltion Improvements 12 / 19

30 Implementing the Trnsltion using BDDs (2/2) r((x ) (b U )) = r(x ) r(b U ) = Next[] (r( ) (P[ ] r(b) Next[b U ])) = Next[] ( Vr[] (P[ ] Vr[b] Next[b U ])) This BDD is then mssged into n irredundnt sum of products: = ( Vr[] Next[]) (Vr[b] Next[] Next[b U ] P[ ]) S. Minto. Fst genertion of irredundnt sum-of-products forms from binry decision digrms. In Proc. of SASIMI'92, pp Alexndre Duret-Lutz LTL Trnsltion Improvements 12 / 19

31 BDD Gins (1/3): Automtic Simplictions Trivil simplictions of ded brnches: Less trivil simplictions: e.g. (( U b) (b U c)). Alexndre Duret-Lutz LTL Trnsltion Improvements 13 / 19

32 BDD Gins (1/3): Automtic Simplictions Trivil simplictions of ded brnches: A trnsition lbeled by cnnot exist Less trivil simplictions: e.g. (( U b) (b U c)). Alexndre Duret-Lutz LTL Trnsltion Improvements 13 / 19

33 BDD Gins (1/3): Automtic Simplictions Trivil simplictions of ded brnches: A trnsition lbeled by cnnot exist Less trivil simplictions: e.g. (( U b) (b U c)). Tbleu rules give four immedite successors: ( ) ( b) ( c) ( ) ( b) ( c) X (b U c) ( b) ( c) (X ( U b)) ( b) ( c) (X ( U b)) X (b U c) BDD rewritings give two successors: Vr[b] ( Vr[] Next[ ( U b)]) Vr[c] ( Vr[b] Next[ (b U c)]) Alexndre Duret-Lutz LTL Trnsltion Improvements 13 / 19

34 BDD Gins (1/3): Automtic Simplictions Trivil simplictions of ded brnches: A trnsition lbeled by cnnot exist Less trivil simplictions: e.g. (( U b) (b U c)). Tbleu rules give four immedite successors: ( ) ( b) ( c) ( ) ( b) ( c) X (b U c) ( b) ( c) (X ( U b)) ( b) ( c) (X ( U b)) X (b U c) BDD rewritings give two successors: Vr[b] ( Vr[] Next[ ( U b)]) Vr[c] ( Vr[b] Next[ (b U c)]) = Vr[b] Vr[c] ( Vr[] Next[ ( U b)]) Alexndre Duret-Lutz LTL Trnsltion Improvements 13 / 19

35 BDD Gins (2/3): Automtic Quotienting BDD rewritings give us n equivlence reltion between LTL formuls. r(g F ) =((Next[F ] P[]) Vr[]) Next[G F ] r(f G F ) =((Next[F ] P[]) Vr[]) Next[G F ] G F (F ) G F G F Could we produce something more deterministic? Alexndre Duret-Lutz LTL Trnsltion Improvements 14 / 19

36 BDD Gin (3/3): Improving Determinism Let's improve the determinism of the previous construction by checking how dierent vrible ssignments inuence destintions: r(g F ) = ((Next[F ] P[]) Vr[]) Next[G F ] r(g F ) Vr[] = Vr[] Next[G F ] r(g F ) Vr[] = Vr[] Next[F ] P[] Next[G F ] G F (F ) G F G F Experiment on 188 LTL formuls from the literture pplied to rndom grphs: 0.4% less sttes, 25% less trnsitions in product. Alexndre Duret-Lutz LTL Trnsltion Improvements 15 / 19

37 Temporl Hierrchy Rectivity G F pi F G q i Recurrence G F p Persistence F G p Obligtion G pi F q i Sfety G p Gurntee F p Z. Mnn nd A. Pnueli. A hierrchy of temporl properties. In Proc. of PODC'90, pp ACM Alexndre Duret-Lutz LTL Trnsltion Improvements 16 / 19

38 Temporl Hierrchy Deterministic Büchi Automt Rectivity G F pi F G q i Wek Büchi Automt Recurrence G F p Persistence F G p Obligtion G pi F q i Wek Det. Büchi Aut. (WDBA) Sfety G p Gurntee F p I. ƒerná nd R. Pelánek. Relting hierrchy of temporl properties to model checking. In Proc. of MFCS'03, vol of LNCS, pp Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 16 / 19

39 Deling with Obligtion Properties 118 out of the 188 formuls (62%) re obligtions properties WDBA cn be minimized in sme wy s DFAs (Löding; 2001) Dx et l. (2007) show how to minimize ny utomton tht cn be expressed s WDBA. In Spot this lgorithm tkes TGBA nd outputs WDBA when the minimiztion is pplicble Although converting to miniml WDBA incurs determiniztion, the number of sttes seldom increses C. Löding. Ecient minimiztion of deterministic wek ω-utomt. Informtion Processing Letters, 79(3):105109, 2001 C. Dx, J. Eisinger, nd F. Kledtke. Mechnizing the powerset construction for restricted clsses of ω-utomt. In Proc. of ATVA'07, vol of LNCS. Springer Alexndre Duret-Lutz LTL Trnsltion Improvements 17 / 19

40 Trnsltion of Litterture Formuls Cumulted sizes of utomt for 188 formuls from the litterture Products with rndom stte-spce of 200 sttes Σ A ϕ Σ A M A ϕ st. tr. st. tr. BA Spin (A 9) LTL2BA Modell ( 1) Spot Spot det Spot WDBA TGBA Spot Spot det Spot WDBA A = 15min timeout = bogus trnsltion Produce more deterministic ut. WDBA minimiztion when pplicble Alexndre Duret-Lutz LTL Trnsltion Improvements 18 / 19

41 Conclusion LTL trnsltion The core trnsltion is ecient nd simple to implement Four tricks: Supporting the wek until opertor (not shown). Choosing pproprite rewriting rules (not shown). Using BDD is importnt (for speed nd determinism) Buiding WDBA when possible is often good ide Try it on-line: Wht's Cooking Support for PSL (Property Speciction Lnguge) Essentilly: LTL + rtionl opertors. { 1[*]; init; busy[=2]; done } []-> (!init U bell) Simultion-bsed reductions (for TGBA) Alexndre Duret-Lutz LTL Trnsltion Improvements 19 / 19