1. LTL – model checking Jonas Kongslund Peter Mechlenborg Christian Plesner Kristian Støvring Sørensen

2. Overview System Negation of property PLTL-formula () Model State space Model checker Normal-form formula Graph Generalised Büchi automaton Büchi automaton (Asys) Büchi automaton (A ) Product automaton (Asys A ) Checking emptiness Yes! No!

3. Büchi Automata • Def.: Labelled Büchi Automaton

4. Büchi Automata 2 • Def.: Run of a LBA

5. (a|d)(bc+)ω {a,d} {b} {c} Büchi Automata 3 • Example: Σ={a,b,c,d,e}

6. Büchi Automata 4 • For each PLTL formula φ one can construct an LBA Aφ s.t. Lω(Aφ) is the sequences of sets of atomic propositions that satisfy φ. • Let Σ=2AP where AP is the set of atomic propositions.

7. Büchi Automata 5 • Def.: Generalised LBA

8. Eliminate F and G operators Make negations adjacent to atomic propositions Example: LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? Getting Normal

9. LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? Getting Normal 2 • Past operators do not add any expressive power to LTL • Why are they useful? • Past operators are not easy expressed with future operators

10. LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? Getting Normal 3 • Past operators does not add any expressive power to LTL • Why are they useful? • Past operators are not easy to translate to normal form • Possible exponential blowup

11. Normal Form → GLBA LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? • Overall idea: A node in the graph represents a state, an edge represent a step forward in time. Each node contains formulas that must be true at this time; view these formulas as proof obligations: • Atomic propositions: check for contradictions • Conjunctions: check both clauses • Disjunctions: split into two nodes and allow a nondeterministic choice • Next: Push proof obligation to the successors • Until and its evil twin: unfold recursively on demand

12. {{q}, {p, q}} Ø {{p}, {p, q}} Accept states 1 LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? Definition of strict p U q: Sooner or later, q must happen! (Remember, every run is accepted, since the set of accept sets is empty)

13. {{q}, {p, q}} Ø {{p}, {p, q}} Accept states 2 LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? Definition of strict p U q: Sooner or later, q must happen! Problem: The automaton accepts pω!

14. Accept states 3 LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? Definition of strict p U q: Sooner or later, q must happen! {{q}, {p, q}} Ø {{p}, {p, q}} Solution: Insert accept states to break the cycle (not needed for U).

15. Un-generalizing GLBAs 1 LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? The generated automaton may have more than one set of accept states (one for each ‘until’ in the original formula):

16. Un-generalizing GLBAs 2 LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? Duplicate automaton. When leaving an accept state, jump to next automaton. Keep only one set of accept states.

17. Un-generalizing GLBAs 3 LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? Duplicate automaton. When leaving an accept state, jump to next automaton. Keep only one set of accept states.

18. Un-generalizing GLBAs 4 LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? Duplicate automaton. When leaving an accept state, jump to next automaton. Keep only one set of accept states.

19. Un-generalizing GLBAs 5 LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? Duplicate automaton. When leaving an accept state, jump to next automaton. Keep only one set of accept states.

20. Combining the two LBAs 1 LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? Wanted: an automaton accepting the intersection of the two languages: x

21. Combining the two LBAs 2 LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? By the ordinary DFA product construction: Problem: Requires accept states to be visited at the same time.

22. Combining the two LBAs 3 LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? Solution: Use a GLBA with two accept sets, then reduce to an LBA.

23. The emptiness problem LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty? How do we do it? Find an appropriate cycle in the LBA – if no such cycle exists, the language is empty. Why does this work? Theorem 17. Seriously, why? In order for the language to be non-empty, there must be an infinite run of the automaton that visits an accept state infinitely often. This means that there has to be a reachable cycle containing an accept state.

24. Overview System Negation of property PLTL-formula () Model State space Model checker Normal-form formula Graph Generalised Büchi automaton Büchi automaton (Asys) Büchi automaton (A ) Product automaton (Asys A ) Checking emptiness Yes! No!

25. The state space • Example int i; proctype P1(){ do ::true -> atomic( if::(i<2) -> i=i+1 fi) od } proctype P2(){ do ::true -> atomic( if::(i!=2) -> i=2 ::else -> i=0 fi) od } init{i=0; run(P1); run(P2);}

26. The state space 2 • A state • all global vars. • local vars. and program counter in all processes • State space: all possible simulations from the initial state • State space must be finite

27. i=0 i=1 i=2 The state space 3 P1 and P2 enabled P1 and P2 enabled P2 enabled

28. State space → LBA • Convert states to proposition tables • Get all propositions from the LTL expression • In each state • Change the lable to the set of all satisfied propositions

29. i=0 p i=1 i=2 r q State space → LBA 2 • Propositions: p:= (i <= 0) q:= (i == 1) r:= (i >= 2)

30. State space → LBA 3 • Make all paths infinite • Make all states accepting • Product is now normal DFA product

31. The rest • Is in chapter 5

32. References • G. J. Holzmann: An improved protocol reachability analysis technique. • O. Lichtenstein, A. Pnueli: The glory of the past. • R. Gerth et al.: Simple on-the-fly automatic verification of linear temporal logic. • K. Etessami, G. J. Holzmann: Optimizing Büchi automata. • A. M. Mikkelsen: On-the-fly model checking in Design/CPN. • G. J. Holzmann: The model checker SPIN.

33. Exercises • Exercises 8, 9, 10 (s3 should be s2), 12 • Derive the semantics of U from the semantics of U, and give an intuitive explanation.