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- 1. Temporal Logic & Model Checking Dr. Jayaraj Poroor
- 2. Formal methods: Why? REAL DEVELOPMENT IDEAL DEVELOPMENT Specification ¯ Design ¯ Coding ¯ Testing ¯ Product …. + DErrors ¯ + DErrors ¯¯ + DBugs ¯¯¯ Limited Coverage ¯ with safety issues
- 3. Formal methods: Where? Life Critical Mission Critical Medical Devices Therac-25 Space Vehicles Ariane-5 Industrial Control Slammer Worm at Davis-Besse Plant • An Investigation of Therac-25 Accidents [Leveson, Turner, 93] • Ariane 5 Flight 501 Failure, Report by Inquiry Board [Lions, 96] • Slammer worm crashed Ohio nuke plant network, News Report [http://www.securityfocus.com/news/6767, 03]
- 4. Sentences (Syntax) and Models (Semantics) “Every PhD student must have an advisor who is a member of faculty” F = "x× phd-student(x) Þ $y × advisor-of(y, x) Ù faculty(y) What does F mean? ◦ Classical Interpretation - A mathematical structure with: ◦ advisor-of mapped to a binary relation on the structure ◦ phd-student, faculty mapped to unary relations ◦ Logical symbols (Ù, Þ) have fixed interpretation ◦ Quantifiers (",$) range over elements of the underlying set Model of F = a satisfying interpretation for F
- 5. Fundamental reasoning tasks M satisfies F ? Model checking problem ? satisfies F Satisfiability problem * satisfies F Validity problem { a : M |= F(a)} Formula (query) evaluation Note: |= denotes satisfies relation
- 6. Modal Logic Modalities: Necessity, knowledge, belief, obligation, tense ◦ Symbolic Logic [Lewis,32] Possible World Semantics ◦ Kripke Structure Temporal Logic ◦ Time and Modality [Prior,57] ◦ Temporal modalities: always (), eventually ()
- 7. Possible Worlds & Kripke Structure for a washing machine idle spinning outlet-open Kripke Structure = (States, ®, washing Accessibility Relation: { } inlet-open outlet-open alarm Ø(spinning Ù washing) Ø(inlet-open Ù outlet-open) L) L : States ® 2AP alarm
- 8. Verification: Sequential Programs ◦ Assigning Meanings to Programs [Floyd,67] ◦ An Axiomatic basis for Computer Programming [Hoare, CACM69] (Hoare Logic) ◦ Guarded commands, non-determinism and formal derivation of programs [Dijstra,75] (GCL)
- 9. Hoare Logic (Assignment Axiom) (Conditional Rule) (Sequencing Rule) ; Proof Tableaux (Pre-strengthening, Post-weakening)
- 10. Hoare Logic Example P = if (x > y) then z := x else z := y Prove that: {true} P {z = max(x,y)} {true} if (x > y) then {x > y} {x = max(x,y)} {z = max(x,y)} z := x else z := y {true} {Ø(x > y)} {z = max(x,y)} {z = max(x,y)} {y = max(x,y)} x > y Þ x = max(x,y) x £ y Þ y = max(x,y)
- 11. Concurrency Simple pre-condition/post-condition assertions insufficient ◦ Deadlocks, Data races, Starvation! ◦ Need a language for expressing concurrency properties Cannot ignore intermediate steps! ◦ P ; Q : Intermediate states of P & Q do not interleave/interact ◦ P || Q : Intermediate states interleave/interact (in exponential number of ways)
- 12. Specifying properties of concurrent systems Language: Temporal Logic The Temporal Logic of Programs [Pnueli,77] (Linear Temporal Logic) Safety ◦ something bad will never happen: ◦ Ø(spinning Ù washing) Liveness ◦ Something good will eventually happen: ◦ alarm Fairness ◦ Always something good will eventually happen ◦ idle
- 13. Why Temporal Logics? Computational System Modal and Temporal Logics Operational Semantics models of Labelled Transition System (Kripke Structure) [Stirling03] http://www.fing.edu.uy/inco/eventos/wssa/
- 14. Producer-Consumer with 1-buffer Producer wtp: while (!isempty); csp: buf = produce(); flp : isempty = false; Consumer wtc: while (isempty); csc: consume(buf); flc : isempty = true;
- 15. State space for 1-buffer system States = control state ´ data state = {wtp,csp,flp} ´ {wtc,csc,flc} ´ {da} ´ {em} wt : wait cs : critical section fl : flag update da : buf has data available em : isempty is true e.g., (wtp, csc, ,em) Î States
- 16. LTS for 1-buffer system wtp, wtc, ,em csp, wtc,da,em flp, wtc,da, wtp, wtc,da, flp, csc, , flp, flc, ,em wtp, csc, , wtp, flc, ,em flp, wtc, ,em csp, flc,da,em flp, flc,da, wtp, flc,da,
- 17. Temporal Properties for 1-buffer system Safety: Ø(csp Ù csc) ◦ Producer and Consumer will never be in the CS at the same time Liveness: (da Ù Ø em) ◦ Eventually data will become available and empty flag reset Fairness: csp ◦ Producer is always given a fair chance to produce
- 18. Model Checking Model checking: M |= F? Design & Synthesis of synchronization skeletons using branching temporal logic [Clarke &Emerson, 81] Specification & Verification of Concurrent Systems in Cesar [Queille & Sifakis, 82] Automatic Verification of Finite-State Concurrent Systems using Temporal Logic Specifications [Clarke, Emerson, Sistla, TOPLAS86]
- 19. Computations of LTS Unfold LTS ® Infinite tree of computations ◦ Interleaved Semantics ◦ Concurrency as non-determinism View of computations: Linear vs Branching ◦ Linear Temporal Logic ◦ Computational Tree Logic
- 20. Computational Tree Logic Path quantifier ◦ A: All paths (inevitably) ◦ E: there Exists a path (possibly) Temporal operator ◦ X: neXt state ◦ F: some Future state (eventually) ◦ G: Globally; all future states ◦ U: Until e.g., AF: for all paths eventually, EG: for some path globally
- 21. CTL semantics s0 s0 M,s0 |= AF p [Clarke, Emerson, Sistla, TOPLAS86] M s0 M,s0 |= AG p M,s0 |= EF p
- 22. CTL examples Logic in Computer Science [Huth, Ryan, 04]
- 23. Computational Tree Logic f ::= T | F | p | Øf | fÙf | fÚf | f®f | AX f | EX f | AF f | EF f | AG f | EG f | A [f U f] | E [f U f] A: inevitably (along all paths) E : possibly (there exists a path) G: globally (always), F: in future (eventually) X: neXt state, U: until
- 24. Model Checking M |= F? SAT(M, f) : 2S ◦ INPUT: CTL model M = (S, ®, L) CTL formula f ◦ OUTPUT: Set of states (Í S)that satisfy f ◦ Complexity: O(f . |S| . (|S| + |®|)) Logic in Computer Science [Huth, Ryan, 04]
- 25. SAT: Conjunction & Inevitability y1, y2,, y1, y2 y1 Ù y2 AFy AFy AFy AFy AFy AFy AFy y y, AFy
- 26. SAT: Possibly Until y1 E[y1Uy2 ] y1, E[y1Uy2 ] E[y1Uy2 ] y2, E[y1Uy2 ] y2
- 27. SAT(f) : 2S begin case(f) T : return S ^ : return Æ Øy : return S – SAT(y) f1Ùf2 : return SAT(f1) Ç SAT(f2) f1Úf2 : return SAT(f1) È SAT(f2) f1Þf2 : return SAT(Øf1Úf2) …
- 28. SAT (case(f) cont.d) AFy : SATAF(y) E[f1Uf2] : SATEU(f1,f2) end case end
- 29. SATAF(f) function f(f,Y) begin if Y = Æ then return SAT(f) else return Y È pre"(Y) end begin Y := Æ; repeat X := Y; Y := f(f,Y); until X = Y end
- 30. Fixpoint characterization Consider, F: 2S ® 2S Formula are identified with their characteristic set ◦ e.g. f denotes set of all states where f is true Subsets of S (Î2S) form complete lattice ◦ Partial order: Í, Join: È, Meet: Ç Knaster-Tarski theorem ◦ “Monotone functions on a complete lattice possess least and greatest fixpoints”: A lattice-theoretical fixpoint theorem and its applications [Tarski,55]
- 31. Fixpoint characterization: Eventually, Until EFf = mZ. f Ú EX Z AFf = mZ. f Ú AX Z E[y1Uy2] = mZ. y2 Ú (y1 Ù EX Z) A[y1Uy2] = mZ. y2 Ú (y1 Ù AX Z)
- 32. Fixpoint characterization: globally AGf = nZ.fÙAX Z EGf = nZ.fÙEX Z
- 33. LTL Model checking The complexity of propositional linear temporal logics [Sistla, Clarke, 85] Checking that finite state concurrent programs satisfy their linear specification, [Lichtenstein, Pnueli, POPL85] An automata-theoretic approach to automatic program verification [Vardi, Wolper, 86] ◦ LTL to Buchi Automata
- 34. State explosion The state explosion problem [Clarke, Grumberg, 87] ◦ Number of concurrent processes ◦ Number of variables Partial-order reduction ◦ An Introduction to Trace Theory [Mazurkiewicz,95] Symbolic Model Checking: 1020 states and beyond [Burch, Clarke, McMillan, Dill, Hwang, 92] ◦ OBDD: Ordered Binary Decision Diagrams [Bryant,86] ◦ m-Calculus: Finiteness is m-ineffable [Park,74] Abstraction Combining theorem-proving & model checking On-the-fly model checking
- 35. Some Tools SPIN/Promela ◦ http://spinroot.com Java Pathfinder ◦ http://javapathfinder.sourceforge.net/ NuSMV ◦ http://nusmv.irst.itc.it/
- 36. Thank You Have a great day!

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