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1 Introduction to Model Checking Ken McMillan Cadence Berkeley Labs

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2 2 Outline l Model checking –Temporal logic –Model checking algorithms –Expressiveness and complexity l Symbolic model checking –The state explosion problem –Binary Decision Diagrams –Computing fixed points with BDDs –Application

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3 3 Propositional Linear Temporal Logic l Express properties of Reactive Systems –interactive, nonterminating l For PLTL, a model is an infinite state sequence l Temporal operators –Globally: G p at t iff p for all t t. ppppppppppp... G p...

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4 4 Temporal operators... –Future: F p at t iff p for some t t. pppppp F p... –Until: p U q at t iff – q for some t t and – p in the range [ t, t ) pppppp p U q... pppq –Next-time: X p at t iff p at t+1

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5 5 Examples l Liveness: if input, then eventually output G (input F output) l Strong fairness: infinitely send implies infinitely recv. GF send GF recv l Weak until: no output before input output W input atomic props infinitely often p W q p U q G p

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6 6 Safety v. Liveness l Safety –Refutable by finite run l Liveness –Refutable only by infinite run –Every finite run extensible to satisfying run

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7 7 PLTL semantics l Given an infinite sequence – if is true in state s i of. – if is true in state s 0 of. – if is valid. l A formula is an atomic proposition, or... true, p q, p, p U q, X p

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8 8 PLTL semantics... l Definition of satisfaction iff Derived operators...

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9 9 Model Checking (Clarke/Emerson, Queille/Sifakis) MC G(p -> F q) yes no p q p q temporal formula finite-state model algorithm counterexample Model must now represent all behaviors

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10 Kripke models l A Kripke model (S,R,L) consists of –set of states S –set of transitions R S S –labeling L S AP l Kripke models from programs p p repeat p := true; p := false; end

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11 Mutual exclusion example N1,N2 turn=0 T1,N2 turn=1 T1,T2 turn=1 C1,N2 turn=1 C1,T2 turn=1 N1,T2 turn=2 T1,T2 turn=2 N1,C2 turn=2 T1,C2 turn=2 N = noncritical, T = trying, C = critical

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12 PLTL on Kripke models l A path in model M = (S,R,L) is a sequence such that (s i,s i +1) R. F p p p p s0s0 s1s1 s2s2 s 3...

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13 Branching time l Model of time is a tree, not a sequence l Path quantifiers AF p p p p

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14 Computation Tree Logic l Every operator F, G, X, U preceded by A or E l Universal modalities... pp p... AG p pppp p pp AF p

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15 CTL, cont... l Existential modalities p p... EG p p p EF p

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16 CTL, cont l Other modalities AX p, EX p, A(p U q), E(p U q) l Some dualities... l Examples: mutual exclusion specs... AG (C 1 C 2 )mutual exclusion AG (T 1 AF C 1 )liveness AG (N 1 EX T 1 )non-blocking

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17 CTL model checking l Model checking problem: –Determine for given M, s 0 and f, whether l Simple algorithm: –Inductive over structure of formula –Backward propagation of formula labels –O(f V(V + E))

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18 Example N1,N2 turn=0 T1,N2 turn=1 T1,T2 turn=1 C1,N2 turn=1 C1,T2 turn=1 N1,T2 turn=2 T1,T2 turn=2 N1,C2 turn=2 T1,C2 turn=2 AG (T 1 AF C 1 )

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19 CES algorithm l Need only modalities EX, EU, EG. –e.g., –Checking E(p U q) by backward BFS –Checking EG p q p BFS p SCC EG p Complexity = O(f (V + E))

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20 CTL* l Contains both CTL and LTL –path formulas p U q, G p, Fp, Xp, p, p q –state formulas A p, E p p in LTL A p in CTL* l Framework for comparing expressiveness –Existential properties not expressible in PLTL e.g., AG EF p –Fairness assumptions not expressible in CTL e.g., A (GF p GF q)

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21 Model checking complexities CTL PLTL O(2 f (V+E)) CTL O(f (V+E)) * = Note: all are linear in model size PSPACE COMPLETE

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22 8 Comparing CTL and LTL l Think of CTL formulas as approximations to LTL –AG EF p is weaker than G F p So, use CTL when it applies... –AF AG p is stronger than F G p p Good for finding bugs... Good for verifying... pp l CTL formulas easier to verify

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23 Symbolic model checking l State explosion problem –State graph exponential in program size l Symbolic model checking approach –Boolean formulas represent sets and relations –Use fixed point characterizations of CTL operators –Model checking without building state graph Sometimes can handle much larger sate space

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24 Binary Decision Diagrams (Bryant) l Ordered decision tree for f = ab + cd d ddddddd c ccc b b a

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25 OBDD reduction l Reduced (OBDD) form: 01 d c b a 0 1 Key idea: combine equivalent sub-cases

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26 OBDD properties l Canonical form (for fixed order) –direct comparison l Efficient apply algorithm –build BDDs for large circuits f g O(|f| |g|) fg l Variable order strongly affects size

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27 Boolean quantification l If v is a boolean variable, then v.f = f | v =0 V f | v =1 l Multivariate quantification w 1,w 2,…,w n ). f l Complexity on BDD representation –worst case exponential –heuristically efficient Example: b,c). (ab cd) = a d

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28 Characterizing sets l Let M = (S,R,L) be a Kripke model l Let S be the set of boolean vectors (v 1,v 2,…,v n ) {0,1} n Represent any P S by its characteristic function P P = {(v 1,v 2,…,v n ) : P } l Set operations – = false S = true – P Q = P V Q P Q = P Q – S \ P = P

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29 Characterizing relations l Transition relation R is a set of state pairs… R = {((v 1,v 2,…,v n ), (v 1,v 2,…,v n )) : R } l Examples –A synchronous sequential circuit v1v1 v0v0 R = (v 0 = v 0 ) (v 1 = v 0 v 1 )

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30 Transition relations, cont... –An asynchronous circuit s r q q –Interleaving model –Simultaneous model

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31 Forward and reverse image l Forward image P R Image(P,R)

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32 Images, cont... l Reverse image P R Image -1 (P,R) = EX P

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33 Symbolic CTL model checking l Equate a formula f with the set of states satisfying it… l Compute BDDs for characteristic functions… – p, p q, p q(use BDD ops) –EX p= Image -1 (p,R) –AX p= EX p l Remaining operators have fixed-point characterization... In fact, this is the least fixed point...

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34 Fixed points of monotonic functions Let be a function S S Say is monotonic when Fixed point of is y such that If monotonic, then it has –least fixed point y. (y) –greatest fixed point y. (y)

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35 Iteratively computing fixed points l Suppose S is finite –The least fixed point y. (y) is the limit of –The greatest fixed point y. (y) is the limit of Note, since S is finite, convergence is finite

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36 Example: EF p l EF p is characterized by l Thus, it is the limit of the increasing series... p p EX p p EX(p EX p)......which we can compute entirely using BDD operations

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37 Example: EG p l EG p is characterized by l Thus, it is the limit of the decreasing series......which we can compute entirely using BDD operations p EX p p p EX(p EX p)...

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38 Remaining operators l Allows CTL model checking with only BDD ops –Avoid building state graph –(Sometimes) avoid state explosion problem Now you can go home and build your own symbolic model checker...

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39 Example: Gigamax cache protocol l Bus snooping maintains local consistency l Message passing protocol for global consistency MPP... cluster bus MPP... global bus UIC...

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40 Protocol example l Cluster B read --> cluster A l Cluster A response --> B and main memory l Clusters A and B end shared MPP... cluster bus MPP... global bus UIC... owned copy read miss ABC

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41 Protocol correctness issues l Protocol issues –deadlock –unexpected messages –liveness l Coherence –each address is sequentially consistent –store ordering (system dependent) l Abstraction is relative to properties specified

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42 One-address abstraction l Cache replacement is nondeterministic l Message queue latency is arbitrary IN OUT ? A??? output of A may or may not occur at any given time

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43 Specifications l Absence of deadlock SPEC AG (EF p.readable & EF p.writable); l Coherence SPEC AG((p.readable & bit -> ~EF(p.readable & ~bit)); { 0 if data < n 1 otherwise bit = Abstraction:

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44 Counterexample: deadlock in 13 steps l Cluster A read --> global (waits, takes lock) l Cluster C read --> cluster B l Cluster B response --> C and main memory l Cluster C read --> cluster A (takes lock) MPP... cluster bus MPP... global bus UIC... owned copy from cluster A ABC

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45 State space explosion l State space growth is exponential

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46 BDD performance l BDD size growth is linear

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47 BDD performance l Run time growth is quadratic

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48 Why does it work?... Many partial states equivalent......implies many subfunctions equivalent... OBDD

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49 When doesnt it work? l Protocols that pass pointers l Linked lists l Anytime one part of the system knows a large amount of information about another part

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50 Summary l Model checking –Automatic verification (or falsification) of finite state systems –Linear v. branching time logics l State explosion problem –Binary Decision Diagrams –Heuristically efficient boolean operations –Image calculations –Fixed point characterization of CTL –Model checking without building state graph l Applications –Find subtle errors in complex protocols

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