Computation Tree Logic
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1 Computation Tree Logic Hao Zheng Department of Computer Science and Engineering University of South Florida Tampa, FL Phone: (813) Fax: (813) Hao Zheng (CSE, USF) Comp Sys Verification 1 / 52
2 Overview 1 Introduction (Section 6.1) 2 Computation Tree Logic (Section 6.2) CTL - Syntax CTL - Semantics CTL Semantics - Equivalences 3 CTL Model Checking (Section 6.4) 4 Fairness in CTL (Section 6.5) Hao Zheng (CSE, USF) Comp Sys Verification 2 / 52
3 Contents 1 Introduction (Section 6.1) 2 Computation Tree Logic (Section 6.2) CTL - Syntax CTL - Semantics CTL Semantics - Equivalences 3 CTL Model Checking (Section 6.4) 4 Fairness in CTL (Section 6.5) Hao Zheng (CSE, USF) Comp Sys Verification 3 / 52
4 Introduction (6.1) Linear temporal logic: Statements about (all) paths starting in a state. s = (x 20) iff for all possible paths starting in s always x 20. Quantifier is implicit. Branching temporal logic: Statements about all or some paths starting in a state. s = (x 20) iff for all paths starting in s always x 20. s = (x 20) iff for some path starting in s always x 20. Nesting of path quantifiers is allowed. Checking ϕ in LTL can be done using ϕ, but this does not work for nested formulas such as a. In any state of every computation ( ), it is possible ( ) to return to the initial state. a vs a, differenece? Hao Zheng (CSE, USF) Comp Sys Verification 4 / 52
5 Computational Tree View of Transition Systems Semantics is based Transition a branching Systems notion and of time. Trees An infinite tree of states obtained by unfolding the transition system. One time instant may have several possible successor time instants. (s 0,0) s 0 s 1 {x = 0} (s 1,1) {x 6= 0} (s 2,2) (s 3,2) s 2 s 3 {x = 0} {x = 1,x 6= 0} (s 3,3) (s 2,3) (s 3,3) (s 2,4) (s 3,4) (s 3,4) (s 2,4) (s 3,4) Hao Zheng (CSE, USF) Comp Sys Verification 5 / 52
6 Branching vs Linear Temporal Logics Incomparable expressiveness: There are properties that can be expressed in LTL, but not in CTL. There are also properties that can be expressed in CTL, but not in LTL. Distinct model-checking algorithms with different time/space complexities. Fairness assumptions require special treatment in CTL. A natural part of LTL. Equivalences and preorders between transition systems based on simulation and bisimulation relations rather than traces. Hao Zheng (CSE, USF) Comp Sys Verification 6 / 52
7 Linear-Time vs. Branching-Time Summary Aspect Linear Time Branching Time behavior path-based: state-based: in a state s trace(s) computation tree of s temporal LTL: path formulas ϕ CTL: state formulas logic s = ϕ iff existential path quantification ϕ π Paths(s). π = ϕ universal path quantification: ϕ complexity of the PSPACE complete PTIME model checking ) problems O ( TS 2 ϕ O ( TS Φ ) implementation- trace inclusion and the like simulation and bisimulation relation (proof is PSPACE-complete) (proof in polynomial time) fairness no special techniques special techniques needed Hao Zheng (CSE, USF) Comp Sys Verification 7 / 52
8 Contents 1 Introduction (Section 6.1) 2 Computation Tree Logic (Section 6.2) CTL - Syntax CTL - Semantics CTL Semantics - Equivalences 3 CTL Model Checking (Section 6.4) 4 Fairness in CTL (Section 6.5) Hao Zheng (CSE, USF) Comp Sys Verification 8 / 52
9 Computational Tree Logic - Syntax (6.2.1) Modal logic over infinite trees [Clarke & Emerson 1981]. Statements over states (Φ): a AP atomic proposition Φ and Φ 1 Φ 2 negation and conjunction ϕ there exists a path fulfilling ϕ ϕ all paths fulfill ϕ Statements over paths (ϕ): Φ the next state fulfills Φ Φ 1 U Φ 2 Φ 1 holds until a Φ 2 -state is reached Note that and U alternate with and : Φ and Φ CTL, but Φ and Φ CTL. Four operators by the syntax rules:,, U, U Check Example 6.2 in the book for some example formulas. Hao Zheng (CSE, USF) Comp Sys Verification 9 / 52
10 Derived Operators potentially Φ: Φ = (true U Φ) inevitably Φ: Φ = (true U Φ) potentially always Φ: Φ = Φ invariantly Φ: Φ = Φ weak until: (Φ 1 W Φ 2 ) = ( (Φ 1 Φ 2 ) U ( Φ 1 Φ 2 ) ) (Φ 1 W Φ 2 ) = ( (Φ 1 Φ 2 ) U ( Φ 1 Φ 2 ) ) The boolean connectives are derived as usual. Hao Zheng (CSE, USF) Comp Sys Verification 10 / 52
11 Example Properties in CTL Mutual exclusion: ( crit 1 crit 2 ) Starvation freedom: ( crit 1 ) ( crit 2 ) Each red light is preceded by a yellow light: (yellow red) Traffic light is infinitely often green: green Every request is eventually granted: (request response) In every reachable state, it is possible to return to the start state: start Hao Zheng (CSE, USF) Comp Sys Verification 11 / 52
12 CTL Semantics Visualization of Semantics 9 red 9 red 9(yellowUred) 8 red 8 red 8(yellowUred) Chris J. Myers (Lecture 6: CTL) Verification of Cyber-Physical Systems 11 / 155 Hao Zheng (CSE, USF) Comp Sys Verification 12 / 52
13 CTL Semantics - State Formulas Defined by a relation = such that s = Φ if and only if formula Φ holds in state s s = a iff a L(s) s = Φ iff (s = Φ) s = Φ Ψ iff (s = Φ) (s = Ψ) s = ϕ iff π = ϕ for some path π that starts in s s = ϕ iff π = ϕ for all paths π that start in s Hao Zheng (CSE, USF) Comp Sys Verification 13 / 52
14 CTL Semantics - Path Formulas Define a relation = such that π = ϕ if and only if path π satisfies ϕ π = Φ π = Φ U Ψ iff π[1] = Φ iff ( j 0. π[j] = Ψ ( 0 k < j. π[k] = Φ)) where π[i] denotes the state s i in the path π Hao Zheng (CSE, USF) Comp Sys Verification 14 / 52
15 CTL Semantics - Transition System Let T S = (S, Act,, I, AP, L) be a transition system. For CTL-state-formula Φ, the satisfaction set Sat(Φ) is defined by: Sat(Φ) = { s S s = Φ } TS satisfies CTL-formula Φ iff Φ holds in all its initial states: TS = Φ if and only if s 0 I. s 0 = Φ This is equivalent to I Sat(Φ). Hao Zheng (CSE, USF) Comp Sys Verification 15 / 52
16 CTL Semantics - Examples s 0 s 1 s 3 { a } { a, b } { a } s 2 { b } (a) a a a a ( a) (a U b) (a U ( a ( a U b))) Hao Zheng (CSE, USF) Comp Sys Verification 16 / 52
17 over transition systems is based on a universal quantifica statement Remark 6.10 TS = The ϕsemantics thus holdsof if and Negation only if there exists ϕ, i.e.,s 0 = ϕ. On the other hand, TS = ϕ req Considerthefollowingtransitionsystem: TS = Φ and TS = Φ is possible due to having multiple initial states, e.g., s 0 = Φ and s 0 = Φ. s 0 { a } s 0 a, whereass 0 = T S = a. aaccordingly,ts and T S = a = a and T Hao Zheng (CSE, USF) Comp Sys Verification 17 / 52
18 6.2.3 CTL Equivalence Definition 6.12 CTL-formulas Φ and Ψ (over AP) are equivalent, denoted Φ Ψ if and only if Sat(Φ) = Sat(Ψ) for all transition systems TS over AP. Φ Ψ iff (TS = Φ if and only if TS = Ψ) Hao Zheng (CSE, USF) Comp Sys Verification 18 / 52
19 Duality Laws Φ Φ Φ Φ Φ Φ Φ Φ (Φ U Ψ) ((Φ Ψ) W ( Φ Ψ)) Hao Zheng (CSE, USF) Comp Sys Verification 19 / 52
20 Expansion Laws Recall in LTL: ϕ U ψ ψ (ϕ (ϕ U ψ)) In CTL: (Φ U Ψ) Ψ (Φ (Φ U Ψ)) Φ Φ Φ Φ Φ Φ (Φ U Ψ) Ψ (Φ (Φ U Ψ)) Φ Φ Φ Φ Φ Φ Hao Zheng (CSE, USF) Comp Sys Verification 20 / 52
21 Distributive Laws (1) Recall in LTL: (ϕ ψ) ϕ ψ and (ϕ ψ) ϕ ψ In CTL: (Φ Ψ) Φ Ψ (Φ Ψ) Φ Ψ Hao Zheng (CSE, USF) Comp Sys Verification 21 / 52
22 Distributive Laws (2) Note that (Φ Ψ) Φ Ψ and (Φ Ψ) Φ Ψ. s s 00 s 0 {a} {b } s = (a b) since for all π Paths(s). π = (a b). s (s ) ω = a but s (s ) ω = b, so s = b. s (s ) ω = b but s (s ) ω = a, so s = a. Therefore, s = a b. Hao Zheng (CSE, USF) Comp Sys Verification 22 / 52
23 6.3 Equivalence of LTL and CTL Formulas Definition 6.17 CTL-formula Φ and LTL-formula ϕ (both over AP) are equivalent, denoted Φ ϕ, if for any transition system TS (over AP): TS = Φ if and only if TS = ϕ Theorem 6.18 Let Φ be a CTL-formula, and ϕ the LTL-formula obtained by eliminating all path quantifiers in Φ. Then: [Clarke & Draghicescu] Φ ϕ or there does not exist any LTL-formula that is equivalent to Φ. Hao Zheng (CSE, USF) Comp Sys Verification 23 / 52
24 LTL and CTL are Incomparable Some LTL-formulas cannot be expressed in CTL, e.g., a (a a) Some CTL-formulas cannot be expressed in LTL, e.g., a (a a) a Cannot be expressed = there does not exist an equivalent formula. Hao Zheng (CSE, USF) Comp Sys Verification 24 / 52
25 Comparing LTL and CTL (2) Comparing LTL and CTL (Lemma 6.19) a a. 8 8 a is not equivalent to a. s 0 s 1 s 2 Chris J. Myers (Lecture 6: CTL) Verification of Cyber-Physical Systems 26 / 155 Hao Zheng (CSE, USF) Comp Sys Verification 25 / 52
26 Comparing LTL and CTL (2) Comparing LTL and CTL (Lemma 6.19) a a. 8 8 a is not equivalent to a. s 0 s 1 s 2 s 0 = a but s 0 = a }{{} path s ω 0 violates it Chris J. Myers (Lecture 6: CTL) Verification of Cyber-Physical Systems 26 / 155 Hao Zheng (CSE, USF) Comp Sys Verification 25 / 52
27 Comparing LTL and CTL (1) Comparing LTL and CTL (Lemma 6.20) 8 (a ^8 (a a) is not a) equivalent (ato (a ^a). a). /0 s 3 s 2 s 1 s 0 /0 {a} {a} s 4 {a} Chris J. Myers (Lecture 6: CTL) Verification of Cyber-Physical Systems 24 / 155 Hao Zheng (CSE, USF) Comp Sys Verification 26 / 52
28 Comparing LTL and CTL (1) Comparing LTL and CTL (Lemma 6.20) 8 (a ^8 (a a) is not a) equivalent (ato (a ^a). a). /0 s 3 s 2 s 1 s 0 /0 {a} {a} s 4 {a} s 0 = (a a) but s 0 = (a a) }{{} path s 0 s 1 (s 2 ) ω violates it Chris J. Myers (Lecture 6: CTL) Verification of Cyber-Physical Systems 24 / 155 Hao Zheng (CSE, USF) Comp Sys Verification 26 / 52
29 Comparing LTL and CTL (3) The CTL-formula 8 9 a cannot be expre The CTL-formula a cannot be expressed in LTL is is shown by contradiction: assume j 8 s s 0 TS /0 {a} = 8 9 a, and thus by assumption TS = ths(ts 0 ) Paths(TS), thus TS 0 = j. Hao0Zheng (CSE, USF) Comp Sys Verification w 27 / 52
30 Contents 1 Introduction (Section 6.1) 2 Computation Tree Logic (Section 6.2) CTL - Syntax CTL - Semantics CTL Semantics - Equivalences 3 CTL Model Checking (Section 6.4) 4 Fairness in CTL (Section 6.5) Hao Zheng (CSE, USF) Comp Sys Verification 28 / 52
31 Existential Normal Form (ENF) Section The set of CTL formulas in existential normal form (ENF) is given by: Φ ::= true a Φ 1 Φ 2 Φ Φ (Φ1 U Φ 2 ) Φ For each CTL formula, there exists an equivalent CTL formula in ENF. Φ Φ (Φ U Ψ) ( Ψ U ( Φ Ψ)) Ψ Handle only Φ, Φ, and (Φ 1 U Φ 2 ). Hao Zheng (CSE, USF) Comp Sys Verification 29 / 52
32 CTL Model Checking How to check whether TS satisfies CTL formula Φ? Convert the formula Φ into the equivalent Φ in ENF. Compute recursively the set Sat(Φ) = { s S s = Φ }. TS = Φ if and only if I Sat(Φ). Recursive bottom-up computation of Sat(Φ): Consider the parse-tree of Φ. Start to compute Sat(Ψ i ), for all leafs, then go one level up in the tree and determine Sat( ) for these nodes, repeat until the root is computed. e.g., Sat( Ψ 1 Ψ }{{} 2 ) = Sat( Ψ 1 ) Sat( Ψ }{{} 2 ) }{{} node at level i node at node at level i+1 level i+1 Hao Zheng (CSE, USF) Comp Sys Verification 30 / 52
33 CTL Model Checking: An Example CTL Model Checking Example Φ = a (b U c). }{{}}{{} = 9Ψ a ^9(bU9 c) {z } Ψ }{{. {z } } Ψ 00 {z } 0 Sat(Φ) Sat(Ψ) U Sat(Ψ ) a b Sat(Ψ ) c Chris J. Myers (Lecture 6: CTL) Verification of Cyber-Physical Systems 34 / 155 Hao Zheng (CSE, USF) Comp Sys Verification 31 / 52
34 Theorem 6.23 Characterization of Sat (1) For all CTL formulas Φ, Ψ over AP it holds: Sat(true) = S Sat(a) = { s S a L(s) }, for any a AP Sat(Φ Ψ) = Sat(Φ) Sat(Ψ) Sat( Φ) = S \ Sat(Φ) Sat( Φ) = { s S Post(s) Sat(Φ) } where TS = (S, Act,, I, AP, L) is a transition system without terminal states. Hao Zheng (CSE, USF) Comp Sys Verification 32 / 52
35 Theorem 6.23 Characterization of Sat (2) Sat( (Φ U Ψ)) is the smallest subset T of S, such that: 1 Sat(Ψ) T and 2 (s Sat(Φ) and Post(s) T ) implies s T Sat( Φ) is the largest subset T of S, such that: 1 T Sat(Φ) and 2 s T implies Post(s) T where TS = (S, Act,, I, AP, L) is a transition system without terminal states. Hao Zheng (CSE, USF) Comp Sys Verification 33 / 52
36 Algorithm 14 Computation of Sat switch(φ): a : return { s S a L(s) };... : Ψ : return { s S Post(s) Sat(Ψ) }; end switch (Φ 1 U Φ 2 ) : T := Sat(Φ 2 ); compute the smallest fixed point while { s Sat(Φ 1 ) \ T Post(s) T } = do let s { s Sat(Φ 1 ) \ T Post(s) T }; T := T { s }; od; return T ; Φ : T := Sat(Φ); compute the greatest fixed point while { s T Post(s) T = } = do let s { s T Post(s) T = }; T := T \ { s }; od; return T ; Hao Zheng (CSE, USF) Comp Sys Verification 34 / 52
37 Computing Sat( (Φ U Ψ)) An Example Example Check ((p = r) (p q)) (true U ((p = r) (p q))) { r } s 5 s 6 { p, q, r } s 3 s 2 { p } s 4 { q } s 7 { p, r } { q, r } s 1 s 0 { p, q } Hao Zheng (CSE, USF) Comp Sys Verification 35 / 52
38 Computing Sat( (Φ U Ψ)) Summary Sat( (Φ U Ψ)) is the smallest set T S such that (1) Sat(Ψ) T and (2) {s T s = Φ P ost(s) T } Initially, T 0 = {Sat(Ψ)}. Iteratively compute T i+1 = T i {s Sat(Φ) P ost(s) T i } for i 0. In other words, computing Sat( (Φ U Ψ)) results in T 0 T 1... T j T j+1... Since we assume TS to be finite, there exists a j 0 such that T j = T j+1 =... = Sat( (Φ U Ψ)) Hao Zheng (CSE, USF) Comp Sys Verification 36 / 52
39 Computing Sat( Φ) An Example Example Check q { r } s 5 s 6 { p, q, r } s 3 s 2 { p } s 4 { q } s 7 { p, r } { q, r } s 1 s 0 { p, q } Hao Zheng (CSE, USF) Comp Sys Verification 37 / 52
40 Computing Sat( Φ) Summary Sat( Φ) is the largest set T S such that (1) T Sat(Φ) and (2) {s T P ost(s) T } Initially, T 0 = Sat(Φ). Then, iteratively compute T i+1 = T i {s Sat(Φ) P ost(s) T i } Thus, computing Sat( Φ) results in T 0 T 1... Since we assume TS to be finite, there exists a j 0 such that T j = T j+1 =... = Sat( Φ) Hao Zheng (CSE, USF) Comp Sys Verification 38 / 52
41 Alternative Algorithm for Computing Sat( Φ) Check Example q { r } { p, q, r } { p } { q } { p, r } { q, r } (a) { p, q } (b) K[q] (c) SCC (d) Hao Zheng (CSE, USF) Comp Sys Verification 39 / 52
42 6.4.3 Time Complexity For transition system TS with N states and M transitions, and CTL formula Φ, the CTL model-checking problem TS = Φ can be determined in time O( Φ (N + M)). This result applies to both algorithms for Φ. Hao Zheng (CSE, USF) Comp Sys Verification 40 / 52
43 Contents 1 Introduction (Section 6.1) 2 Computation Tree Logic (Section 6.2) CTL - Syntax CTL - Semantics CTL Semantics - Equivalences 3 CTL Model Checking (Section 6.4) 4 Fairness in CTL (Section 6.5) Hao Zheng (CSE, USF) Comp Sys Verification 41 / 52
44 Fairness in CTL Cannot express (fair ϕ) or (fair ϕ) in CTL. fair and ϕ are a fairness constraint and a path formula. Also, a cannot be expressed in CTL. Therefore, fairness in CTL requires special treatment. Hao Zheng (CSE, USF) Comp Sys Verification 42 / 52
45 Definition 6.32 CTL Fairness Constraints A CTL fairness constraint is an LTL formula over CTL state formulas. sfair = ( Φ i Ψ i ) ufair = 0<i k 0<i k Ψ i wfair = 0<i k ( Φ i Ψ i ) where Φ i and Ψ i (for 0 < i k) are CTL state formulas over AP. fair = ufair sfair wfair. Note that s = Φ i and s = Ψ i refer to standard (unfair!) CTL semantics. A path π in TS is fair w.r.r fair if π = fair. Hao Zheng (CSE, USF) Comp Sys Verification 43 / 52
46 Definition 6.33 Semantics of fair CTL s = fair a iff a Label(s) s = fair Φ iff (s = fair Φ) s = fair Φ Ψ iff (s = fair Φ) (s = fair Ψ) s = fair ϕ iff π = fair ϕ for some fair path π that starts in s i.e. π = fair and π = ϕ. s = fair ϕ iff π = fair ϕ for all fair paths π that start in s i.e. π = fair and π = ϕ. π = fair Φ π = fair Φ U Ψ iff π[1] = fair Φ iff ( j 0. π[j] = fair Ψ ( 0 k < j. π[k] = fair Φ)) Hao Zheng (CSE, USF) Comp Sys Verification 44 / 52
47 Definition 6.34 Transition System Semantics For CTL-state-formula Φ, and fairness assumption fair, the satisfaction set Sat fair (Φ) is defined by: Sat fair (Φ) = { s S s = fair Φ } TS satisfies CTL-formula Φ iff Φ holds in all its initial states: TS = fair Φ if and only if s 0 I. s 0 = fair Φ This is equivalent to I Sat fair (Φ). Hao Zheng (CSE, USF) Comp Sys Verification 45 / 52
48 Example 6.15 Check (a b). {b} s 3 Computation Tr s 0 s 1 s 2 {a} {a} {a} s 4 {a} TS = (a b) Figure 6.15: An example of a transition system. Cex: s 0 s 1 (s 2 s 4 ) ω Hao Zheng (CSE, USF) Comp Sys Verification 46 / 52
49 Example 6.15 Check (a b). {b} s 3 Computation Tr s 0 s 1 s 2 {a} {a} {a} s 4 {a} Let fair = ( s 2 a) ( s 2 b). Both loops Figure should 6.15: be An visited example fairly. of a transition system. TS = fair (a b) Hao Zheng (CSE, USF) Comp Sys Verification 46 / 52
50 CTL Model Checking with Fairness (basic idea) Determine Sat fair ( true) = { s S FairPaths(s) }. Introduce an atomic proposition a fair such that: a fair L(s) if and only if s Sat fair ( true) Compute the sets Sat fair (Ψ) for all subformulas Ψ of Φ (in ENF) by: Sat fair (a) = { s S a L(s) } Sat fair ( a) = S \ Sat fair (a) Sat fair (a a ) = Sat fair (a) Sat fair (a ) Sat fair ( a) = Sat ( (a a fair )) Sat fair ( (a U a )) = Sat ( (a U (a a fair ))) Sat fair ( a) = Thus: model checking CTL under fairness constraints is: CTL model checking + algorithm for computing Sat fair ( a)! Hao Zheng (CSE, USF) Comp Sys Verification 47 / 52
51 Characterization of Sat fair ( a) Given a strong fairness constraint sfair = state s, s = sfair a 0<i k ( a i b i ), and a iff there exists a finite path fragment s 0... s n and a cycle s 0... s r such that 1 s 0 = s and s n = s 0 = s r 2 s i = a, for 0 i n, and s j = a, for 0 j r, and 3 Sat(a i ) { s 0,..., s r } = or Sat(b i ) { s 0,..., s r } for 0 < i k Cycles like s 0... s r exist in SCCs. Hao Zheng (CSE, USF) Comp Sys Verification 48 / 52
52 Computing Sat fair ( a) under Strong Fairness Consider only state s if s = a, otherwise eliminate s. Change TS into TS[a] = (S, Act,, I, AP, L ) with S = Sat(a), = (S Act S ), I = I S, and L (s) = L(s) for s S Each infinite path fragment in TS[a] satisfies a. Determine all non-trivial strongly connected components in TS[a]. s = sfair a iff there exists a non-trivial SCC C reachable from s such that 0 < i k, C Sat(a i ) = or C Sat(b i ) ( ) Sat sfair ( a) = { s S Reach TS[a] (s) T } T is the union of all non-trivial SCCs satisfying (*); Notice that computing Sat fair ( a) is essential for other CTL formulas. How to compute the set T of SCCs? Hao Zheng (CSE, USF) Comp Sys Verification 49 / 52
53 Computing Sat fair ( a) under Uncond. Fairness Given an unconditional fairness constraint ufair b j, 0<i k Let T be the set union of all non-trivial SCCs C of TS[a] such that each SCC C T satisfies It now follows: 0 < i k, C Sat(b i ) s = ufair a if and only if Reach TS[a] (s) T T can be determined by a simple graph analysis (DFS). Hao Zheng (CSE, USF) Comp Sys Verification 50 / 52
54 Example 6.41 Unconditional Fairness ansition systems TS (upper) and TS (lower) in which it is implicitly tates are labeled with the atomic proposition a: Assume all states labeled with atomic proposition a: { b 1 } s 3 s 1 s 0 s 2 s 4 { b 2 } s 3 s 1 s 0 s 2 { b 1 } s 4 { b 2 } ufair = b 1 b 2 sumption, TS[a] =TS and TS [a] =TS Hao Zheng (CSE, USF) Comp Sys Verification.) Consider the unconditio 51 / 52
55 Strong Fairness sfair = a 1 b 1, i.e., k=1 s = sfair a iff C is a non-trivial SCC in TS[a] reachable from s with: (1) C Sat(b 1 ), or (2) D Sat(a 1 ) =, for some non-trivial SCC D in C. D is a non-trivial SCC in the graph that is obtained from C[ a 1 ]. For T, the union of non-trivial SCCs that satisfy (1) and (2): s = sfair a if and only if Reach G[a] (s) T For several strong fairness constraints (k > 1), this is applied recursively. T is determined by standard graph analysis (DFS). Hao Zheng (CSE, USF) Comp Sys Verification 52 / 52
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