Model Checking Basics Dr. Eng. Amr T. Abdel-Hamid Netw 703 Winter 2012 Slides based on slides of: K. Havelund & Agroce, Reliable Software: Testing and Monitoring, CMU. E. Clarke, Formal Methods, to be updated by course name S. Tahar, E. Cerny and X. Song, Formal Verification of Systems. Network Protocols Dr. Amr Talaat Netw 703 Network Protocols Branching Time Temporal Logic (BTTL) Structure of time: an infinite tree, each instant may have many successor insta nts Along each path in the tree, the corresponding timeline is isomorphic to N State quantifiers: Xp, Fp, Gp, pUq (like in linear temporal logic) Path quantifiers: for All paths (A) and there Exists a path (E) from a given state A = E = In linear time logic, temporal operators are provided for describing events alon g a single future, however, when a linear formula is used for specification, ther e is usually an implicit universal quantification over all possible futures (linear tr aces) In contrast, in branching time logic the operators usually reflect the branching n ature of time by allowing explicit quantification over possible futures in any stat e Dr. Amr Talaat Netw 703 Network Protocols CTL Formulas are constructed from path quantifiers and temp oral operators: Path quantifier: A: for every path E: there exists a path Temporal Operator: F: holds sometime in the future X: holds next time G: holds globally in the future p U q: p holds until q hold i.e. AF, AG, AX, A( U ), EF, EG, EX, E( U ) Dr. Amr Talaat Netw 703 Network Protocols CTL Dr. Amr Talaat Netw 703 Network Protocols CTL: Computation Tree Logic Dr. Amr Talaat Netw 703 Network Protocols CTL Example Structure M : A Kripke structure: triple M = S: set of states R S S: transition relation L: S 2 AP : (Truth valuation) set of atomic propositions true in ea ch state S = {1,2,3,4,5}, AP = {a,b,c}, R = {(1,2), (2,3), (5,3), (5,5), (5,1), (2,4), (4,2), (1,4), (3,4)} L(1) = {b}, L(2) = {a}, L(3) = {a,b,c}, L(4) = {b,c}, L(5) = {c} 6/80 Dr. Amr Talaat Netw 703 Network Protocols Example: Two input Muller C-element (assuming finite discrete delays): Specification in CTL: Liveness: If inputs remain equal, then eventually the output will c hange to this value. AG( A( ( a=0 b=0 ) U ( out=0 a=1 b=1 ) ) ) AG( A( ( a=1 b=1 ) U ( out=1 a=0 b=0 ) ) ) Safety: If all inputs and the output have the same value then the output should not change until all inputs change their values. AG( ( a=0 b=0 out=0 ) A( out=0 U (a=1 b=1 ) ) ) AG( ( a=1 b=1 out=1 ) A( out=1 U (a=0 b=0 ) ) ) 7/80 Dr. Amr Talaat Netw 703 Network Protocols Model Checking Problem Given an FSM M (equivalent Kripke structure) and a tem poral logic formula p, does M define a model of p? Determine the truth of a formula with respect to a give n (initial) state in M Find all states s of M such that (M, s) p For any propositional temporal logic, the model chec king problem is decidable: exhaustive search of all pat hs through the finite input structure 8/80 Dr. Amr Talaat Netw 703 Network Protocols Model Checking Problem 9/80 Dr. Amr Talaat Netw 703 Network Protocols Theoretical Background Theorem [Wolper, 1986]: The model checking for CTL is in deterministic polynomial time Theorem [Sistla & Clark, 1985]: The model checking pr oblem for PLTL is PSPACE complete Theorem [Emerson & Lei, 1987]: Given any model-chec king algorithm for a linear logic LTL, there is a model che cking algorithm for the corresponding branching logic BT L, whose basic modalities are defined by the LTL, of the s ame order of complexity Theorem [Clark, Emerson & Sistla, 1986]: The model checking problem for CTL* is PSPACE-complete 10/80 Dr. Amr Talaat Netw 703 Network Protocols PSPACE-complete 11/80 Dr. Amr Talaat Netw 703 Network Protocols Fixpoints Algorithm Model Checking Algorithms Original algorithm described in terms of labeling the CTL structur e (Clark83) Required explicit representation of the whole state space Better algorithm based on fixed point calculations Algorithm amenable to symbolic formulation Symbolic evaluation allows implicit enumeration of states Significant improvement in maximum size of systems that can be verified Dr. Amr Talaat Netw 703 Network Protocols 13 Symbolic model checking Symbolic model checking approach Boolean formulas represent sets and relations Use fixed point characterizations of CTL operators Model checking without building complete state graph CTL model checking problem can be solved in O(|f| (|S|+|R|)) Note that the complexity is linear in the size of the formula and the transition system Recall that the size of the transition system is exponential in the num ber of variables (this is called the state space explosion problem) Dr. Amr Talaat Netw 703 Network Protocols CTL Model Checking Algorithm Translate the formula to a formula which uses the basis EX p, EG p, E(p U q) Start from the innermost subformulas Label the states in the transition system with the subformulas that hold in that state Initially states are labeled with atomic properties Each (temporal or boolean) operator has to be processed once Processing of each operator takes O(|S|+|R|) Dr. Amr Talaat Netw 703 Network Protocols CTL Model Checking Algorithm Boolean operators are easy p : Each state which is not labeled with p should be labeled with p p q : Each state which is labeled with both p and q should be la beled with p q p q : Each state which is labeled with p or q should be labeled with p q Dr. Amr Talaat Netw 703 Network Protocols CTL Model Checking Algorithm: EX p EX p is easy to do in O(|S|+|R|) All the nodes which have a next state labeled with p should be label ed with EX p s2s1s4s3 pp p, EX p EX p s2s1s4s3 Dr. Amr Talaat Netw 703 Network Protocols CTL Model Checking Algorithm: E(pUq) E(pUq): Find the states which are the source of a path where p U q holds Find the nodes that reach a node that is labeled with q by a path where each node is labeled with p Label such nodes with E(pUq) It is a reachability problem which can be solved in O(|S|+|R|) First label the nodes which satisfy q with E(pUq) For each node labeled with E(pUq), label all its pre vious states that are labeled with p with E(p U q) Dr. Amr Talaat Netw 703 Network Protocols CTL Model Checking Algorithm: E(pUq) s2s1s4s3 pp p, E(pUq) s2s1s4s3 q q, E(pUq) Dr. Amr Talaat Netw 703 Network Protocols CTL Model Checking Algorithm: EG p EG p: Find infinite paths where each node on the path is label ed with p, and label nodes in such paths with EG p First remove all the states which do not satisfy p from the transitio n graph Compute the strongly connected components of the remaining gr aph, and then find the nodes which can reach the strongly conne cted components (both of which can be done in O(|S|+|R|) Label the nodes in the strongly connected components and the n odes that can reach the strongly connected components with EG p Dr. Amr Talaat Netw 703 Network Protocols CTL Model Checking Algorithm: EG p s2s1s4s3 pp p, EG p s2s1s4s3 p p, EG p s2s4s3 pp p A strongly connected component Dr. Amr Talaat Netw 703 Network Protocols Verification vs. Falsification Verification: Show: initial states truth set of p Falsification: Find: a state initial states truth set of p Generate a counter-example starting from that state CTL model checking algorithm can also generate a counter-ex ample path if the property is not satisfied without increasing the complexity The ability to find counter-examples is one of the biggest stren gths of the model checkers Dr. Amr Talaat Netw 703 Network Protocols Temporal Properties Fixpoints [Emerson and Clarke 80] Here are some interesting CTL equivalences: AG p = p AX AG p EG p = p EX EG p AF p = p AX AF p EF p = p EX EF p A(p U q) = q (p AX A(p U q)) p EU q = q (p EX E(p U q)) Note that we wrote the CTL temporal operators in terms of thems elves and EX and AX operators Dr. Amr Talaat Netw 703 Network Protocols Functionals Given a transition system M=(S, L, R), we will define functions (Properties) from sets of states to sets of states F : 2 S 2 S For example, one such function is the EX operator (which com putes the precondition of a set of states) EX : 2 S 2 S which can be defined as: EX(p) = { s | (s,s) R and s p } Image calculation Dr. Amr Talaat Netw 703 Network Protocols CTL Example Structure M : A Kripke structure: triple M = S: set of states R S S: transition relation L: S 2 AP : (Truth valuation) set of atomic propositions true in ea ch state S = {1,2,3,4,5}, AP = {a,b,c}, R = {(1,2), (2,3), (5,3), (5,5), (5,1), (2,4), (4,2), (1,4), (3,4)} L(1) = {b}, L(2) = {a}, L(3) = {a,b,c}, L(4) = {b,c}, L(5) = {c} 24/80 Dr. Amr Talaat Netw 703 Network Protocols Fixpoint Characterizations Fixpoint CharacterizationEquivalences AG p = y. p AX y AG p = p AX AG p EG p = y. p EX y EG p = p EX EG p AF p = y. p AX y AF p = p AX AF p EF p = y. p EX y EF p = p EX EF p A(p U q) = y. q (p AX (y)) p AU q=q (p AX (p AU q)) E(pUq) = y. q (p EX (y)) p EU q = q (p EX (p EU q)) Dr. Amr Talaat Netw 703 Network Protocols Functionals Now, we can think of all temporal operators also as functions fr om sets of states to sets of states For example: AX p = EX( p) or if we use the set notation AX p = (S - EX(S - p)) LogicSet p q p q p q p q p S p False TrueS Dr. Amr Talaat Netw 703 Network Protocols Lattice The set of states of the transition system forms a lattice: lattice2 S partial order bottom element top elementS Least upper bound (lub) Greatest lower bound (glb) Dr. Amr Talaat Netw 703 Network Protocols Lattice (lub): y P is a least upper bound of S in P means y is an upper bound of S and z P which is an upper bound of S, y z (glb) y P is a greatest lower bound of S in P means y is a lower bound of S and z P which is a lower bound of S, z y 28/80 Dr. Amr Talaat Netw 703 Network Protocols Temporal Properties Fixpoints Based on the equivalence EF p = p EX EF p EF p is a fixpoint of function (F): where F =>F y = p EX y F (EF p) = EF p In fact, EF p is the least fixpoint of F, which is written as: EF p = y. p EX y ( means least fixpoint) Dr. Amr Talaat Netw 703 Network Protocols EF Fixpoint Computation p EF(p)states that can reach p p EX(p) EX(EX(p))... EF(p) states that can reach p p EX(p) EX(EX(p)) ... EF(p) Dr. Amr Talaat Netw 703 Network Protocols Temporal Properties Fixpoints Based on the equivalence EG p = p AX EG p EG p is a fixpoint of function (F): where F => F y = p EX y F (EG p) = EG p In fact, EG p is the greatest fixpoint of F, which is written as: EG p = y. p EX y ( means greatest fixpoint) Dr. Amr Talaat Netw 703 Network Protocols EG Fixpoint Computation EG(p) EG(p) states that can avoid reaching pp EX(p) EX(EX(p))... EG(p) states that can avoid reaching p p EX(p) EX(EX(p)) ... Dr. Amr Talaat Netw 703 Network Protocols Least Fixpoint Given a monotonic function F, its least fixpoint is the greatest low er bound (glb) of all the reductive elements : y. F y = { y | F y y } The least fixpoint y. F y is the limit of the following sequence (a ssuming F is -continuous): , F , F 2 , F 3 ,... If S is finite, then we can compute the least fixpoint using the abo ve sequence Dr. Amr Talaat Netw 703 Network Protocols EF Fixpoint Computation EF p = y. p EX y is the limit of the sequence: , p EX , p EX(p EX ), p EX(p EX(p EX )),... which is equivalent to , p, p EX p, p EX (p EX (p) ),... Dr. Amr Talaat Netw 703 Network Protocols EF Fixpoint Computation s2s1s4s3 p p Start 1 st iteration p EX = {s1,s4} EX( )= {s1,s4} ={s1,s4} 2 nd iteration p EX(p EX ) = {s1,s4} EX({s1,s4})= {s1,s4} {s3}={s1,s3,s4} 3 rd iteration p EX(p EX(p EX )) = {s1,s4} EX({s1,s3,s4})= {s1,s4} {s2,s3,s4}={s1,s2,s3,s4} 4 th iteration p EX(p EX(p EX(p EX ))) = {s1,s4} EX({s1,s2,s3,s4})= {s1,s4} {s1,s2,s3,s4} = {s1,s2,s3,s4} Dr. Amr Talaat Netw 703 Network Protocols EF Fixpoint Computation p EF(p)states that can reach p p EX(p) EX(EX(p))... EF(p) states that can reach p p EX(p) EX(EX(p)) ... EF(p) Dr. Amr Talaat Netw 703 Network Protocols Greatest Fixpoint Given a monotonic function F, its greatest fixpoint is the least upp er bound (lub) of all the extensive elements: y. F y = { y | F y y } The greatest fixpoint y. F y is the limit of the following sequenc e (assuming F is -continuous): S, F S, F 2 S, F 3 S,... If S is finite, then we can compute the greatest fixpoint using the above sequence Dr. Amr Talaat Netw 703 Network Protocols EG Fixpoint Computation Similarly, EG p = y. p EX y is the limit of the sequence: S, p EX S, p EX(p EX S), p EX(p EX (p EX S)),... which is equivalent to S, p, p EX p, p EX (p EX (p) ),... Dr. Amr Talaat Netw 703 Network Protocols EG Fixpoint Computation s2s1s4s3 pp p Start S = {s1,s2,s3,s4} 1 st iteration p EX S = {s1,s3,s4} EX({s1,s2,s3,s4})= {s1,s3,s4} {s1,s2,s3,s4}={s1,s3,s4} 2 nd iteration p EX(p EX S) = {s1,s3,s4} EX({s1,s3,s4})= {s1,s3,s4} {s2,s3,s4}={s3,s4} 3 rd iteration p EX(p EX(p EX S)) = {s1,s3,s4} EX({s3,s4})= {s1,s3,s4} {s2,s3,s4}={s3,s4} Dr. Amr Talaat Netw 703 Network Protocols EG Fixpoint Computation EG(p) EG(p) states that can avoid reaching pp EX(p) EX(EX(p))... EG(p) states that can avoid reaching p p EX(p) EX(EX(p)) ... Dr. Amr Talaat Netw 703 Network Protocols Example Structure M : S = {1,2,3,4,5}, AP = {a,b,c}, R = {(1,2), (2,3), (5,3), (5,5), (5,1), (2,4), (4,2), (1,4), (3,4)} L(1) = {b}, L(2) = {a}, L(3) = {a,b,c}, L(4) = {b,c}, L(5) = {c} Check if AG(a c) holds 41/80 Dr. Amr Talaat Netw 703 Network Protocols Example (cont.) Remember that: AG p = y. p AX y AX p = EX( p) H(a c) = H(a) H(c) ={2,3} {3,4,5} = {2,3,4,5} I0 S = {1,2,3,4,5} I1 {2,3,4,5} S = {2,3,4,5} {1,2,3,4,5} = {2,3,4,5} I2 {2,3,4,5} AX(2,3,4,5) = {2,3,4,5} {1,2,3,4} = {2,3,4} I3 {2,3,4,5} AX(2,3,4) = {2,3,4,5} {1,2,3,4} = {2,3,4} I3 = I2 H(AG(a c)) = {2,3,4} To verify that f holds in state s, check if s H(f) 42/80 Dr. Amr Talaat Netw 703 Network Protocols Example 43/ a,b c b,c a dc For the FSM below, formally check the following properties, using Fixpoint Theorm: AG(a c b) AF(a b) If failed show the subset of the design the property holds for as well as the counter example S = {1,2,3,4,5,6}, AP = {a,b,c,d}, R = {(1,2), (1,3),(2,3), (3,4), (4,4), (4,5), (5,2), (2,6), (6,1)} L(1) = {a,b}, L(2) = {c}, L(3) = {b,c}, L(4) = {a}, L(5) = {c}, L(6) = {d} Dr. Amr Talaat Netw 703 Network Protocols Example (cont.) Remember that: H(a b) = H(a) H(b) H(c) ={1,4} {2,3,5} {1,3} = {1,2,3,4,5} AG(a c b) = AG p = y. p AX y = y. p AX y AX p = EX( p) I0 S = {1,2,3,4,5,6} I1 {1,2,3,4,5} S = {1,2,3,4,5} {1,3,4,5,6} = {1,2,3,4,5} I2 {1,2,3,4,5} AX(1,2,3,4,5) = {1,2,3,4,5} {1,3,4,5,6} = {1,3,4,5} This is because that : AX(1,2,3,4,5) = EX( (1,2,3,4,5)) = EX(6) = (2) = S- {2 } = {1,3,4,5,6} I3 {1,2,3,4,5} AX(1,3,4,5) = {1,3,4,5} This is because that : AX(1,3,4,5) = EX( (1,3,4,5)) = EX(2,6) = (2,6) = S- {2,6} = {1,3,4,5} I3 = I2 H(AG(a b c)) = {1,3,4,5} The property does not hold, except for the above states, and it is clear that s tates {2,6} can be considered as counter examples. state 6 does not contain neither a,c,b and state 2 does not have a proceedin g one on one of its pathes path (2,6) 44/80 Dr. Amr Talaat Netw 703 Network Protocols Example (AF(ab)) Remember that: H(a b) = H(a) H(b) ={1,4} {1,3}= {1} AF(a b) = AG p = y. p AX y = y. p AX y AX p = EX( p) I0 I1 {1} AX( ) = {1} = This is because that : AX( ) = EX( ( )) = EX(1,2,3,4,5,6) = The property does not hold for any state. 45/80 Dr. Amr Talaat Netw 703 Network Protocols Model Checking Tools (non-commercial) SMV (Symbolic Model Verifier) A tool for checking finite state systems against specifications in the temporal logic CTL. Developed at Carnegie Mellon University by E. Clarke, K. McMillan et. al. Supports a simple input language: SMV For more information: Cadence SMV Updated version of SMV by K. McMillan at Berkeley Cadence Labs Input languages: extended SMV and synchronous Verilog Supports temporal logics CTL and LTL, finite automata, embedded assertions, an d refinement specifications. Features compositional reasoning, link with a simple theorem prover, an easy-to-u se graphical user interface and source level debugging capabilities For more information:46/80 Dr. Amr Talaat Netw 703 Network Protocols Model Checking Tools (commercial) FormalCheck Supports the synthesizable subsets of Verilog and VHDL hardware d esign languages. User supplies FormalCheck with a set of queries (properties and con straints) Each property is defined using semantics of the class of omega auto mata. Tool provides powerful model reduction options. For more information:ck.html 47/80 Dr. Amr Talaat Netw 703 Network Protocols 48 Model Checking Automatic verification (or falsification) of finite state systems Explicit State Representation State Explosion Pr oblem (about 10 8 states maximum) Breakthrough: Implicit State Representation using ROBDD (about states). Use Boolean characteristic functions represented by RO BDDs to encode sets of states and transition relations. Dr. Amr Talaat Netw 703 Network Protocols Binary Decision Diagrams Ordered binary decision diagrams (OBDDs) are a canonical fo rm for Boolean formulas. OBDDs are often substantially more compact than traditional n ormal forms. Moreover, they can be manipulated very efficiently. Introduced at: R. E. Bryant. Graph-based algorithms for boolean function manip ulation. IEEE Transactions on Computers, C-35(8), 1986. Dr. Amr Talaat Netw 703 Network Protocols Binary Decision Trees A Binary decision tree is a rooted, directed tree with two types of vertices, terminal vertices and nonterminal vertices. Each nonterminal vertex v is labeled by a variable var(v) and h as two successors: low (v) corresponding to the case where the variable is assign ed 0, and high (v) corresponding to the case where the variabl e is assigned 1. Each terminal vertex v is labeled by value(v) which is either 0 or 1 Dr. Amr Talaat Netw 703 Network Protocols Example BDT for a two-bit comparator, f(a1,a2,b1,b2) = (a1 b1) (a 2 b2) Dr. Amr Talaat Netw 703 Network Protocols Binary Decision Diagram i.e. exactly like decision TREE Dr. Amr Talaat Netw 703 Network Protocols Reduced Ordered BDDs In practical applications, it is desirable to have a canonical repr esentation for Boolean functions. This simplifies tasks like checking equivalence of two formulas and deciding if a given formula is satisfiable or not. Such a representation must guarantee that two Boolean functi ons are logically equivalent if and only if they have isomorphic representations. Dr. Amr Talaat Netw 703 Network Protocols Reduced Ordered BDD Canonical Form property A canonical representation for Boolean functions is desirable: two Boolean functions are logically equivalent iff they have isomo rphic representations This simplifies checking equivalence of two formulas and deciding if a formula is satisfiable Two BDDs are isomorphic if there exists a bijection h between the g raphs such that Terminals are mapped to terminals and nonterminals are mapped to nonterminals For every terminal vertex v, value(v) = value(h(v)), and For every nonterminal vertex v: var(v) = var(h(v)), h(low(v)) = low( h(v)), and h(high(v)) = high(h(v)) Dr. Amr Talaat Netw 703 Network Protocols Canonical Form property Bryant (1986) showed that BDDs are a canonical repr esentation for Boolean functions under two restrictions: the variables appear in the same order along each path from th e root to a terminal there are no isomorphic subtrees or redundant vertices Dr. Amr Talaat Netw 703 Network Protocols Reduced Ordered Binary Decision Diagr ams (ROBDDs): CREATION Canonical Form Property Requirement (1): Impose total order