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Black Box Checking Book: Chapter 9

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Model Checking Finite state description of a system B. LTL formula. Translate into an automaton P. Check whether L(B) L(P)=. If so, S satisfies. Otherwise, the intersection includes a counterexample. Repeat for different properties.

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Buchi automata ( -automata) S - finite set of states. (B has l n states) S 0 S - initial states. (P has m states) - finite alphabet. (contains p letters) S S - transition relation. F S - accepting states. Accepting run: passes a state in F infinitely often. System automata: F=S, deterministic.

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Example: check a a, a a a <> a

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Example: check <> a a a a a <> a

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Example: check <>a Use automatic translation algorithms, e.g., [Gerth,Peled,Vardi,Wolper 95] a a a, a <> ~a

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System cb a

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Every element in the product is a counter example for the checked property. b a a a a s2s2 c a s1s1 s3s3 q2q2 q1q1 s 1,q 1 s 1,q 2 s 3,q 2 s 2,q 1 a b c a Acceptance is determined by automaton P.

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Testing Unknown deterministic finite state system B. Known: n states and alphabet. An abstract model C of B. C satisfies all the properties we want from B. Check conformance of B and C. Another version: only a bound n on the number of states l is known.

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Model Checking / Testing Given Finite state system B. Transition relation of B known. Property represent by automaton P. Check if L(B) L(P)=. Graph theory or BDD techniques. Complexity: polynomial. Unknown Finite state system B. Alphabet and number of states of B or upper bound known. Specification given as an abstract system C. Check if B C. Complexity: polynomial if number states known. Exponential otherwise.

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Black box checking Property represent by automaton P. Check if L(B) L(P)=. Graph theory techniques. Unknown Finite state system B. Alphabet and Upper bound on Number of states of B known. Complexity: exponential.

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Combination lock automaton Accepts only words with a specific suffix (cdab in the example). s1s1 s2s2 s3s3 s4s4 s5s5 bdca

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Conformance testing Cannot distinguish if reduced or not. a b a a b b a b a b

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Conformance testing (cont.) When the black box is nondeterministic, we might never test some choices. b a a

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Conformance testing (cont.) a bb a a a a b b b a Need: bound on number of states of B. a

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Need reliable RESET s1s1 s3s3 s2s2 a a a b b

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Vasilevskii algorithm Known automaton A has l states. Black box automaton has up to n states. Check each transition. Check that there are no "combination lock" errors. Complexity: O(l 2 n p n-l+1 ). When n=l: O(l 3 p).

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Experiments aa bb cc reset a a b b c c try b a a b b c c try c fail

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Simpler problem: deadlock? Nondeterministic algorithm: guess a path of length n from the initial state to a deadlock state. Linear time, logarithmic space. Deterministic algorithm: systematically try paths of length n, one after the other (and use reset), until deadlock is reached. Exponential time, linear space.

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Deadlock complexity Nondeterministic algorithm: Linear time, logarithmic space. Deterministic algorithm: Exponential (p n-1 ) time, linear space. Lower bound: Exponential time (use combination lock automata). How does this conform with what we know about complexity theory?

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Modeling black box checking Cannot model using Turing machines: not all the information about B is given. Only certain experiments are allowed. We learn the model as we make the experiments. Can use the model of games of incomplete information.

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Games of incomplete information Two players: player, player (here, deterministic). Finitely many configurations C. Including: Initial C i, Winning : W + and W -. An equivalence relation on C (the player cannot distinguish between equivalent states). Labels L on moves (try a, reset, success, fail). The player has the moves labeled the same from configurations that are equivalent. Strategy for the player: will lead to a configuration in W + W -. Cannot distinguish equivalent conf. Nondet. strategy: ends with W +. Can distinguish.

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Modeling BBC as games Each configuration contains an automaton and its current state (and more). Moves of the player are labeled with try a, reset... Moves of the -player with success, fail. c 1 c 2 when the automata in c 1 and c 2 would respond in the same way to the experiments so far.

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A naive strategy for BBC Learn first the structure of the black box. Then apply the intersection. Enumerate automata with n states (without repeating isomorphic automata). For a current automata and new automata, construct a distinguishing sequence. Only one of them survives. Complexity: O((n+1) p (n+1) /n!)

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On-the-fly strategy Systematically (as in the deadlock case), find two sequences v 1 and v 2 of length <=m n. Applying v 1 to P brings us to a state t that is accepting. Applying v 2 to P brings us back to t. Apply v 1 (v 2 ) n+1 to B. If this succeeds, there is a cycle in the intersection labeled with v 2, with t as the P (accepting) component. Complexity: O(n 2 p 2mn m).

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Learning an automaton Use Angluins algorithm for learning an automaton. The learning algorithm queries whether some strings are in the automaton B. It can also conjecture an automaton M i and asks for a counterexample. It then generates an automaton with more states M i+1 and so forth.

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A strategy based on learning Start the learning algorithm. Queries are just experiments to B. For a conjectured automaton M i, check if M i P = If so, we check conformance of M i with B (Vasilevskii algorithm). If nonempty, it contains some v 1 (v 2 ). We test B with v 1 (v 2 ) n+1. If this succeeds: error, otherwise, this is a counterexample for M i.

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Black Box Checking Strategy Incremental learning Comparing counterexample Model Checking Report error No error found black box testing counterexample no counterexample false negative actual error discrepancy conformance established System PathModel

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Complexity l - real size of B. n - an upper bound of size of B. p - size of alphabet. Lower bound: reachability is similar to deadlock. O(l 3 p l + l 2 mn) if there is an error. O(l 3 p l + l 2 n p n-l+1 + l 2 mn) if there is no error. If n is not known, check while time allows.

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Some experiments Basic system written in SML (by Alex Groce, CMU). Experiment with black box using Unix I/O. Allows model-free model checking of C code with inter-process communication. Compiling tested code in SML with BBC program as one process.

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Conclusions Black box checking is a combination of testing and model checking. If a tight bound on size of B is given: learn B first, then do model checking. Tight lower bound on complexity, up to polynomial factor. Use of games of incomplete information to model testing problems.

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