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Compatibility between shared variable valuations in timed automaton network model- checking Zhao Jianhua, Zhou Xiuyi, Li Xuandong, Zheng Guoliang Presented by ZHAO Jianhua

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Background (Time Automata) A timed automaton can be viewed as a conventional finite state automaton plus some clock variables ， which are used to constraint time distances between events. AB Clocks: x, y E1:x < 5, y := 0 E2: y < 8, x := 0 x < 5y < 8

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Background (timed automaton network) A timed automaton network is a finite set of timed automata which interact with each other. These timed automata may interact with each other through a finite set of shared variables. For each timed automaton network, an equivalent timed automaton can be built.

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Background (timed automaton network) An example: AB Clocks: x E11:x < 5, x:=0 v:=1 E12: x < 8, x := 0 v==0 x < 5x < 8 12 E21:y < 8, y:=0 v==1 E12: y < 3, y := 0 v:=0 y<8 y < 3 Clocks: y

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Background (reachability analysis 1) Many interesting properties (for example, safety) can be expressed as reachability of locations of timed automata. Because the state spaces of timed automata are infinite, model checking techniques can not be applied to timed automaton directly. –Symbolic representation of states are used in automatically reachability analysis.

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Background (Symbolic States) A symbolic state of a timed automaton network is a tuple (l,s, D) –l is the global location of the network. –s is the valuation of the set of shared variables. –D is a conjunction of formulas like x-y<c. A symbolic state (l,s, D ) represents a set of concrete states (l,s,v), where v satisfies D. Given a symbolic state S, the set of concrete states which are reachable from a concrete state in S through a given transition t can also be represented as a symbolic state. We call it as the successor of S w.r.t. t.

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Background (Basic reachability analysis algorithm 1) Wait = { S 0 }, Passed = {}, where S 0 is the initial symbolic state while (Wait != {} ) do { S = a symbolic state in Wait; Wait = Wait – {S} for each transition t leaving S do {S’ = successor of S w.r.t. t; if (S’!= Φ and S’ is not contained by any state in Passed) Wait = Wait + {S’} if (the location of S’ is the target location) return true; } Passed = Passed + {S} }

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Background (Basic reachability analysis algorithm 2) The algorithm explores the state space by generating successors of generated states continuously. The algorithm will not generated the successors of a generated symbolic state (l,s, D 1 ) only if –another symbolic state (l, s, D 2 ) containing (l,s, D 1 ) has already been generated. –a symbolic state S 1 contains another one S 2, if the set of concrete states represented by S 1 contains the one represented by S 2.

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Compatibility between shared variable valuations A shared variable valuations s 1 is compatible with s 2 on a tuple (l,D) if for each transition e leaving l, one of the following conditions holds. –s 1 and s 2 are identical. –The conjunction of D and g is false, where g is the time guard of e. –Neither s 1 nor s 2 satisfies the shared variable guards of e. –The variable guard of e is satisfied by s 1, and the transition e sets s 1 and s 2 to two compatible variable valuations.

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An example of Compatibility (v1 = 3; v2 = 3) is compatible with (v1 = 2; v2 = 3) on ((A,M), (x > 3 ^ y < 10)) A B Clocks: x MN Clocks: y Shared variables: v1, v2 B C e11 : x > 5; v2 = 3 x:=0, v1:=0 e12 : x < 3; v1 = 3 x:=0, v1:=v1+1 e21 : y < 10; v1:=v2+1, y:= 0

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Compatibility contain Definition 3. Let (l, s 1, D 1 ) and (l, s 2, D 2 ) be two symbolic states of a timed automaton network. We say (l, s 1, D 1 ) compatibility contains (l, s 2,D 2 ) –if s 1 is compatible with s 2 on (l, D 1 ) and –D 1 contains D 2.

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A lemma about the compatibility contain Lemma –Let S 1, and S 2 be two symbolic states of a timed automaton network. We have that all the locations reachable from S 2 are also reachable from S 1 if S 1 compatibility contains S 2. Intuitively, (l, s 1, D 1 ) is more like to reach the target location than (l, s 2, D 2 ) is. The algorithm can avoid generating successors of a generated symbolic state (l, s, D 1 ) if –another symbolic state which compatibility-contains (l, s, D) has already been generated. This condition is weaker than the basic one.

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Find the compatible valuations During the reachability analysis, if a symbolic state (l,s,D) is generated, an algorithm can be used to find valuations with which s is compatible on (l,D). This algorithm uses a backward propagation method to compute such valuations based on the definition of compatibility. All these valuations are recorded in valuation sets attached to the generated states. For each generated state (l, s’,D’), it is compatibility contained by (l,s,D) if D’ is contained by D and s is found to be compatible with s’.

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A compact data structure Let v 1, v 2, …, v n be a set of shared variables. We proved that the attached valuation sets can be represented as Cartesian products s 1 × s 2 × … × s n This observation leads to a compact data structure to record the compatible shared variable valuations.

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The optimization The algorithm is optimized as follows –A shared variable valuation set is attached to each generated state. (using the compact data structure) –Avoid generating successor of (l, s, D) if there is another generated state (l, s’, D’) such that s is in the attached set of (l, s’, D’) and D’ contains D –During the reachability analysis, the attached sets are continuously expanded by backward propagation.

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The performance (1) (The bounded retransmission protocol)

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The performance (2) (the Bang&Olufsion audio protocol) The optimized algorithm uses only about 40% memories as the original one does.

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