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Tree Regular Model Checking P. Abdulla, B. Jonsson, P. Mahata and J. d’Orso Uppsala University

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Presentation Overview Aim Regular model checking Trees, tree relations Transitive closures Results, conclusions

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Aim With counters With stacks With channels parameterized A uniform analysis framework for systems : These systems can be characterized by finite-state automata.

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Regular model checking Most important operation: computing transitive closures. Why ? Because it allows many analysis: Reachability safety properties fairness properties

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Example A simple token passing protocol:

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Example A simple token passing protocol:

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Example A simple token passing protocol:

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Example A simple token passing protocol:

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Our Contribution Regular Model Checking is nice, but… … it’s only valid for linear or circular topologies ! Idea: extend to trees instead !

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Trees 01 00 01 10 010 a b c d a b d node label this node is called “root”

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Tree Automata 01 or and or and q1q2 state input symbol

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Tree Automata Input: and or 101 Run:

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Tree Automata Input: and or 101 Run: q1 Transition: 0 q1

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Tree Automata Input: and or 101 Run: q1q2 Transition: 1 q2

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Tree Automata Input: and or 101 Run: q1 q2 Transition: or q2 q1

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Tree Automata Input: and or 101 Run: q1 q2 Transition: and q2

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Tree Automata Input: and or 101 Run: q1 q2 Accept ! q2

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Tree Relations a bc d ef

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We represent a pair of “similar” trees by a tree on an alphabet with pairs: a bc d ef (a,d) (b,e)(c,f)

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History Automata x Tx input

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History Automata x Tx T xx input intermediate

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History Automata x Tx x xT T xx inputoutput intermediate

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History Automata In (word) regular model checking, concept of columns Transpose this to trees: represent a sequence of runs into single run. x Tx q1 q2q3 x xT q4 q5q6 T xx inputoutput intermediate run 1run 2

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History Automata In (word) regular model checking, concept of columns Transpose this to trees: represent a sequence of runs into single run. x Tx x xT inputoutput run 1+2 q1.q4 q2.q5q3.q6

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Symbolic Automata Property of H.A.’s: accept the transitive closure. Problem: infinite size ! Approach: apply standard subset construction (determinization) Supporting data structure for sets: regular expressions.

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Results We have run a prototype tool on several examples: Token-passing protocol (1&2 ways) “percolate” protocol (compute disjunctions) Tree arbiter (mutual exclusion) Paper accepted at CAV’02.

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Future work Change underlying automata representation (matrices instead of BDDs) Expand to new types of systems (e.g. pushdown systems) Create a graphical interface

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