Model Checking Revision: Model Checking for Infinite Systems Revision: Traffic Light Controller (TLC) Revision: 1.12
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1 Model Checking mc Revision:.2 Model Checking for Infinite Systems mc 2 Revision:.2 check algorithmically temporal / sequential properties fixpoint algorithms with symbolic representations: systems are originally finite state timed automata (clocks) simple model: finite state automaton hybrid automata (differential equations) termination guaranteed if finite quotient structure exists comparison of automata can be seen as model checking check that the output streams of two finite state systems match simply run model checker for some time, e.g. Java Pathfinder process algebra: simulation and bisimulation checking run time verification. example: add checker synthesized from temporal spec temporal logics as specification mechanism 2. example: run all schedules for one test case safety, liveness and more general temporal operators, fairness check programs (incl. loops and recursion) over finite domains, e.g. SLAM Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz mc 3 Revision:.2 mc 4 Revision:.2 Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz
2 mc 5 Revision:.2 mc 6 Revision:.2 Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz mc 7 Revision:.2 mc 8 Revision:.2 Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz
3 mc 9 Revision:.2 mc Revision:.2 Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz mc Revision:.2 Safety mc 2 Revision:.2 the two traffic lights should never show a light at the same time Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz
4 State Space mc 3 Revision:.2 Safety mc 4 Revision:.2 state space is the set of assignments to variables of the system safety properties specify invariants of the system state space is finite if the range of variables is finite this notion works for inifinite state spaces as well simple generic algorithm for checking safety properties:. iteratively generate all reachable states TLC example: 2. check for violation of invariant for newly reached states single assignment σ:{southnorth, eastwest} {,, } 3. terminate if all newly reached states can be found set of assignments is isomorphic to {,,} 2 eg state space is isomorphic to the crossproduct of variable ranges compare with assertions used in run time checking: assert in C and VHDL not all states are reachable: (, ) contract checking: require, ensure, etc. in Eiffel Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz Unsafe TLC in SMV mc 5 Revision:.2 SMV mc 6 Revision:.2 MODULE trafficlight (enable) VAR light : {,, }; back : boolean; ASSIGN init (light) := ; next (light) := case light = &!enable : ; light = & enable : ; light = & back : ; light = &!back : ; : ; esac; next (back) := case light = & enable : ; light = : ; : back; esac; MODULE main VAR southnorth : trafficlight (); eastwest : trafficlight (); SPEC AG!(southnorth.light = & eastwest.light = ) symbolic model checker implemented by Ken McMillan at CMU (early 9 ies) input language: finite models + temporal specification hierarchical description, similar to hardward description language (HDL) integer and enumeration types, arithmetic operations heavily relies on the data structure Binary Decision Diagrams (BDDs) Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz
5 Reachable States of One Traffic Light mc 7 Revision:.2 Reachable States of Unsafe TLC mc 8 Revision:.2 2 reachable states out of 2 states enable light back 7 reachable states out of 44 states (or 72 if only one enable ) unfortunately buggy state is among them Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz Safe TLC in SMV mc 9 Revision:.2 Liveness mc 2 Revision:.2 MODULE main VAR turn : { ew, sn }; southnorth : trafficlight (enablesouthnorth); eastwest : trafficlight (enableeastwest); DEFINE enableeastwest := southnorth.light = & turn = ew; enablesouthnorth := eastwest.light = & turn = sn; SPEC AG!(southnorth.light = & eastwest.light = ) idea: disable traffic light as long the other is not and its not the others turn traffic lights showing should eventually show Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz
6 Liveness mc 2 Revision:.2 Liveness mc 22 Revision:.2 traffic lights showing should eventually show traffic lights showing should eventually show Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz Boolean Encoding mc 23 Revision:.2 Result of Boolean Encoding mc 24 Revision:.2 compilation of finite model into pure propositional domain initial state picate I represented as boolean formula first step is to flatten the hierarchy!eastwest.light@ & eastwest.light@ recursive instantiation of all submodules (equivalent to init(eastwest.light) := ) name and parameter substitution may increase program size exponentially second step is to encode variables with boolean variables light light@ light@ transition relation T represented as boolean formula encoding of atomic picates p as boolean formulae!eastwest.light@ &!eastwest.light@ (equivalent to eastwest.light!= ) Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz Systemtheory 2 Formal Systems 2 #3422 SS 26 Armin Biere JKU Linz
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