Game-theoretic approach to the simulation checking problem

Game-theoretic approach to the simulation checking problem Peter Bulychev Vladimir Zakharov Lomonosov Moscow State University

Model Checking • The main goal of model checking is to verify whether a given model satisfies a required property (specification). • Simulation relations preserve satisfiability of specifications given in the form of temporal logic formulas. • Simulation is used to prove that one model is a refinement/abstraction of the other.

Varieties of simulation • Various types of simulation • Strong simulation (preserves CTL*) • Weak simulation (preserves LTL-X) • Quasi-block simulation (is monotonic w.r.t. parallel composition) • Stuttering simulation (preserves CTL*-X) • Equivalence relations and preorders (simulations and bisimulations) • Models with fair constraints

(Bi)simulation checking approaches • Relational coarsest partition (bisimulations only) • Fixed-point approach • Game-theoretic • Universal (fair/unfair, simulation/bisimulation) • Efficient (strong simulation)

Game-theoretic approach

Game for strong simulation

Reduction to game rules • In some cases reduction can be obtained automatically, • For more complex relations it is necessary to write game rules by hand. We have written game rules for stuttering (bi)simulation and proved their correctness.

Theoretical results

Game-theoretic language • Observation: • Games for computing different kinds of simulation have much in common. • Result: • We designed the language for describing rules of simulation checking games.

Game-theoretic language: example {The game for checking strong simulation on LTS with labeled transitions} types S: (S1, S2); D: (S1, S2, A); rules (A s1)(E s2) S(s1, s2); steps S(s1, s2) -> D(s1', s2, a) : t(s1,a,s1'); D(s1, s2', a) -> S(s1, s2) : t(s2',a,s2);

Game-theoretic language • We have described a number of (bi)simulations in our language: • Strong • Weak • Block • Stuttering

Simulation checking tool • Our simulation checking tool checks whether there exists a simulation defined in game-theoretic terms between two models. Models Model’s BDDs Game’s BDD Game solver Answer (counterexample) Game rules

BDD • We have used BDD to describe symbolically game graph and models to be checked. • When we tested our tool with models that consist of 105 states, we ran out of memory: BDD of the game was too large • Therefore, we decided to construct BDD of the game on-the-fly. • However, BDD of the models must be in explicit form.

Where are we now? • We are trying to answer the following questions: • If there is a winning strategy, how can we find it as fast as possible? • Otherwise, how can we maximally fast find a counterexample? • What is the optimal order for BDD variables?

Timed automatons • Timed automatons are used to model continuous and monotonous processes • UPPAAL tool developed by K.G.Larsen group at Aaalborg University (Denmark) can be used to analyze timed automatons press? Model of two-level light controller: user should press it twice quickly to turn on bright light or press once to turn on soft light. press? X:=0 press? X<=3 Off Soft Bright press? X>3

Timed simulations • We defined several timed simulations and proposed game-theoretic algorithms for solving them jointly with the K.G.Larsen group • The sets of winning clock valuations are stored in the symbolic form in the game states • These algorithms will be implemented in the UPPAAL tool

Questions?

Game-theoretic approach to the simulation checking problem