Game-theoretic simulation checking tool Peter Bulychev, Vladimir Zakharov, Igor Konnov Moscow State University

What is (bi)simulation? Simulation is a relation between models (labeled transition systems). Intuitively M_1 simulates M_2 iff M_1 can match the moves of M_2 Bisimulation is an equivalence relation There are different types of (bi)simulations

Where can (bi)simulations be applied? (Bi)simulations preserve branching-time and linear-time logics Invariant-based approach to the verification Abstraction method

What have we done? We designed the language for describing (bi)simulations in game- theoretic form We developed the tool for checking different (bi)simulation relations

Outline Different notions of (bi)simulations and checking Game-theoretic approach to (bi)simulation checking On the universal simulation-checking tool

Example of strong simulation

Different notions of simulation Tau-actions Different types of simulation Strong (preserves CTL*) Weak (preserves LTL_X) Quasi-block (is monotonic w.r.t. parallel composition) Stuttering (preserves CTL*_X) Simulation and bisimulation Models with fair constraints

Different approaches to computing relations Relational coarsest partition (bisimulations only) Fixed-point approach Game-theoretic Universal (fair/unfair, simulation/bisimulation) Efficient (strong simulation)

Game-theoretic approach There are two players in the game, Spoiler and Duplicator, Duplicator tries to show that chosen relation is fulfilled, Spoiler tries to stuck Duplicator If Duplicator player wins then one model simulates the other

Game for strong simulation

Game-theoretic approach

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

Stuttering simulation

Theoretical results Stuttering simulation Stuttering bisimulation Non fairO(m 2 ) time O(m 2 ) space O(mn) time O(m) space FairO(m 2 n 2 ) time O(m 2 ) space O(m 2 n 2 ) time O(m 2 ) space

Game-theoretic language Observation: Games for computing different notions of simulation have a common framework. Result: We designed the language for describing rules of simulation checking game.

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

Game-theoretic language : example {The game for computing usual similarity on LTS with labeled transitions} types S: (S1, S2); D: (S1, S2, A); initial (A s1(I1))(E s2(I2)) 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);

Tool for simulation checking We have developed the universal simulation-checking tool. This tool checks whether simulation given in theoretic-game form exists between two models.

BDD We have used BDD to store game graph and models When we tested our tool with models consisting of 10^5 states, we ran out of memory: BDD of the game was too large We decided to construct BDD of the game on-the-fly However BDD of the models should be given in explicit form

Models We debugged our tool on small models with 10^3 states such as an abstraction of dining philosophers system and Milner scheduler Then we used our tool to check stuttering and block simulation between RSVP models with different topology Universal simulation checker tool is less efficient than block simulation checker tool written by Igor Konnov, but it is universal

Results of tests ModelSizeSpaceTime RSVP (r3c4 vs r2c3) 25000x M1m24s Milner (M4 vs M3) 3000x50025M1s

Implemetation details Language: python BDD library: Cudd (Colorado University Decision Diagram) OS: Linux

Future work To apply our tool to checking whether one model is an abstraction of other Fair (bi)simulation To speed up our tool To compare with the similar tool developed in University of Freiburg, Germany

Thank you Your questions