Static analysis of parity games: alternating reachability under parity

Transcription

1 8 January 2016, DTU Denmark Static analysis of parity games: alternating reachability under parity Michael Huth, Imperial College London Nir Piterman, University of Leicester Jim Huan-Pu Kuo, Imperial College London (at the time) Jim was kindly supported by an UK EPSRC Doctoral Training Award

2 Motivation Data-Flow Analysis as Model Checking, B. Steffen 1991 Program Analysis Model Checking Model checking is static analysis of modal logic, F. & H.R. Nielson 2010 Parity Games = Model Checking Mu-Calculus Local model checking games, C. Stirling 1995

3 Idea behind this work Statically analyse parity game to under-approximate its solution in PTIME Use fixed-points of monotone function to that end Program Analysis Model Checking Parity Games = Model Checking Mu-Calculus

4 Primer on parity games 2-player, turn-based games player 0 moves in circled nodes, player 1 in squared nodes plays are infinite paths in game graph player 0 wins play iff minimal infinitely often occurring color is even; e.g. player 0 wins v 0 (v 1 v 2 )(v 1 v 2 ) but loses v 0 v 1 v 2 (v 1 v 0 )(v 1 v 0 ) player 0 wins node v if she has a strategy to force a win on all plays beginning at v Determinacy: all nodes won by one of these players; pure, memoryless strategies suffice

5 Alternating reachability under parity for (X,p) 2-player, turn-based game for function played on game graph of parity game same plays as in parity game but different winning condition: given non-empty node set X and p in {0,1}, player p wins play v 0 v 1 v 2 iff there is j > 0 such that v j is in X and we have min {c(v i ) 0 <= i <= j} % 2 = p omega-regular winning condition, so this is determined game by [Martin 1975]

6 An Invariant let X and p be such that all nodes in X are won by player p for alternating reachability under parity then all nodes in X are won by player p in the underlying parity game (argument combines paths from X to X with minimal parity p)

7 Monotone function for (X,p) defined over the power set of all nodes of color parity p it returns the set of nodes in X won by player p for alternating reachability under parity!! this function is monotone in X! winning region of player p for alternating reachability under parity invariant implies that non-empty (greatest) fixed points are node sets won by player p in parity game G

8 Static analysis psolc

9 Finite memory needed? for X = {v0} and p=0, player 0 requires memory: she moves from v1 to v0 only if v2 already occurred; otherwise she moves from v1 to v2 however, for the greatest fixed-point X = {v0,v2} and p=0, no memory is needed for player 0 Does our static analysis encounter non-empty greatest fixed-points X that require finite memory?

10 Static analysis incomplete all fixedpoints empty 1-player game won entirely by player 0! For p=0, initial X is {v0, v2, v3, v4, v5, v6}. Fixed-point computation removes the following node sets from X, in that order: {v0}, {v4, v5}, {v3}, {v2}, {v6} leaving X empty.

11 Related work Fatal Attractors in Parity Games, M.H., J. H.-P. Kuo, and N. Piterman, FOSSACS 2013 more restrictive node sets and monotone function: all nodes in X have color k and p = k%2 player p has to reach X whilst avoiding nodes of color < k if player p can do this from all nodes in X, then X is a fatal attractor static analysis psolb based on fatal attractors: cubic in number of nodes psolc refines psolb; more precision over psolb occurs mostly on random games with low edge density

12 Conclusions A static analysis of parity games based on fixed-point computation of a monotone function gives partial solutions in PTIME. This generalized Fatal Attractors to more canonical games: alternating reachability under parity This omega-regular acceptance condition gives rise to the monotone function for the above static analysis Future work: gain better understanding of sources of incompleteness of this static analysis

13 Appendix: Complexity psolc time complexity: linear in number of edges and colours, quadratic in number of nodes psolc space complexity: linear in number of edges and colours time complexity of solving alternating reachability under parity: same as psolc space complexity