Concurrent reachability games

  • Authors:

  • Luca de Alfaro;Thomas A. Henzinger;Orna Kupferman

  • Affiliations:

  • Department of Computer Engineering, University of California, Santa Cruz, USA;Computer and Communication Sciences, EPFL, Lausanne, Switzerland;School of Computer Science and Engineering, Hebrew University, Jerusalem, Israel

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2007

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Abstract

We consider concurrent two-player games with reachability objectives. In such games, at each round, player 1 and player 2 independently and simultaneously choose moves, and the two choices determine the next state of the game. The objective of player 1 is to reach a set of target states; the objective of player 2 is to prevent this. These are zero-sum games, and the reachability objective is one of the most basic objectives: determining the set of states from which player 1 can win the game is a fundamental problem in control theory and system verification. There are three types of winning states, according to the degree of certainty with which player 1 can reach the target. From type-1 states, player 1 has a deterministic strategy to always reach the target. From type-2 states, player 1 has a randomized strategy to reach the target with probability 1. From type-3 states, player 1 has for every real @e0 a randomized strategy to reach the target with probability greater than 1-@e. We show that for finite state spaces, all three sets of winning states can be computed in polynomial time: type-1 states in linear time, and type-2 and type-3 states in quadratic time. The algorithms to compute the three sets of winning states also enable the construction of the winning and spoiling strategies.