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A symbolic algorithm for lazy synthesis of eager strategies


We present an algorithm for solving two-player safety games that combines a mixed forward/backward search strategy with a symbolic representation of the state space. By combining forward and backward exploration, our algorithm can synthesize strategies that are eager in the sense that they try to prevent progress towards the error states as soon as possible, whereas standard backwards algorithms often produce permissive solutions that only react when absolutely necessary. We provide experimental results for two classes of crafted benchmarks, the benchmark set of the Reactive Synthesis Competition (SYNTCOMP) 2017, as well as a set of randomly generated benchmarks. The results show that our algorithm in many cases produces more eager strategies than a standard backwards algorithm, and solves a number of benchmarks that are intractable for existing tools. Finally, we observe a connection between our algorithm and a recently proposed algorithm for the synthesis of controllers that are robust against disturbances, pointing to possible future applications.

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  1. This part is the light-weight backward search: unlike UPRE in the standard backward algorithm, preimage does not contain any quantifier alternation.

  2. This is the only place where our algorithm uses image, and it is only included to keep the definitions and correctness argument simple - the algorithm also works if the model checker omits this last image computation step, see Sect. 5.

  3. It may be a subsequence due to the merging of error levels from different iterations of the main loop.


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We thank Bernd Finkbeiner and Martin Zimmermann for fruitful discussions. This work was supported by the German Research Foundation (DFG) under the project ASDPS (JA 2357/2-1).

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Jacobs, S., Sakr, M. A symbolic algorithm for lazy synthesis of eager strategies. Acta Informatica 57, 81–106 (2020).

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