Synthesis for Probabilistic Environments

Abstract

In synthesis we construct finite state systems from temporal specifications. While this problem is well understood in the classical setting of non-probabilistic synthesis, this paper suggests the novel approach of open synthesis under the assumptions of an environment that chooses its actions randomized rather than nondeterministically. Assuming a randomized environment inspires alternative semantics both for linear-time and branching-time logics. For linear-time, natural acceptance criteria are almost-sure and observable acceptance, where it suffices if the probability measure of accepting paths is 1 and greater than 0, respectively.

We distinguish 0-environments, which can freely assign probabilities to each environment action, from ε-environments, where the probabilities assigned by the environment are bound from below by some ε>0. While the results in case of 0-environments are essentially the same as for nondeterministic environments, the languages occurring in case of ε-environments are topologically different from the results for nondeterministic and 0-environments (in case of LTL, recognizable by weak alternating automata vs. recognizable by deterministic automata). The complexity of open synthesis is, in both cases, EXPTIME and 2EXPTIME-complete for CTL and LTL specifications, respectively.

This work was partly supported by the German Research Foundation (DFG) as part of the Transregional Collaborative Research Center “Automatic Verification and Analysis of Complex Systems” (SFB/TR 14 AVACS).