On the Termination Problem for Probabilistic Higher-Order Recursive Programs

Naoki Kobayashi, Ugo Dal Lago, Charles Grellois

Abstract: In the last two decades, there has been much progress on model checking of both probabilistic systems and higher-order programs. In spite of the emergence of higherorder probabilistic programming languages, not much has been done to combine those two approaches. As a first step towards model checking of probabilistic higher-order programs, we introduce PHORS, a probabilistic extension of higher-order recursion schemes (HORS), as a model of probabilistic higherorder programs. The model of PHORS may alternatively be viewed as a higher-order extension of recursive Markov chains. We then investigate the probabilistic termination problem - or, equivalently, the probabilistic reachability problem. We prove that almost sure termination of order-2 PHORS is undecidable. We also provide a fixpoint characterization of the termination probability of PHORS, and develop a sound (but possibly incomplete) procedure for approximately computing the termination probability.