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We introduce a system of monadic affine sized types, which substantiallygeneralise usual sized types, and allows this way to capture probabilistichigher-order programs which terminate almost surely. Going beyond plain, strongnormalisation without losing soundness turns out to be a hard task, whichcannot be accomplished without a richer, quantitative notion of types, but alsowithout imposing some affinity constraints. The proposed type system ispowerful enough to type classic examples of probabilistically terminatingprograms such as random walks. The way typable programs are proved to be almostsurely terminating is based on reducibility, but requires a substantialadaptation of the technique.

Probabilistic Termination by Monadic Affine Sized Typing (Long Version)