Intersection types and (positive) almost-sure termination

Randomized higher-order computation can be seen as being captured by a lambda calculus endowed with single algebraic operation, namely construct for binary probabilistic choice. What matters about such computations is the probability of obtaining any given result, rather than possibility or necessity it, like in (non)deterministic computation. Termination, arguably simplest kind reachability problem, spelled out at least two ways, depending on whether it talks convergence expected evaluation time, second one providing stronger guarantee. In this paper, we show that intersection types are capable precisely characterizing both notions termination inside system types: lambda-term underapproximated its type, while underlying derivation's weight gives lower bound to term's number steps normal form. Noticeably, approximations tight -- not only soundness but also completeness holds. The crucial ingredient non-idempotency, without which would impossible reason reduction necessary completely evaluate term. Besides, approximation obtain proved optimal recursion theoretically: no recursively enumerable formal do better that.