A Calculus for Amortized Expected Runtimes

We develop a weakest-precondition-style calculus à la Dijkstra for reasoning about amortized expected runtimes of randomized algorithms with access to dynamic memory — the aert calculus. Our is truly quantitative, i.e. instead Boolean valued predicates, it manipulates real-valued functions. En route calculus, we study ert Kaminski et al. [2018] extended by capabilities handling memory, thus enabling compositional and local data structures . This extension employs runtime separation logic , which has been foreshadowed Matheja [2020] then implemented in Isabelle/HOL Haslbeck [2021]. In addition Haslbeck’s results, further prove soundness so-extended respect an operational Markov decision process model featuring countably-branching nondeterminism, provide extensive intuitive explanations, proof rules logic-style verification upper bounds on runtimes. Finally, build so-called potential method analysis into obtaining Soundness obtained from some probabilistic form telescoping. Since one needs be able handle changes can principle both positive or negative, essentially capable certain signed random variables. A particularly pleasing feature our solution that, unlike e.g. Kozen [1985], obtain loop rule variables, furthermore, Katoen [2017], makes do without need involved technical machinery keeping track integrability present case studies, including formal delete-insert-find-any set structure [Brodal 1996], yields constant per operation, whereas no deterministic algorithm achieve this.