This paper presents a wp-style calculus for obtaining bounds on the expected run-time of probabilistic programs. Its application includes determining the (possibly infinite) expected termination time of a probabilistic program and proving positive almost-sure terminationdoes a program terminate with probability one in finite expected time?We provide several proof rules for bounding the run-time of loops, and prove the soundness of the approach with respect to a simple operational model. We show that our approach is a conservative extension of Nielson's approach for reasoning about the run-time of deterministic programs. We analyze the expected run-time of some example programs including a one-dimensional random walk and the coupon collector problem.

This paper presents a wp-style calculus for obtaining expectations on the outcomes of (mutually) recursive probabilistic programs. We provide several proof rules to derive one-and two-sided bounds for such expectations, and show the soundness of our wp-calculus with respect to a probabilistic pushdown automaton semantics. We also give a wp-style calculus for obtaining bounds on the expected runtime of recursive programs that can be used to determine the (possibly infinite) time until termination of such programs.

Symbolic-Heap Separation logic is a popular formalism for automated reasoning about heap-manipulating programs, which allows the user to give customized data structure definitions.In this paper, we give a new decidability proof for the separation logic fragment of Iosif, Rogalewicz and Simacek. We circumvent the reduction to MSO from their proof and provide a direct model-theoretic construction with elementary complexity. We implemented our approach in the Harrsh analyzer and evaluate its effectiveness. In particular, we show that Harrsh can decide the entailment problem for data structure definitions for which no previous decision procedures have been implemented.

We present quantitative separation logic (QSL). In contrast to classical separation logic, QSL employs quantities which evaluate to real numbers instead of predicates which evaluate to Boolean values. The connectives of classical separation logic, separating conjunction and separating implication, are lifted from predicates to quantities. This extension is conservative: Both connectives are backward compatible to their classical analogs and obey the same laws, e.g. modus ponens, adjointness, etc. Furthermore, we develop a weakest precondition calculus for quantitative reasoning about probabilistic pointer programs in QSL. This calculus is a conservative extension of both Ishtiaq's, O'Hearn's and Reynolds' separation logic for heap-manipulating programs and Kozen's / McIver and Morgan's weakest preexpectations for probabilistic programs. Soundness is proven with respect to an operational semantics based on Markov decision processes. Our calculus preserves O'Hearn's frame rule, which enables local reasoning. We demonstrate that our calculus enables reasoning about quantities such as the probability of terminating with an empty heap, the probability of reaching a certain array permutation, or the expected length of a list.

This paper presents a wp-style calculus for obtaining bounds on the expected runtime of randomized algorithms. Its application includes determining the (possibly infinite) expected termination time of a randomized algorithm and proving positive almost-sure termination-does a program terminate with probability one in finite expected time? We provide several proof rules for bounding the runtime of loops, and prove the soundness of the approach with respect to a simple operational model. We show that our approach is a conservative extension of Nielson's approach for reasoning about the runtime of deterministic programs. We analyze the expected runtime of some example programs including the coupon collector's problem, a one-dimensional random walk and a randomized binary search.