How long, O Bayesian network, will I sample thee? A program analysis perspective on expected sampling times

Bayesian networks (BNs) are probabilistic graphical models for describing complex joint probability distributions. The main problem for BNs is inference: Determine the probability of an event given observed evidence. Since exact inference is often infeasible for large BNs, popular approximate inference methods rely on sampling. We study the problem of determining the expected time to obtain a single valid sample from a BN. To this end, we translate the BN together with observations into a probabilistic program. We provide proof rules that yield the exact expected runtime of this program in a fully automated fashion. We implemented our approach and successfully analyzed various real-world BNs taken from the Bayesian network repository

A Deductive Verification Infrastructure for Probabilistic Programs

This paper presents a quantitative program verification infrastructure for discrete probabilistic programs. Our infrastructure can be viewed as the probabilistic analogue of Boogie: its central components are an intermediate verification language (IVL) together with a real-valued logic. Our IVL provides a programming-language-style for expressing verification conditions whose validity implies the correctness of a program under investigation. As our focus is on verifying quantitative properties such as bounds on expected outcomes, expected run-times, or termination probabilities, off-the-shelf IVLs based on Boolean first-order logic do not suffice. Instead, a paradigm shift from the standard Boolean to a real-valued domain is required. Our IVL features quantitative generalizations of standard verification constructs such as assume- and assert-statements. Verification conditions are generated by a weakest-precondition-style semantics, based on our real-valued logic. We show that our verification infrastructure supports natural encodings of numerous verification techniques from the literature. With our SMT-based implementation, we automatically verify a variety of benchmarks. To the best of our knowledge, this establishes the first deductive verification infrastructure for expectation-based reasoning about probabilistic programs

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

Quantitative Separation Logic - A Logic for Reasoning about Probabilistic Programs

We present quantitative separation logic ($\mathsf{QSL}$). In contrast to classical separation logic, $\mathsf{QSL}$ 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 $\mathsf{QSL}$. This calculus is a conservative extension of both 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

A Deductive Verification Infrastructure for Probabilistic Programs

This paper presents a quantitative program verification infrastructure for discrete probabilistic programs. Our infrastructure can be viewed as the probabilistic analogue of Boogie: its central components are an intermediate verification language (IVL) together with a real-valued logic. Our IVL provides a programming-language-style for expressing verification conditions whose validity implies the correctness of a program under investigation. As our focus is on verifying quantitative properties such as bounds on expected outcomes, expected run-times, or termination probabilities, off-the-shelf IVLs based on Boolean first-order logic do not suffice. Instead, a paradigm shift from the standard Boolean to a real-valued domain is required. Our IVL features quantitative generalizations of standard verification constructs such as assume- and assert-statements. Verification conditions are generated by a weakest-precondition-style semantics, based on our real-valued logic. We show that our verification infrastructure supports natural encodings of numerous verification techniques from the literature. With our SMT-based implementation, we automatically verify a variety of benchmarks. To the best of our knowledge, this establishes the first deductive verification infrastructure for expectation-based reasoning about probabilistic programs

SL-COMP: Competition of Solvers for Separation Logic

International audienceSL-COMP aims at bringing together researchers interested on improving the state of the art of the automated deduction methods for Separation Logic (SL). The event took place twice until now and collected more than 1K problems for different fragments of SL. The input format of problems is based on the SMT-LIB format and therefore fully typed; only one new command is added to SMT-LIB's list, the command for the declaration of the heap's type. The SMT-LIB theory of SL comes with ten logics, some of them being combinations of SL with linear arithmetics. The competition's divisions are defined by the logic fragment, the kind of decision problem (satisfiability or entailment) and the presence of quantifiers. Until now, SL-COMP has been run on the StarExec platform, where the benchmark set and the binaries of participant solvers are freely available. The benchmark set is also available with the competition's documentation on a public repository in GitHub

Automated reasoning and randomization in separation logic

We study three aspects of program verification with separation logic:1. Reasoning about quantitative properties, such as the probability of memory-safe termination, of randomized heap-manipulating programs.2. Automated reasoning about the robustness of and entailments between formulas in the symbolic heap fragment of separation logic itself.3. Automated reasoning about pointer programs by combining abstractions based on separation logic with the above techniques and model checking. Regarding the first item, we extend separation logic to reason about quantities, which evaluate to real numbers, instead of predicates, which evaluate to Boolean values. Based on the resulting quantitative separation logic, we develop a weakest precondition calculus à la Dijkstra for quantitative reasoning about randomized heap-manipulating programs. We show that this calculus is a sound and conservative extension of both separation logic and McIver and Morgan's weakest preexpectations which preserves virtually all properties of classical separation logic. We demonstrate its applicability by several case studies. Regarding the second item, we develop an algorithmic framework based on heap automata to compositionally check robustness properties, e.g., satisfiability or acyclicity, of symbolic heaps with inductive predicate definitions. We consider two approaches to discharge entailments for fragments of separation logic. In particular, this includes a pragmatic decision procedure with nondeterministic polynomial-time complexity for entailments between graphical symbolic heaps. Regarding the third item, we introduce Attestor - an automated verification tool for analyzing Java programs operating on dynamic data structures. The tool involves the generation of an abstract state space employing inductive predicate definitions in separation logic. Properties of individual states are defined by heap automata. LTL model checking is then applied to this state space, supporting the verification of both structural and functional correctness properties. Attestor is fully automated, procedure modular, and provides informative visual feedback including counterexamples for violated properties

Flexible Refinement Proofs in Separation Logic

Refinement transforms an abstract system model into a concrete, executable program, such that properties established for the abstract model carry over to the concrete implementation. Refinement has been used successfully in the development of substantial verified systems. Nevertheless, existing refinement techniques have limitations that impede their practical usefulness. Some techniques generate executable code automatically, which generally leads to implementations with sub-optimal performance. Others employ bottom-up program verification to reason about efficient implementations, but impose strict requirements on the structure of the code, the structure of the refinement proofs, as well as the employed verification logic and tools. In this paper, we present a novel refinement technique that removes these limitations. Our technique uses separation logic to reason about efficient concurrent implementations. It prescribes only a loose coupling between an abstract model and the concrete implementation. It thereby supports a wide range of program structures, data representations, and proof structures. We make only minimal assumptions about the underlying program logic, which allows our technique to be used in combination with a wide range of logics and to be automated using off-the-shelf separation logic verifiers. We formalize the technique, prove the central trace inclusion property, and demonstrate its usefulness on several case studies.Comment: 35 pages, submitted to 31st European Symposium on Programmin