Christoph Matheja

Weakest Precondition Reasoning for Expected Run---Times of Probabilistic Programs

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 termination--does 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 Nielsons 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.

Reasoning about Recursive Probabilistic Programs

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.

Unified Reasoning about Robustness Properties of Symbolic-Heap Separation Logic

We introduce heap automata, a formalism for automatic reasoning about robustness properties of the symbolic heap fragment of separation logic with user-defined inductive predicates. Robustness properties, such as satisfiability, reachability, and acyclicity, are important for a wide range of reasoning tasks in automated program analysis and verification based on separation logic. Previously, such properties have appeared in many places in the separation logic literature, but have not been studied in a systematic manner. In this paper, we develop an algorithmic framework based on heap automata that allows us to derive asymptotically optimal decision procedures for a wide range of robustness properties in a uniform way.

Inferring Covariances for Probabilistic Programs

We study weakest precondition reasoning about the (co)variance of outcomes and the variance of run–times of probabilistic programs with conditioning. For outcomes, we show that approximating (co)variances is computationally more difficult than approximating expected values. In particular, we prove that computing both lower and upper bounds for (co)variances is \(\varSigma _2^0\)–complete. As a consequence, neither lower nor upper bounds are computably enumerable. We therefore present invariant–based techniques that do enable enumeration of both upper and lower bounds, once appropriate invariants are found. Finally, we extend this approach to reasoning about run–time variances.

How long, O Bayesian network, will I sample thee?

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.

Tree-Like Grammars and Separation Logic

Separation Logic with inductive predicate definitions (\(\texttt {SL}\)) and hyperedge replacement grammars (HRG) are established formalisms to describe the abstract shape of data structures maintained by heap-manipulating programs. Fragments of both formalisms are known to coincide, and neither the entailment problem for \(\texttt {SL}\) nor its counterpart for HRGs, the inclusion problem, are decidable in general.

Rule-Based Conditioning of Probabilistic Data

Data interoperability is a major issue in data management for data science and big data analytics. Probabilistic data integration (PDI) is a specific kind of data integration where extraction and integration problems such as inconsistency and uncertainty are handled by means of a probabilistic data representation. This allows a data integration process with two phases: (1) a quick partial integration where data quality problems are represented as uncertainty in the resulting integrated data, and (2) using the uncertain data and continuously improving its quality as more evidence is gathered. The main contribution of this paper is an iterative approach for incorporating evidence of users in the probabilistically integrated data. Evidence can be specified as hard or soft rules (i.e., rules that are uncertain themselves).

Graph-Based Shape Analysis Beyond Context-Freeness.

We develop a shape analysis for reasoning about relational properties of data structures. Both the concrete and the abstract domain are represented by hypergraphs. The analysis is parameterized by user-supplied indexed graph grammars to guide concretization and abstraction. This novel extension of context-free graph grammars is powerful enough to model complex data structures such as balanced binary trees with parent pointers, while preserving most desirable properties of context-free graph grammars.

Let this Graph Be Your Witness

We present a graph-based tool for analysing Java programs operating on dynamic data structures. It involves the generation of an abstract state space employing a user-defined graph grammar. LTL model checking is then applied to this state space, supporting both structural and functional correctness properties. The analysis is fully automated, procedure-modular, and provides informative visual feedback including counterexamples in the case of property violations.

Weakest Precondition Reasoning for Expected Runtimes of Randomized Algorithms

This article 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.