4,763 research outputs found
Finding polynomial loop invariants for probabilistic programs
Quantitative loop invariants are an essential element in the verification of
probabilistic programs. Recently, multivariate Lagrange interpolation has been
applied to synthesizing polynomial invariants. In this paper, we propose an
alternative approach. First, we fix a polynomial template as a candidate of a
loop invariant. Using Stengle's Positivstellensatz and a transformation to a
sum-of-squares problem, we find sufficient conditions on the coefficients.
Then, we solve a semidefinite programming feasibility problem to synthesize the
loop invariants. If the semidefinite program is unfeasible, we backtrack after
increasing the degree of the template. Our approach is semi-complete in the
sense that it will always lead us to a feasible solution if one exists and
numerical errors are small. Experimental results show the efficiency of our
approach.Comment: accompanies an ATVA 2017 submissio
Lower Bounds for Possibly Divergent Probabilistic Programs
We present a new proof rule for verifying lower bounds on quantities of probabilistic programs. Our proof rule is not confined to almost-surely terminating programs -- as is the case for existing rules -- and can be used to establish non-trivial lower bounds on, e.g., termination probabilities and expected values, for possibly divergent probabilistic loops, e.g., the well-known three-dimensional random walk on a lattice
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 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
Bounded Model Checking for Probabilistic Programs
In this paper we investigate the applicability of standard model checking
approaches to verifying properties in probabilistic programming. As the
operational model for a standard probabilistic program is a potentially
infinite parametric Markov decision process, no direct adaption of existing
techniques is possible. Therefore, we propose an on-the-fly approach where the
operational model is successively created and verified via a step-wise
execution of the program. This approach enables to take key features of many
probabilistic programs into account: nondeterminism and conditioning. We
discuss the restrictions and demonstrate the scalability on several benchmarks
Bounded Expectations: Resource Analysis for Probabilistic Programs
This paper presents a new static analysis for deriving upper bounds on the
expected resource consumption of probabilistic programs. The analysis is fully
automatic and derives symbolic bounds that are multivariate polynomials of the
inputs. The new technique combines manual state-of-the-art reasoning techniques
for probabilistic programs with an effective method for automatic
resource-bound analysis of deterministic programs. It can be seen as both, an
extension of automatic amortized resource analysis (AARA) to probabilistic
programs and an automation of manual reasoning for probabilistic programs that
is based on weakest preconditions. As a result, bound inference can be reduced
to off-the-shelf LP solving in many cases and automatically-derived bounds can
be interactively extended with standard program logics if the automation fails.
Building on existing work, the soundness of the analysis is proved with respect
to an operational semantics that is based on Markov decision processes. The
effectiveness of the technique is demonstrated with a prototype implementation
that is used to automatically analyze 39 challenging probabilistic programs and
randomized algorithms. Experimental results indicate that the derived constant
factors in the bounds are very precise and even optimal for many programs
Intersection types and (positive) almost-sure termination
Randomized higher-order computation can be seen as being captured by a λ-calculus endowed with a single algebraic operation, namely a construct for binary probabilistic choice. What matters about such computations is the probability of obtaining any given result, rather than the possibility or the necessity of obtaining it, like in (non)deterministic computation. Termination, arguably the simplest kind of reachability problem, can be spelled out in at least two ways, depending on whether it talks about the probability of convergence or about the expected evaluation time, the second one providing a stronger guarantee. In this paper, we show that intersection types are capable of precisely characterizing both notions of termination inside a single system of types: the probability of convergence of any λ-term can be underapproximated by its type, while the underlying derivation's weight gives a lower bound to the term's expected number of steps to normal form. Noticeably, both approximations are tight-not only soundness but also completeness holds. The crucial ingredient is non-idempotency, without which it would be impossible to reason on the expected number of reduction steps which are necessary to completely evaluate any term. Besides, the kind of approximation we obtain is proved to be optimal recursion theoretically: no recursively enumerable formal system can do better than that
- …