13 research outputs found
Counterexample-Guided Polynomial Loop Invariant Generation by Lagrange Interpolation
We apply multivariate Lagrange interpolation to synthesize polynomial
quantitative loop invariants for probabilistic programs. We reduce the
computation of an quantitative loop invariant to solving constraints over
program variables and unknown coefficients. Lagrange interpolation allows us to
find constraints with less unknown coefficients. Counterexample-guided
refinement furthermore generates linear constraints that pinpoint the desired
quantitative invariants. We evaluate our technique by several case studies with
polynomial quantitative loop invariants in the experiments
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
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
PrIC3: Property Directed Reachability for MDPs
IC3 has been a leap forward in symbolic model checking. This paper proposes
PrIC3 (pronounced pricy-three), a conservative extension of IC3 to symbolic
model checking of MDPs. Our main focus is to develop the theory underlying
PrIC3. Alongside, we present a first implementation of PrIC3 including the key
ingredients from IC3 such as generalization, repushing, and propagation
A program logic for union bounds
International audienceWe propose a probabilistic Hoare logic aHL based on the union bound, a tool from basic probability theory. While the union bound is simple, it is an extremely common tool for analyzing randomized algorithms. In formal verification terms, the union bound allows flexible and compos-itional reasoning over possible ways an algorithm may go wrong. It also enables a clean separation between reasoning about probabilities and reasoning about events, which are expressed as standard first-order formulas in our logic. Notably, assertions in our logic are non-probabilistic, even though we can conclude probabilistic facts from the judgments. Our logic can also prove accuracy properties for interactive programs, where the program must produce intermediate outputs as soon as pieces of the input arrive, rather than accessing the entire input at once. This setting also enables adaptivity, where later inputs may depend on earlier intermediate outputs. We show how to prove accuracy for several examples from the differential privacy literature, both interactive and non-interactive. 1998 ACM Subject Classification D.2.4 Software/Program Verification 1 Introduction Probabilistic computations arise naturally in many areas of computer science. For instance, they are widely used in cryptography, privacy, and security for achieving goals that lie beyond the reach of deterministic programs. However, the correctness of probabilistic programs can be quite subtle, often relying on complex reasoning about probabilistic events. Accordingly, probabilistic computations present an attractive target for formal verification. A long line of research, spanning more than four decades, has focused on expressive formalisms for reasoning about general probabilistic properties both for purely probabilistic programs and for programs that combine probabilistic and non-deterministic choice (see, e.g., [29, 34, 35]). More recent research investigates specialized formalisms that work with more restricted assertions and proof techniques, aiming to simplify formal verification. As perhaps the purest examples of this approach, some program logics prove probabilistic properties by working purely with non-probabilistic assertions; we call such systems lightweight logics. Examples include probabilistic relational Hoare logic [3] for proving the reductionist security of cryptographic constructions, and the related approximate probabilistic relational Hoare logic [4] for reasoning about differential privacy. These logics rely on the powerful abstraction of probabilistic couplings to derive probabilistic facts from non-probabilistic assertions [7]
Strong Invariants Are Hard: On the Hardness of Strongest Polynomial Invariants for (Probabilistic) Programs
We show that computing the strongest polynomial invariant for single-path
loops with polynomial assignments is at least as hard as the Skolem problem, a
famous problem whose decidability has been open for almost a century. While the
strongest polynomial invariants are computable for affine loops, for polynomial
loops the problem remained wide open. As an intermediate result of independent
interest, we prove that reachability for discrete polynomial dynamical systems
is Skolem-hard as well. Furthermore, we generalize the notion of invariant
ideals and introduce moment invariant ideals for probabilistic programs. With
this tool, we further show that the strongest polynomial moment invariant is
(i) uncomputable, for probabilistic loops with branching statements, and (ii)
Skolem-hard to compute for polynomial probabilistic loops without branching
statements. Finally, we identify a class of probabilistic loops for which the
strongest polynomial moment invariant is computable and provide an algorithm
for it
Stochastic Invariants for Probabilistic Termination
Termination is one of the basic liveness properties, and we study the
termination problem for probabilistic programs with real-valued variables.
Previous works focused on the qualitative problem that asks whether an input
program terminates with probability~1 (almost-sure termination). A powerful
approach for this qualitative problem is the notion of ranking supermartingales
with respect to a given set of invariants. The quantitative problem
(probabilistic termination) asks for bounds on the termination probability. A
fundamental and conceptual drawback of the existing approaches to address
probabilistic termination is that even though the supermartingales consider the
probabilistic behavior of the programs, the invariants are obtained completely
ignoring the probabilistic aspect.
In this work we address the probabilistic termination problem for
linear-arithmetic probabilistic programs with nondeterminism. We define the
notion of {\em stochastic invariants}, which are constraints along with a
probability bound that the constraints hold. We introduce a concept of {\em
repulsing supermartingales}. First, we show that repulsing supermartingales can
be used to obtain bounds on the probability of the stochastic invariants.
Second, we show the effectiveness of repulsing supermartingales in the
following three ways: (1)~With a combination of ranking and repulsing
supermartingales we can compute lower bounds on the probability of termination;
(2)~repulsing supermartingales provide witnesses for refutation of almost-sure
termination; and (3)~with a combination of ranking and repulsing
supermartingales we can establish persistence properties of probabilistic
programs.
We also present results on related computational problems and an experimental
evaluation of our approach on academic examples.Comment: Full version of a paper published at POPL 2017. 20 page
Formal verification of higher-order probabilistic programs
Probabilistic programming provides a convenient lingua franca for writing
succinct and rigorous descriptions of probabilistic models and inference tasks.
Several probabilistic programming languages, including Anglican, Church or
Hakaru, derive their expressiveness from a powerful combination of continuous
distributions, conditioning, and higher-order functions. Although very
important for practical applications, these combined features raise fundamental
challenges for program semantics and verification. Several recent works offer
promising answers to these challenges, but their primary focus is on semantical
issues.
In this paper, we take a step further and we develop a set of program logics,
named PPV, for proving properties of programs written in an expressive
probabilistic higher-order language with continuous distributions and operators
for conditioning distributions by real-valued functions. Pleasingly, our
program logics retain the comfortable reasoning style of informal proofs thanks
to carefully selected axiomatizations of key results from probability theory.
The versatility of our logics is illustrated through the formal verification of
several intricate examples from statistics, probabilistic inference, and
machine learning. We further show the expressiveness of our logics by giving
sound embeddings of existing logics. In particular, we do this in a parametric
way by showing how the semantics idea of (unary and relational) TT-lifting can
be internalized in our logics. The soundness of PPV follows by interpreting
programs and assertions in quasi-Borel spaces (QBS), a recently proposed
variant of Borel spaces with a good structure for interpreting higher order
probabilistic programs