14 research outputs found
Static Analysis of Programs with Imprecise Probabilistic Inputs
International audienceHaving a precise yet sound abstraction of the inputs of numerical programs is important to analyze their behavior. For many programs, these inputs are probabilistic, but the actual distribution used is only partially known. We present a static analysis framework for reasoning about programs with inputs given as imprecise probabilities: we define a collecting semantics based on the notion of previsions and an abstract semantics based on an extension of Dempster-Shafer structures. We prove the correctness of our approach and show on some realistic examples the kind of invariants we are able to infer.Il est important de disposer d'une abstraction précise mais correcte des entrées d'un programme numérique pour analyser ses comportements. Pour de nombreux programmes, ces entrées sont probabilistes, mais la distribution réellement utilisée n'est connue que partiellement. Nous présentons un cadre d'analyse statique permettant le raisonnement sur des programmes dont les entrées sont données sous forme de probabilités imprécises: nous définissons une sémantique collectrice fondée sur la notion de prévisions et une sémantique abstraite fondée sur une extension des structures de Dempster-Shafer. Nous démontrons la correction de notre approche et montrons sur des exemples réalistes le genre d'invariants que nous sommes capables d'inférer
Healthiness Conditions for Predicate Transformers
AbstractThe behavior of a program can be modeled by describing how it transforms input states to output states, the state transformer semantics. Alternatively, for verification purposes one is interested in a 'predicate transformer semantics' which, for every condition on the output, yields the weakest precondition on the input that guarantees the desired property for the output.In the presence of computational effects like nondeterministic or probabilistic choice, a computation will be modeled by a map t:X→TY, where T is an appropriate computational monad. The corresponding predicate transformer assigns predicates on Y to predicates on X. One looks for necessary and, if possible, sufficient conditions (healthiness conditions) on predicate transformers that correspond to state transformers t:X→TY.In this paper we propose a framework for establishing healthiness conditions for predicate transformers. As far as the author knows, it fits to almost all situations in which healthiness conditions for predicate transformers have been worked out. It may serve as a guideline for finding new results; but it also shows quite narrow limitations
Mixed powerdomains for probability and nondeterminism
We consider mixed powerdomains combining ordinary nondeterminism and
probabilistic nondeterminism. We characterise them as free algebras for
suitable (in)equation-al theories; we establish functional representation
theorems; and we show equivalencies between state transformers and
appropriately healthy predicate transformers. The extended nonnegative reals
serve as `truth-values'. As usual with powerdomains, everything comes in three
flavours: lower, upper, and order-convex. The powerdomains are suitable convex
sets of subprobability valuations, corresponding to resolving nondeterministic
choice before probabilistic choice. Algebraically this corresponds to the
probabilistic choice operator distributing over the nondeterministic choice
operator. (An alternative approach to combining the two forms of nondeterminism
would be to resolve probabilistic choice first, arriving at a domain-theoretic
version of random sets. However, as we also show, the algebraic approach then
runs into difficulties.)
Rather than working directly with valuations, we take a domain-theoretic
functional-analytic approach, employing domain-theoretic abstract convex sets
called Kegelspitzen; these are equivalent to the abstract probabilistic
algebras of Graham and Jones, but are more convenient to work with. So we
define power Kegelspitzen, and consider free algebras, functional
representations, and predicate transformers. To do so we make use of previous
work on domain-theoretic cones (d-cones), with the bridge between the two of
them being provided by a free d-cone construction on Kegelspitzen
Monads and Quantitative Equational Theories for Nondeterminism and Probability
The monad of convex sets of probability distributions is a well-known tool for modelling the combination of nondeterministic and probabilistic computational effects. In this work we lift this monad from the category of sets to the category of extended metric spaces, by means of the Hausdorff and Kantorovich metric liftings. Our main result is the presentation of this lifted monad in terms of the quantitative equational theory of convex semilattices, using the framework of quantitative algebras recently introduced by Mardare, Panangaden and Plotkin
The Theory of Traces for Systems with Nondeterminism, Probability, and Termination
This paper studies trace-based equivalences for systems combining
nondeterministic and probabilistic choices. We show how trace semantics for
such processes can be recovered by instantiating a coalgebraic construction
known as the generalised powerset construction. We characterise and compare the
resulting semantics to known definitions of trace equivalences appearing in the
literature. Most of our results are based on the exciting interplay between
monads and their presentations via algebraic theories.Comment: This paper is an extended version of a LICS 2019 paper "The Theory of
Traces for Systems with Nondeterminism and Probability". It contains all the
proofs, additional explanations, material, and example
The theory of traces for systems with nondeterminism and probability
This paper studies trace-based equivalences for systems combining nondeterministic and probabilistic choices. We show how trace semantics for such processes can be recovered by instantiating a coalgebraic construction known as the generalised powerset construction. We characterise and compare the resulting semantics to known definitions of trace equivalences appearing in the literature. Most of our results are based on the exciting interplay between monads and their presentations via algebraic theories