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## Static Analysis of Programs with Imprecise Probabilistic Inputs

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3134 |
A mathematical theory of evidence.
- Shafer
- 1976
(Show Context)
Citation Context ... and F is ω-continuous), which encodes the set of all the measures µ such that ∫ x∈X h(x)dµ ≥ F (h) for every h. Implementations of imprecise probabilities. P-boxes [17] and DempsterShafer structures =-=[36]-=-, which are both related to capacities, are used to propagate both probabilistic and non-deterministic information in numerical simulation for instance. Arithmetic rules on P-boxes were defined in e.g... |

3087 |
An Introduction to Probability Theory and Its Applications,
- FELLER
- 1968
(Show Context)
Citation Context ...y distributions (but not imprecise probabilities) are considered on transitions of a transition model (and not on data, as we do here). The models used are mostly based on discrete time Markov chains =-=[14]-=-. In static analysis of programs by abstract interpretation, which is the subject of this paper, several abstract semantics have been considered. Monniaux [32] automatically constructs a probabilistic... |

1072 |
Statistical Reasoning with Imprecise Probabilities.
- Walley
- 1991
(Show Context)
Citation Context ...ies. There is a vast literature on the subject, and there are several mathematical notions that model imprecise probabilities, among which those based on capacities [8], and those based on previsions =-=[41]-=-. Capacities are simply monotone functions that map each measurable subset to its measure, such that the measure of the empty set is 0; but the measure of the disjoint union of two sets A and B does n... |

376 |
Theory of capacities. Annales de l’Institut Fourier (Grenoble).
- Choquet
- 1953
(Show Context)
Citation Context ...with so-called imprecise probabilities. There is a vast literature on the subject, and there are several mathematical notions that model imprecise probabilities, among which those based on capacities =-=[8]-=-, and those based on previsions [41]. Capacities are simply monotone functions that map each measurable subset to its measure, such that the measure of the empty set is 0; but the measure of the disjo... |

237 | Temporal abstract interpretation, in:
- Cousot, Cousot
- 2000
(Show Context)
Citation Context ...bility over the intended value v as the image measure of pi by some measurable map ha l-0 09 42 12 6,sv er sio ns1s-sf from Ω to the space of values. This is the approach taken by Cousot and Monereau =-=[10]-=-, where Ω is the space of infinite sequences of coin flips, each coin flip being independent and unbiased. A probability distribution on a space X is then encoded by a measurable map f : Ω → X, and th... |

236 | PRISM 4.0: Verification of probabilistic real-time systems.
- Kwiatkowska, Norman, et al.
- 2011
(Show Context)
Citation Context ... analysis of probabilistic systems, some in abstract interpretation but most notably in model-checking. Our work is orthogonal to the one in probabilistic model-checking (as implemented in e.g. PRISM =-=[27]-=-) where probability distributions (but not imprecise probabilities) are considered on transitions of a transition model (and not on data, as we do here). The models used are mostly based on discrete t... |

119 |
Construction probability boxes and Dempster-Shafer structures,
- Ferson, Kreinovich, et al.
- 2003
(Show Context)
Citation Context ...amely, F (h) + F (h′) ≤ F (h + h′), and F is ω-continuous), which encodes the set of all the measures µ such that ∫ x∈X h(x)dµ ≥ F (h) for every h. Implementations of imprecise probabilities. P-boxes =-=[17]-=- and DempsterShafer structures [36], which are both related to capacities, are used to propagate both probabilistic and non-deterministic information in numerical simulation for instance. Arithmetic r... |

108 |
Probabilistic Arithmetic I: Numerical Methods for Calculating Convolutions and Dependency Bounds,
- Williamson, Downs
- 1990
(Show Context)
Citation Context ...which are both related to capacities, are used to propagate both probabilistic and non-deterministic information in numerical simulation for instance. Arithmetic rules on P-boxes were defined in e.g. =-=[42]-=-, and implementations are available, for instance the DSI Toolbox [2] based on Matlab and INTLAB [34], Statool [4] implementing the arithmetic of [3] and RiskCalc [16]. They were not designed to be us... |

79 | Affine arithmetic and its applications to computer graphics. Anais do VII SIBGRAPI
- Comba, Stolfi
- 1993
(Show Context)
Citation Context ... non-linear relations (due to non-linear operations in the program) are over-approximated by an additional linear term. More formally, perturbed affine forms [20, 21] are an extension of affine forms =-=[9]-=- in which each variable x is over-approximated by an expression of the form x̂ = αx0 + ∑n i=1 α x i εi + ∑m j=1 β x j ηj where the noise symbols εi or ηj are formal variables ranging over [−1, 1] just... |

70 |
Static analysis of digital filters
- Feret
- 2004
(Show Context)
Citation Context ...s as they are present in (almost) every software that must handle data coming from sensors. Computing the range of values reachable by the output variable y is a challenge adressed by many techniques =-=[15]-=-. However, all these methods assume that the inputs x (given by the function input() in the program) are bounded within a certain range and do not assume any distribution of the values within this ran... |

67 |
Bounding the Results of Arithmetic Operations on Random Variables of Unknown Dependency Using Intervals,
- Berleant, Goodman-Strauss
- 1998
(Show Context)
Citation Context ... Arithmetic rules on P-boxes were defined in e.g. [42], and implementations are available, for instance the DSI Toolbox [2] based on Matlab and INTLAB [34], Statool [4] implementing the arithmetic of =-=[3]-=- and RiskCalc [16]. They were not designed to be used for static analysis of programs (there is no consideration on semantics of programs nor join operators defined, typically) as we do in this paper ... |

50 |
Nonlinear Signal Processing: A Statistical Approach”,
- Arce
- 2005
(Show Context)
Citation Context ...hysical sensors. Imagine a signal processing software that filters out thermal noise [29] from images given by a digital camera, for example with nonlinear filtering techniques such as median filters =-=[1]-=-. Thermal noise is such that each pixel has an independent Gaussian noise, with zero mean and a standard deviation varying according to Nyquist law [29]. In particular, the standard deviation depends ... |

49 | Statool: A Tool for Distribution Envelope Determination DEnv, An Interval-Based Algorithm for Arithmetic on Random Variables,”
- Berleant, Xie, et al.
- 2003
(Show Context)
Citation Context ... numerical simulation for instance. Arithmetic rules on P-boxes were defined in e.g. [42], and implementations are available, for instance the DSI Toolbox [2] based on Matlab and INTLAB [34], Statool =-=[4]-=- implementing the arithmetic of [3] and RiskCalc [16]. They were not designed to be used for static analysis of programs (there is no consideration on semantics of programs nor join operators defined,... |

44 |
RAMAS Risk Calc 4.0 Software: Risk Assessment with Uncertain Numbers
- Ferson
- 2002
(Show Context)
Citation Context ... on P-boxes were defined in e.g. [42], and implementations are available, for instance the DSI Toolbox [2] based on Matlab and INTLAB [34], Statool [4] implementing the arithmetic of [3] and RiskCalc =-=[16]-=-. They were not designed to be used for static analysis of programs (there is no consideration on semantics of programs nor join operators defined, typically) as we do in this paper but are rather des... |

37 | Abstract interpretation of probabilistic semantics
- Monniaux
- 2000
(Show Context)
Citation Context ...tly based on discrete time Markov chains [14]. In static analysis of programs by abstract interpretation, which is the subject of this paper, several abstract semantics have been considered. Monniaux =-=[32]-=- automatically constructs a probabilistic abstract domain as a collection of abstract elements with an associated weight. This is very similar to Dempster-Shafer structures where focal elements are el... |

32 | and G.Plotkin. Semantic domains for combining probability and non-determinism.
- Tix
- 2009
(Show Context)
Citation Context ...assing. Capacity-based semantics fail because we cannot even define sequential composition there [24]; sequential composition is defined by complex formulas in other models, such as convex powercones =-=[40]-=-, where this involves unique extensions of maps to sets of non-empty closed convex subsets. There are variations in what a prevision on a space X of values is. To us, a prevision on X will be any map ... |

29 | Nondeterminism and probabilistic choice: Obeying the laws.
- Mislove
- 2000
(Show Context)
Citation Context ... probabilities were studied by several authors, among which one of the authors of this paper [24, 23, 26, 25]. In particular, the convex powerdomains of spaces of measures on X was studied by Mislove =-=[31]-=-, by Tix et al. [39, 40], and by Morgan and McIver [30]. Static analysis of probabilistic systems. There is a large literature in static analysis of probabilistic systems, some in abstract interpretat... |

22 | Unifying practical uncertainty representations: I. Generalized p-boxes,
- Destercke, Dubois, et al.
- 2008
(Show Context)
Citation Context ...] and [5]. Let us mention as well Neumaier’s clouds [33] as another way to formalize imprecise probabilities (used in [19]). A unification of the different uncertainty representations was proposed in =-=[11, 12]-=- with comparisons between P-boxes and clouds. The domain theoretic foundations of imprecise probabilities were studied by several authors, among which one of the authors of this paper [24, 23, 26, 25]... |

22 | Static analysis of finite precision computations.
- Goubault, Putot
- 2011
(Show Context)
Citation Context ...al numbers can be handled at the level of the abstract semantics. One can extend probabilistic affine forms to handle rounding errors, as quickly described in [5], in the same way as for affine forms =-=[22]-=-, and we intend to invest in that direction in the future. 4 Abstract semantics We now formally define our abstract semantics. It is based on an abstract domain that extends the probability affine for... |

18 | Potential based clouds in robust design optimization.
- Fuchs, Neumaier
- 2009
(Show Context)
Citation Context ...of programs (there is no consideration on semantics of programs nor join operators defined, typically) as we do in this paper but are rather designed for making numerical simulations or optimizations =-=[19]-=- for instance for risk assessment [18]. Several recent papers proposed extensions of these arithmetics that either increase the precision or the efficiency of this arithmetic, as in e.g. [7], [37], [3... |

18 | C.: Demonic, angelic and unbounded probabilistic choices in sequential programs
- McIver, Morgan
- 2001
(Show Context)
Citation Context ...hich one of the authors of this paper [24, 23, 26, 25]. In particular, the convex powerdomains of spaces of measures on X was studied by Mislove [31], by Tix et al. [39, 40], and by Morgan and McIver =-=[30]-=-. Static analysis of probabilistic systems. There is a large literature in static analysis of probabilistic systems, some in abstract interpretation but most notably in model-checking. Our work is ort... |

18 |
Continuous D-Cones: Convexity and Powerdomain Constructions.
- Tix
- 1999
(Show Context)
Citation Context ...studied by several authors, among which one of the authors of this paper [24, 23, 26, 25]. In particular, the convex powerdomains of spaces of measures on X was studied by Mislove [31], by Tix et al. =-=[39, 40]-=-, and by Morgan and McIver [30]. Static analysis of probabilistic systems. There is a large literature in static analysis of probabilistic systems, some in abstract interpretation but most notably in ... |

16 | S.: A logical product approach to zonotope intersection
- Ghorbal, Goubault, et al.
- 2010
(Show Context)
Citation Context ...e the arithmetic operations. The potential non-linear relations (due to non-linear operations in the program) are over-approximated by an additional linear term. More formally, perturbed affine forms =-=[20, 21]-=- are an extension of affine forms [9] in which each variable x is over-approximated by an expression of the form x̂ = αx0 + ∑n i=1 α x i εi + ∑m j=1 β x j ηj where the noise symbols εi or ηj are forma... |

16 | Static analysis of probabilistic programs: Inferring whole program properties from finitely many executions.
- Sankaranarayanan, Chakarov, et al.
- 2013
(Show Context)
Citation Context ...recise semantic invariants from the code of [f ]. Our approach, based on P-box approximants to actual sets of distribution laws, is more direct. Another approach, which is very promising, is taken in =-=[35]-=- that presents an approach for finding interval bounds on the probability of assertions over program variables by examining finitely many paths and performing a standard symbolic execution along each ... |

13 | HybridFluctuat: A Static Analyzer of Numerical Programs Within a Continuous Environment,” Computer Aided Verification:
- Bouissou, Goubault, et al.
- 2009
(Show Context)
Citation Context ...ear in embedded systems and only treated them as open-loop programs, i.e. we ignored the feedback from the program to its plant. In the future, we shall extend this work to treat hybrid systems as in =-=[6]-=-. This will require to be able to handle ODEs with initial values given as probabilistic affine forms. As shown by our second benchmark, we think that we can extend guaranteed ODE solvers to make them... |

13 | Continuous capacities on continuous state spaces
- Goubault-Larrecq
- 2007
(Show Context)
Citation Context ...osed in [11, 12] with comparisons between P-boxes and clouds. The domain theoretic foundations of imprecise probabilities were studied by several authors, among which one of the authors of this paper =-=[24, 23, 26, 25]-=-. In particular, the convex powerdomains of spaces of measures on X was studied by Mislove [31], by Tix et al. [39, 40], and by Morgan and McIver [30]. Static analysis of probabilistic systems. There ... |

13 |
Clouds, fuzzy sets and probability intervals. Reliable Computing
- Neumaier
- 2004
(Show Context)
Citation Context ...t papers proposed extensions of these arithmetics that either increase the precision or the efficiency of this arithmetic, as in e.g. [7], [37], [38] and [5]. Let us mention as well Neumaier’s clouds =-=[33]-=- as another way to formalize imprecise probabilities (used in [19]). A unification of the different uncertainty representations was proposed in [11, 12] with comparisons between P-boxes and clouds. Th... |

9 | S.: A zonotopic framework for functional abstractions
- Goubault, Putot
- 2009
(Show Context)
Citation Context ...rithmetic operations were defined in [5]. Intuitively, a probabilistic affine form encodes both the linear dependency between every program variable and the input (as with classical affine forms, see =-=[21]-=-), and an abstraction of the inputs as a DSI. We can thus compute the DSI associated with each variable (it is a linear transformation of the inputs), and we use the linear correlations between variab... |

9 | Prevision domains and convex powercones - Goubault-Larrecq - 2008 |

9 |
B.: Op Amps for Everyone
- Mancini, Carter
- 2009
(Show Context)
Citation Context ... to imprecise inputs. Typically, these inputs will be numerical values ha l-0 09 42 12 6,sv er sio ns1s-sgiven by physical sensors. Imagine a signal processing software that filters out thermal noise =-=[29]-=- from images given by a digital camera, for example with nonlinear filtering techniques such as median filters [1]. Thermal noise is such that each pixel has an independent Gaussian noise, with zero m... |

7 | Continuous previsions
- Goubault-Larrecq
- 2007
(Show Context)
Citation Context ... of two sets A and B does not necessarily coincide with the sum of their measures. Previsions [41], on the other hand, are more abstract objects, but are better suited to giving semantics to programs =-=[24]-=-, in a style akin to continuationpassing. Capacity-based semantics fail because we cannot even define sequential composition there [24]; sequential composition is defined by complex formulas in other ... |

7 | Validated Solution of Initial Value Problems for ODEs with Interval Parameters”.
- Lin, Stadtherr
- 2006
(Show Context)
Citation Context ...the parameters θ1 and θ2 are uncertain: they are given by a normal distribution with mean 3 and 1, resp., but with an unknown standard deviation in the range [−0.01, 0.01]. As in [13], we used VSPODE =-=[28]-=- to obtain a Taylor model polynomial that expresses the solution at tf = 20 as an order 5 polynomial of θ1 and θ2. We then used the probabilistic affine forms to evaluate the Horner form of this polyn... |

5 | K.: Choquet-Kendall-Matheron theorems for nonHausdorff spaces
- Goubault-Larrecq, Keimel
- 2011
(Show Context)
Citation Context ...osed in [11, 12] with comparisons between P-boxes and clouds. The domain theoretic foundations of imprecise probabilities were studied by several authors, among which one of the authors of this paper =-=[24, 23, 26, 25]-=-. In particular, the convex powerdomains of spaces of measures on X was studied by Mislove [31], by Tix et al. [39, 40], and by Morgan and McIver [30]. Static analysis of probabilistic systems. There ... |

3 | P.: A verified matlab toolbox for the dempster-shafer theory
- Auer, Luther, et al.
- 2010
(Show Context)
Citation Context .... We then graphically represent d by the graphs of the two functions [ Pd, Pd ] . Example 1. Let d1 = {〈[−1, 0.25], 0.1〉 ; 〈[−0.5, 0.5], 0.2〉 ; 〈[0.25, 1], 0.3〉 ; 〈[0.5, 1], 0.1〉 ; 〈[0.5, 2], 0.1〉 ; 〈=-=[1, 2]-=-, 0.2〉}. Then [P2, P2] = ζ(d1) is plotted on the graph below. −1 −0.5 0.25 0.5 1 2 1 Join and meet on DS structures The join of two DSI dX and dY is defined as the union of all focal elements from dX ... |

2 |
What Monte-Carlo Methods Cannot Do
- Ferson
- 1996
(Show Context)
Citation Context ... on semantics of programs nor join operators defined, typically) as we do in this paper but are rather designed for making numerical simulations or optimizations [19] for instance for risk assessment =-=[18]-=-. Several recent papers proposed extensions of these arithmetics that either increase the precision or the efficiency of this arithmetic, as in e.g. [7], [37], [38] and [5]. Let us mention as well Neu... |

2 |
P.D.: Approximate interval method for epistemic uncertainty propagation using polynomial chaos and evidence theory
- Terejanu, Singla, et al.
- 2010
(Show Context)
Citation Context ...9] for instance for risk assessment [18]. Several recent papers proposed extensions of these arithmetics that either increase the precision or the efficiency of this arithmetic, as in e.g. [7], [37], =-=[38]-=- and [5]. Let us mention as well Neumaier’s clouds [33] as another way to formalize imprecise probabilities (used in [19]). A unification of the different uncertainty representations was proposed in [... |

1 |
S.: A generalization of pboxes to affine arithmetic. Computing
- Bouissou, Goubault, et al.
- 2011
(Show Context)
Citation Context ...stance for risk assessment [18]. Several recent papers proposed extensions of these arithmetics that either increase the precision or the efficiency of this arithmetic, as in e.g. [7], [37], [38] and =-=[5]-=-. Let us mention as well Neumaier’s clouds [33] as another way to formalize imprecise probabilities (used in [19]). A unification of the different uncertainty representations was proposed in [11, 12] ... |

1 | O.: A faster algorithm for computing the sum of p-boxes
- Busaba, Suwan, et al.
- 2010
(Show Context)
Citation Context ...izations [19] for instance for risk assessment [18]. Several recent papers proposed extensions of these arithmetics that either increase the precision or the efficiency of this arithmetic, as in e.g. =-=[7]-=-, [37], [38] and [5]. Let us mention as well Neumaier’s clouds [33] as another way to formalize imprecise probabilities (used in [19]). A unification of the different uncertainty representations was p... |

1 | Probability bounds analysis for nonlinear dynamic process models
- Enszer, Lin, et al.
- 2011
(Show Context)
Citation Context ... distance between the lower and upper probabilities, in the abstract which is about twice as much as in our simulations, which is still quite precise. 6.2 Ferson polynomial We now use an example from =-=[13]-=- to test the precision and performance of our abstract domain on arithmetic operations. The problem is to compute bounds on the solution of the differential equations ẋ1 = θ1x1(1− x2) ẋ2 = θ2x2(x1 −... |

1 |
J.M.: Chebyshev affine arithmetic based parametric yield prediction under limited descriptions of uncertainty
- Sun, Huang, et al.
(Show Context)
Citation Context ...ons [19] for instance for risk assessment [18]. Several recent papers proposed extensions of these arithmetics that either increase the precision or the efficiency of this arithmetic, as in e.g. [7], =-=[37]-=-, [38] and [5]. Let us mention as well Neumaier’s clouds [33] as another way to formalize imprecise probabilities (used in [19]). A unification of the different uncertainty representations was propose... |

1 |
09:35 pm (CET). Version 1.1 B Operational semantics We start with a small-step operational semantics, given in Figure 6. Its states are pairs ( Λ, ρ ) , where Λ is a finite list of statements, to be executed sequentially. The grammar for such lists is: Λ
- Friday
- 2013
(Show Context)
Citation Context .... We then graphically represent d by the graphs of the two functions [ Pd, Pd ] . Example 1. Let d1 = {〈[−1, 0.25], 0.1〉 ; 〈[−0.5, 0.5], 0.2〉 ; 〈[0.25, 1], 0.3〉 ; 〈[0.5, 1], 0.1〉 ; 〈[0.5, 2], 0.1〉 ; 〈=-=[1, 2]-=-, 0.2〉}. Then [P2, P2] = ζ(d1) is plotted on the graph below. −1 −0.5 0.25 0.5 1 2 1 Join and meet on DS structures The join of two DSI dX and dY is defined as the union of all focal elements from dX ... |