Uncertainty in circuit performance due to manufacturing and environmental variations is increasing with each new generation of technology. It is therefore important to predict the performance of a chip as a probabilistic quantity. This paper proposes three novel algorithms for statistical timing analysis and parametric yield prediction of digital integrated circuits. The methods have been implemented in the context of the -42660 static timing analyzer. Numerical results are presented to study the strengths and weaknesses of these complementary approaches. Across-the-chip variability continues to be accommodated by 39516 's "Linear Combination of Delay (LCD)" mode. Timing analysis results in the face of statistical temperature and V dd variations are presented on an industrial ASIC part on which a bounded timing methodology leads to surprisingly wrong results.

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Abstract. Given a basic compact semi-algebraic set K ⊂ R n, we introduce a methodology that generates a sequence converging to the volume of K. This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs. Not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on K can be approximated as closely as desired, and so permits to approximate the integral on K of any given polynomial; extension to integration against some weight functions is also provided. Finally, some numerical issues associated with the algorithms involved are briefly discussed. Key words. Computational geometry; volume; integration; K-moment problem; semidefinite programming AMS subject classifications. 14P10, 11E25, 12D15, 90C25 1. Introduction. Computing

Towards predictable deep-submicron manufacturing

this paper, we show that a similar approach can also be used for linear programming problems, where the alternatives are only implicitly given via constraints on decision variables.

In this dissertation we present two techniques on the topic of digital interface design: a probabilistic timing verification and a timing analysis for synthesis, both rooted in a formal specification. Interface design arises when two digital components (e.g., a processor and a memory device) are to be interconnected to build up a system. We have extended a Petri net specification to describe the temporal behavior of the interface protocols of digital components. The specification describes circuit delays as random variables thus making it suitable to model process variations and timing correlation. Interface probabilistic timing verification checks that a subsystem, composed of components to be interconnected and the associated interface logic, satisfies the timing constraints specified by the components' specifications. Our verification technique not only yields tighter results than previous techniques that do not take timing correlation into consideration but also, if the timing cons...

Probabilistic logic programs (PLPs) define a set of probability distribution functions (PDFs) over the set of all Herbrand interpretations of the underlying logical language. When answering a query Q, a lower and upper bound on Q is obtained by optimizing (min and max) an objective function subject to a set of linear constraints whose solutions are the PDFs mentioned above. A common critique not only of PLPs but many probabilistic logics is that the difference between the upper bound and lower bound is large, thus often providing very little useful information in the query answer. In this paper, we provide a new method to answer probabilistic queries that tries to come up with a histogram that “maps ” the probability that the objective function will have a value in a given interval, subject to the above linear constraints. This allows the system to return to the user a histogram where he can directly “see ” what the most likely probability range for his query will be. We prove that computing these histograms is #P-hard, and show that computing these histograms is closely related to polyhedral volume computation. We show how existing randomized algorithms for volume computation can be adapted to the computation of such histograms. A prototype experimental implementation is discussed. 1

Abstract. We provide two algorithms for computing the volume of the convex polytope Ω: = {x ∈ R n + | Ax ≤ b}, for A ∈ R m×n, b ∈ R n. Both algorithms have a O(n m) computational complexity which makes them especially attractive for large n and relatively small m, when the other methods with O(m n) complexity fail. The methodology which differs from previous existing methods uses a Laplace transform technique that is well suited to the half-space representation of Ω. 1.