512 research outputs found
Combinatorial methods for the evaluation of yield and operational reliability of fault-tolerant systems-on-chip
In this paper we develop combinatorial methods for the evaluation of yield and operational reliability of fault-tolerant systems-on-chip. The method for yield computation assumes that defects are produced according to a model in which defects are lethal and affect given components of the system following a distribution common to all defects; the method for the computation of operational reliability also assumes that the fault-tree function of the system is increasing.
The distribution of the number of defects is arbitrary. The methods are based on the formulation of, respectively, the yield and the operational reliability as the probability that a given boolean function with multiple-valued variables has value 1. That probability is computed by analyzing
a ROMDD (reduced ordered multiple-value decision diagram) representation of the function.
For efficiency reasons, a coded ROBDD (reduced ordered binary decision diagram) representation of the function is built first and, then, that coded ROBDD is transformed into the ROMDD required by the methods. We present numerical experiments showing that the methods are able to cope with quite large systems in moderate CPU times.Postprint (published version
Improving Optimization Bounds using Machine Learning: Decision Diagrams meet Deep Reinforcement Learning
Finding tight bounds on the optimal solution is a critical element of
practical solution methods for discrete optimization problems. In the last
decade, decision diagrams (DDs) have brought a new perspective on obtaining
upper and lower bounds that can be significantly better than classical bounding
mechanisms, such as linear relaxations. It is well known that the quality of
the bounds achieved through this flexible bounding method is highly reliant on
the ordering of variables chosen for building the diagram, and finding an
ordering that optimizes standard metrics is an NP-hard problem. In this paper,
we propose an innovative and generic approach based on deep reinforcement
learning for obtaining an ordering for tightening the bounds obtained with
relaxed and restricted DDs. We apply the approach to both the Maximum
Independent Set Problem and the Maximum Cut Problem. Experimental results on
synthetic instances show that the deep reinforcement learning approach, by
achieving tighter objective function bounds, generally outperforms ordering
methods commonly used in the literature when the distribution of instances is
known. To the best knowledge of the authors, this is the first paper to apply
machine learning to directly improve relaxation bounds obtained by
general-purpose bounding mechanisms for combinatorial optimization problems.Comment: Accepted and presented at AAAI'1
LEO: Learning Efficient Orderings for Multiobjective Binary Decision Diagrams
Approaches based on Binary decision diagrams (BDDs) have recently achieved
state-of-the-art results for multiobjective integer programming problems. The
variable ordering used in constructing BDDs can have a significant impact on
their size and on the quality of bounds derived from relaxed or restricted BDDs
for single-objective optimization problems. We first showcase a similar impact
of variable ordering on the Pareto frontier (PF) enumeration time for the
multiobjective knapsack problem, suggesting the need for deriving variable
ordering methods that improve the scalability of the multiobjective BDD
approach. To that end, we derive a novel parameter configuration space based on
variable scoring functions which are linear in a small set of interpretable and
easy-to-compute variable features. We show how the configuration space can be
efficiently explored using black-box optimization, circumventing the curse of
dimensionality (in the number of variables and objectives), and finding good
orderings that reduce the PF enumeration time. However, black-box optimization
approaches incur a computational overhead that outweighs the reduction in time
due to good variable ordering. To alleviate this issue, we propose LEO, a
supervised learning approach for finding efficient variable orderings that
reduce the enumeration time. Experiments on benchmark sets from the knapsack
problem with 3-7 objectives and up to 80 variables show that LEO is ~30-300%
and ~10-200% faster at PF enumeration than common ordering strategies and
algorithm configuration. Our code and instances are available at
https://github.com/khalil-research/leo
New developments in the theory of Groebner bases and applications to formal verification
We present foundational work on standard bases over rings and on Boolean
Groebner bases in the framework of Boolean functions. The research was
motivated by our collaboration with electrical engineers and computer
scientists on problems arising from formal verification of digital circuits. In
fact, algebraic modelling of formal verification problems is developed on the
word-level as well as on the bit-level. The word-level model leads to Groebner
basis in the polynomial ring over Z/2n while the bit-level model leads to
Boolean Groebner bases. In addition to the theoretical foundations of both
approaches, the algorithms have been implemented. Using these implementations
we show that special data structures and the exploitation of symmetries make
Groebner bases competitive to state-of-the-art tools from formal verification
but having the advantage of being systematic and more flexible.Comment: 44 pages, 8 figures, submitted to the Special Issue of the Journal of
Pure and Applied Algebr
Chain Reduction for Binary and Zero-Suppressed Decision Diagrams
Chain reduction enables reduced ordered binary decision diagrams (BDDs) and
zero-suppressed binary decision diagrams (ZDDs) to each take advantage of the
others' ability to symbolically represent Boolean functions in compact form.
For any Boolean function, its chain-reduced ZDD (CZDD) representation will be
no larger than its ZDD representation, and at most twice the size of its BDD
representation. The chain-reduced BDD (CBDD) of a function will be no larger
than its BDD representation, and at most three times the size of its CZDD
representation. Extensions to the standard algorithms for operating on BDDs and
ZDDs enable them to operate on the chain-reduced versions. Experimental
evaluations on representative benchmarks for encoding word lists, solving
combinatorial problems, and operating on digital circuits indicate that chain
reduction can provide significant benefits in terms of both memory and
execution time
Advances in Functional Decomposition: Theory and Applications
Functional decomposition aims at finding efficient representations for Boolean functions. It is used in many applications, including multi-level logic synthesis, formal verification, and testing.
This dissertation presents novel heuristic algorithms for functional decomposition. These algorithms take advantage of suitable representations of the Boolean functions in order to be efficient.
The first two algorithms compute simple-disjoint and disjoint-support decompositions. They are based on representing the target function by a Reduced Ordered Binary Decision Diagram (BDD). Unlike other BDD-based algorithms, the presented ones can deal with larger target functions and produce more decompositions without requiring expensive manipulations of the representation, particularly BDD reordering.
The third algorithm also finds disjoint-support decompositions, but it is based on a technique which integrates circuit graph analysis and BDD-based decomposition. The combination of the two approaches results in an algorithm which is more robust than a purely BDD-based one, and that improves both the quality of the results and the running time.
The fourth algorithm uses circuit graph analysis to obtain non-disjoint decompositions. We show that the problem of computing non-disjoint decompositions can be reduced to the problem of computing multiple-vertex dominators. We also prove that multiple-vertex dominators can be found in polynomial time. This result is important because there is no known polynomial time algorithm for computing all non-disjoint decompositions of a Boolean function.
The fifth algorithm provides an efficient means to decompose a function at the circuit graph level, by using information derived from a BDD representation. This is done without the expensive circuit re-synthesis normally associated with BDD-based decomposition approaches.
Finally we present two publications that resulted from the many detours we have taken along the winding path of our research
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