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On the NonApproximability of Boolean Functions by OBDDs and ReadKTimes Branching Programs
"... Branching programs are considered as a nonuniform model of computation in complexity theory as well as a data structure for boolean functions in several applications. In many applications (e. g., verification), exact representations are required. For learning boolean functions f on the basis of clas ..."
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Branching programs are considered as a nonuniform model of computation in complexity theory as well as a data structure for boolean functions in several applications. In many applications (e. g., verification), exact representations are required. For learning boolean functions f on the basis of classified examples, it is sufficient to produce the representation of a function g approximating f . This motivates the investigation of the size of the smallest branching program approximating f . Although several nonapproximability results are contained in the papers on randomized branching programs, these results often do not work for the uniform distribution (which is the most important one in applications). Here, the following nonapproximability results are presented. (1) It is proven that a simple functions from the branching program literature requires exponential size to be approximated with respect to the uniform distribution by OBDDs, which are the most important type of branching programs in applications. (2) The first truly exponential lower bound on the size of approximating syntactic readktimes branching programs with respect to the uniform distribution and error probability 1/22 # n) , n the input size, is shown. In order to improve upon the so far best results for error probabilities smaller than 1/3, a strong combinatorial lemma from a recent paper of Ajtai on linearlength branching programs is exploited. Keywords: Computational complexity, branching programs, binary decision diagrams, approximations, lower bounds. # Supported in part by DFG We 1066/9. 1
Approximability and NonApproximability by Binary Decision Diagrams (Extended Abstract)
, 1995
"... The usual applications of BDDs (binary decision diagrams), e. g., in verification and for CAD problems, require an exact representation of the considered boolean functions. However, if BDDs are used for learning boolean functions f on the basis of classified examples (e. g., in the environment of ge ..."
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The usual applications of BDDs (binary decision diagrams), e. g., in verification and for CAD problems, require an exact representation of the considered boolean functions. However, if BDDs are used for learning boolean functions f on the basis of classified examples (e. g., in the environment of genetic programming), it is sufficient to produce the representation of a function g approximating f . This motivates the investigation of the size of the smallest BDD approximating f . Here exponential lower bounds for several BDD variants are proven and the relations between the size of approximating BDDs, randomized BDDs, communication complexity and general approximation techniques are revealed.