Consider a decision problem that cannot be 1 ? approximated by circuits of a given size in the sense that any such circuit fails to give the correct answer on at least a fraction of instances. We show that for any such problem there is a speciic \hard-core" set of inputs which is at least a fraction of all inputs and on which no circuit of a slightly smaller size can get even a small advantage ...

This paper reviews some of the work in the area of statistical circuit design. While space does not permit a more then superficial survey of the entire field, specific emphasis is placed on those techniques more familiar to the authors.

We present a number of, somewhat unusual, ways of describing what Craig’s interpolation theorem achieves, and use them to identify some open problems and further directions.

BACKGROUND CONTEXT The X-STOP interspinous decompression device, as a treatment for neurogenic intermittent claudication (NIC) because of lumbar spinal stenosis (LSS), has been shown to be superior to nonoperative control treatment. Current Food and Drug Administration labeling limits X-STOP use to NIC patients with a maximum of 25° concomitant lumbar scoliosis. This value was arrived at arbitr...