Unless I am somehow forgetting some clever interpretation, I was wrong about the transitivity.
Let me try to reason from an example: an experimenter defines a litany of tests for deciding how well a collection of things can be relied upon when treated as computational objects. For instance, an audiophile may have a box of capacitors that they wish to rank according to how well the caps filter out hum without suppressing the dynamic range of the music. This process defines a partition function on the box of capacitors. In a limiting case, we can imagine having only two buckets, one with caps that are good enough and the other with those that are not. In this coarse way, transitivity holds because we either grabbed 3 caps that are from the *good enough* bucket or we did not. What I think I found confusing has to do with the distance function d:: C_t x R_t(H) -> K, with K some ring. Here, allowing the C_t param to vary has the effect of allowing the problem dependence to vary, or as in the example above, allowing the hum tolerance to vary. Fixing a problem domain fixes the C_T and this is rather instead like providing a space equipped with a fixed origin. From that the more familiar distance function d':: R_T(H) x R_T(H) -> K can easily be formed with nice transitivity features and all. Now that I am reoriented a bit, I think an interpretation in terms of V-profunctors and the closed monoidal categories we discussed in the linear logic discussions could be fruitful. In effect, the function d as defined in the paper is effectively a profunctor interpreted via a Cost quantale, covariant in the Abstract category parameter, and contravariant in the Physical category parameter. Dang, I hope some part of this makes any sense :) -- Sent from: http://friam.471366.n2.nabble.com/ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/
