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Discrete or Continuous

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  • Jens
    In mathematics trees are considered being a special relation R on a well defined set S, i.e. they are a subset of of the cartesian set S x S with constraints.
    Message 1 of 6 , May 16, 2012
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      In mathematics trees are considered being a special relation R on a well defined set S, i.e. they are a subset of of the cartesian set S x S with constraints. There is no way around one property: the relation R is a discrete one regardless of the ground set S.

      Can anyone (in this list) imagine a continous relation on sets, i.e. continous trees?

      Jens
    • David Hobby
      On 5/16/2012 10:51 AM, Jens wrote: In mathematics trees are considered being a special relation R on a well defined set S, i.e. they are a subset of of the
      Message 2 of 6 , May 16, 2012
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        On 5/16/2012 10:51 AM, Jens wrote:
         

        In mathematics trees are considered being a special relation R on a well defined set S, i.e. they are a subset of of the cartesian set S x S with constraints. There is no way around one property: the relation R is a discrete one regardless of the ground set S.

        Can anyone (in this list) imagine a continous relation on sets, i.e. continous trees?

        Jens

        Hi.  You could certainly treat R as a fuzzy subset of SxS, giving a fuzzy
        relation.  I'm not sure it's been developed beyond the basics in the literature,
        but that's the language I'd use to describe "continuous trees".

                    ---David
      • mwinter@brocku.ca
        Hi Jens, ... I am not sure what you actually mean by continuous but here are some ideas/suggestions. First of all, you might want to consider relation algebras
        Message 3 of 6 , May 16, 2012
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          Hi Jens,

          > In mathematics trees are considered being a special relation R on a well defined set S, i.e. > they are a subset of of the cartesian set S x S with constraints. There is no way around one > property: the relation R is a discrete one regardless of the ground set S.
          >
          > Can anyone (in this list) imagine a continous relation on sets, i.e. continous trees?

          I am not sure what you actually mean by continuous but here are some ideas/suggestions.
          First of all, you might want to consider relation algebras (or similar structures). In that
          language you can define trees without refering to the elements of the underlying set S
          (see for example [1]). Such a definition will also work in the case of fuzzy relations which
          were mentioned before, if that is what you are interested in. You will find some material
          about a relation algebraic treatment of fuzzy relations in [2].

          On the other hand, continuity normally refers to some kind of underlying topology. Some
          considerations about these kind of relations (in particular with repsect to computabiltiy)
          you'll find in [3].

          I hope that helps :)
          Michael

          [1] Schmidt, Stroehlein: Relations and Graphs. Springer 1993
          [2] Winter: Goguen categories. Springer 2007
          [3] Brattka, Hertlling: Continuity and computability of relations. University Hagen 1994.
        • Vaughan Pratt
          Posted by: Jens jd@cococo.de
          Message 4 of 6 , May 17, 2012
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            Posted by: "Jens" jd@... <mailto:jd@...?Subject=
            > Can anyone (in this list) imagine a continuous relation on sets,

            Sure. 2^R in DCPO*.

            Here DCPO* is the cartesian closed category of *pointed*
            directed-join-complete posets, or DCPOs with bottom, and their
            directed-join-preserving functions. 2 is the "flat" or V-shaped DCPO
            {_|_,0,1}, with _|_ < 0 and _|_ < 1 but with 0 and 1 incomparable. R
            however is not a flat domain in that sense but rather is the linearly
            ordered reals including -oo (which maps must send to bottom) and oo
            (which maps can send anywhere, but any map from R to 2 that sends it to
            0 or 1 must send some finite real to the same value).

            Claim: 2^R is a tree-shaped DCPO whose root is its bottom (the function
            that maps every real to bottom, i.e. are nowhere defined) and whose
            leaves are its total elements (the functions that map no real to bottom,
            i.e. are everywhere defined), namely the subsets of R.

            To understand 2^R precisely it helps to consider 1^R first where 1 is
            the flat singleton {_|_,0}. This is just R since it amounts to Dedekind
            cuts in R ordered by inclusion of order filters (reverse inclusion of
            order ideals). (R is the nearest thing to the linearly ordered poset Q
            of rationals in DCPO* because Q is not directed-join-complete as a
            partial order.)

            Bottom in 1^R is the everywhere-undefined function. This function can't
            be isolated in 1^R (i.e. 1^R is atomless) because if a function is
            undefined at all finite reals it is undefined at top by
            directed-join-completeness.

            Top in 1^R is the function defined everywhere except at bottom This
            function can't be isolated (1^R is coatomless) because a coatom would
            have to be undefined at some finite x, and the function that is defined
            precisely at x and all larger reals would be a strictly larger function
            that is still not defined everywhere.

            Viewing 2^R as a tree, each path can be understood as a copy of 1^R.
            This set of paths would be a nice example of a set of Brouwerian choice
            sequences for which choices are made continuously working from left to
            right along the real axis, with the end product of a sequence being a
            fully determined subset of R.

            R being the unique complete (in the order interval topology rather than
            the Scott topology) dense linear order, there is something canonical
            about this example of a continuous tree. I suppose one could say it was
            the unique complete dense binary tree, if that makes sense.

            Vaughan Pratt
          • Vaughan Pratt
            ... Correction, that should read right to left. The earlier choices are made on the larger reals Vaughan Pratt
            Message 5 of 6 , May 17, 2012
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              On 5/17/2012 1:32 PM, Vaughan Pratt wrote:
              > a set of Brouwerian choice sequences for which choices are made
              > continuously working from left to right along the real axis

              Correction, that should read "right to left." The earlier choices are
              made on the larger reals

              Vaughan Pratt
            • Jens
              Thanks for the answers. The origin of my question was in the area of automated reasoning and also in classification schemes. I ll continue reasoning about
              Message 6 of 6 , May 21, 2012
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                Thanks for the answers. The origin of my question was in the area of automated reasoning and also in classification schemes. I'll continue reasoning about reasoning with fuzzy sets and DCPOs in mind...
                Jens
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