On 08/06/2017 11:41 PM, Linas Vepstas wrote:
Search becomes a problem. Say I want to find (using the pattern matcher)
all sets that contain, as members, A B C. With set notation, I have to
search for (Set X A B C) where X==(VariableNode "X") for 4-elelemnt
sets, and then (Set X Y A B C) for 5-element sets, and so on. To search
all N-element sets, I can use the GlobeNode G and search for (Set G A B
C) ... and I believe this works correctly today. However...
However, there's a problem. Sets are UnorderedLinks, and so performing
the search for all six-element sets (Set X Y Z A B C) requires searching
6!=120 permutations. Well, since A B C are fixed, only 6!/3!= 20
permutations have to be searched. Of the 20, many are duplicates, since
(Set X Y Z A B C) == (Set Y C X B Z A) and so on. The returned results
will include all possible matchups of X Y Z to the other three members
-- 3! in all -- so of the 20 permutations, only 3!=6 are actually different.
For the GlobNode, it would be similar, Now consider a set with 20
elements -- this requires 20! permutations and 20!=2.4e18 which would be
months or years of CPU time. Which is absurd.. well, we could stop
after finding exactly one permutation, because that is all we really
want... (not sure if the current glob code does this -- probably not.
Man Hin is reworking the glob code now.).
This is in contrast to the much easier problem of finding all values of
X for which (Member A X) AND (Member B X) AND (Member C X) which can be
done easily and quickly, without requiring permutation exploration.
I understand. In the first case, you ask to produce all assignments to
complete the set(s), which grows exponentially. While in the second case
you only asks sets containing A, B and C.
But these are 2 different queries.
Anyway, I see your point, I suppose we want to have both representations
Set .... A B C
and
Member
A
...
Member
B
...
...
and use which ever one fits best. Converting from one to another should
be easy using PLN.
Regarding other set-like operators, And, Or, etc, I think I agree with
your proposal to be able to break them up. So for instance we may turn
And A B C
into
AndMember
A
And A B C
AndMember
B
And A B C
...
However I don't think the more "atomic" representation should replace
the more "holistic" one, both should be allowed in parallel, right?
Would that be for backward compatibility at least. Then if it becomes
obvious that the atomic form is superior we can deprecate the holistic form.
EquivalenceScope (stv 1 1)
<vardecl>
P
Q
is merely sugar syntax for
Equivalence (stv 1 1)
Lambda <vardecl> P
Lambda <vardecl> Q
This sugar syntax is mostly useful for humans, because <vardecl> is
not duplicated in the AtomSpace anyway.
Well, but that's not true! (Scope X P(X)==Q(X)) is not the same as
(Scope X P(X)) == (Scope X Q(X)) because the latter can be
alpha-rewritten as (Scope X P(X)) == (Scope Y Q(Y)) which is something
completely different!
Agreed, they're not the same in the logic that you describe, but in PLN
they are, because the former
(Scope X P(X)==Q(X))
is just *syntactic sugar* for the latter,
(Scope X P(X)) == (Scope X Q(X))
which is what really matters in a probabilistic interpretation.
I recall that the probabilistic interpretation of
Equivalence (stv 1 1) A B
is
P(A inter B|A union B)
It is explained here but unfortunately the wiki can't render it
http://wiki.opencog.org/w/ExtensionalSimilarityLink
Nil
-------
But whatever. All I am saying is that we should allow the following form:
EquivalenceScope (stv 1 1)
<empty vardecl>
P
Q
That's all, nothing more.
--linas
Nil
--linas
Nil
An alternate way of thinking about partitions is as
"coloring".
Pick a set, pick N colors, and then insist that every
member of
the set must be colored with one of the N colors. Then
coloring
is a lot like partitioning. e.g.
ColorLink
ColorNode "Red"
SomeAtom
or maybe
EvaluationLink
ColorNode "red"
SomeAtom
Color names could, of course, be anything: e.g. the
names of the
partitions.
In one sense, colorings are identical to partitions; on the
other hand, they can feel "more general" because you
can insist
or demand that certain properties of colorings hold,
e.g. ramsey
theory and reverse mathematics.
You could *force* aka gaurantee uniqueness of color
assignment
by using a StateLink:
StateLink
Some Atom
ColorNode "red"
The atomspace automatically gaurantees that one and
only one
color can be assigned. (although it can be changed) The
UniqueLink allows only one assignment, and it cannot be
changed. These are nice, because they help avoid
programmer
error. by offering automatic guarantees.
You don't have to use atoms for this, either. You could use
values. Recall, values are almost just like atoms,
except that
you can't put them into the atomspace, and you cannot
pattern-match or patttern-mine them. But you can
store color
or partition data in values, if you wanted to. Note
that values
*can* hold atoms! There is a LinkValue that is like a
link, but
it can hold atoms or values or a mixture of both.
--linas
This
semantics is implicit in PartitionNode, whereas if
you just use
MemberLink you'd need to spell out this "partition"
semantics using a
bunch of AndLinks each time...
As a world-class advocate of the partition
function I think
you may
like PartitionNode after you reflect on it
infinitesimally
more...
-- ben
On Tue, Aug 1, 2017 at 5:54 AM, Linas Vepstas
<[email protected]
<mailto:[email protected]> <mailto:[email protected]
<mailto:[email protected]>>
<mailto:[email protected]
<mailto:[email protected]> <mailto:[email protected]
<mailto:[email protected]>>>>
wrote:
> Hi Ben, Mike,
>
> On Fri, Jul 21, 2017 at 9:41 PM, Ben Goertzel
<[email protected] <mailto:[email protected]>
<mailto:[email protected] <mailto:[email protected]>>
<mailto:[email protected] <mailto:[email protected]>
<mailto:[email protected] <mailto:[email protected]>>>> wrote:
>>
>> Some interesting representational issues have
come up
in the context
>> of Atomspace representation of pathways, which
appear
to have more
>> general implications…
>>
>> It seems the semantics we want for a
biological pathway
is sort of
>> like “the pathway P is a set of relationships
R1, R2,
…, R20” in
kinda
>> the same sense that “the human body is a set
of organs:
brain,
heart,
>> lungs, legs, etc.”
>>
>> First of all it seems what we have here is a
part of
relationship… maybe
>> we want
>>
>> PartLink
>> ConceptNode “heart”
>> ConceptNode “human-body”
>>
>> and
>>
>> PartLink
>> >relationship<
>> >pathway<
>>
>> PartLink and PartOfLink have come and gone in
>> OpenCog/Novamente/Webmind history...
>>
>> An argument that PartLink should have fundamental
status and a
>> well-defined fuzzy truth value is given in
this paper:
>>
>>
https://www.academia.edu/1016959/Fuzzy_mereology
<https://www.academia.edu/1016959/Fuzzy_mereology>
<https://www.academia.edu/1016959/Fuzzy_mereology
<https://www.academia.edu/1016959/Fuzzy_mereology>>
<https://www.academia.edu/1016959/Fuzzy_mereology
<https://www.academia.edu/1016959/Fuzzy_mereology>
<https://www.academia.edu/1016959/Fuzzy_mereology
<https://www.academia.edu/1016959/Fuzzy_mereology>>>
>>
>> However what we need for biological pathways
and human
bodies seems
>> like a bit more. We want to say that a human
body
consists of a
>> certain set of parts... not just that each of
them is a
part... We're
>> doing a decomposition.
>>
>> One way to do this would be
>>
>> PartitionLink
>> ConceptNode “human-body”
>> ListLink
>> ConceptNode “legs”
>> ConceptNode “arms”
>> ConceptNode “brain”
>> etc.
>>
>> Relatedly, we could also have
>
>
> As mentioned earlier, there are several
problems with this
format. One is
> the "oops I forgot to mention xyz in the list"
or "gosh
I should
have left
> out pqr" and this becomes a big problem: you
have to
delete the
> PartitionLink, delete the ListLink, create a
new list and
partition. In the
> meanwhile, some other subsystem might be holding a
handle to the old,
> now-wrong PartitionLink, and there is no
effective way of
announcing "hey
> stop using that old thing, get my new thing now".
>
> A second problem is that the above doesn't have
anywhere
to hang
addtional
> data: e.g. "legs are a big part of the human body,
having a mas
of nearly
> half of the body." You can't just slap that on as a
(truth)value,
cause
> there's no where to put that value.
>
> Third problem is that large list-links are hard to
handle in the
pattern
> matcher. Its much much harder to write a query
of the
form "find
me all
> values of $X where
>
> PartitionLink
> ConceptNode “human-body”
> ListLink
> ConceptNode “legs”
> VariableNode “$X”
> ConceptNode “brain”
>
> because, ... well the ListLink is an ordrerd
link, not an
unordered link. If
> you forget to include the pqr (added above)
then the
search will
fail. You
> could try to use unordered links and globnodes, but
these lead to
other
> difficulties, including the n! possible
permutations of an
unordered link
> become large n-factorial large when the
unordered link has n
items in it.
> Recall that old factorial-70 trick used to make
calculators overflow.
>
> In general, any link with more than 3 or 4 or 5
items in
it is
bad news.
> This is a generic statement about knowledge
representation in
opencog.
>
>
>> OverlappingPartitionLink
>> C
>> L
>>
>> if we want to encompass cases where the partition
elements in L can
>> overlap; or
>>
>> CoveringLink
>> C
>> L
>>
>> if we want to encompass cases where the partition
elements in L can
>> overlap, AND the elements in L may encompass
some stuff
that’s
not in
>> C
>>
>> For the pathway case, we could then say
>>
>> PartitionLink
>> ConceptNode “Krebs cycle”
>> ListLink
>> >relationship 1<
>> >relationship 2<
>> etc.
>>
>>
>> Now this solves the semantics problem but
doesn’t solve the
problem of
>> having a long ListLink…. A biological pathway
might
have 100s or
>> 1000s of relationships in it, and we don't
usually want
to make
lists
>> that big in the Atomspace...
>>
>> To solve this we could do something like (for
the human
body case)
>>
>> PartitionLink
>> ConceptNode “human-body”
>> PartitionNode “body-partition-1”
>>
>> PartitionElementLink
>> PartitionNode “body-partition-1"
>> ConceptNode “legs”
>>
>> PartitionElementLink
>> PartitionNode “body-partition-1"
>> ConceptNode “arms”
>>
>> etc.
>>
>> and similarly (for the biological pathway case)
>>
>> PartitionLink
>> ConceptNode “Krebs cycle”
>> PartitionNode “krebs-partition-1”
>>
>> PartitionElementLink
>> PartitionNode “krebs-partition-1"
>> >relationship 1<
>>
>> PartitionElementLink
>> PartitionNode “krebs-partition-1”
>> >relationship 2<
>
>
>
> Yeah, sure. Not sure why the existing
MemberLink is not
sufficient for your
> purposes. The MemberLink has
reasonably-well-defined
semantics,
there are
> already rules for handling it in PLN (or there
will be
rules -- I
think its
> something Nil has thought about) I'm not
clear on why
you'd
want to invent
> something that is just like MemberLink but is
different.
>
>>
>>
>> ...
>>
>> There could be some nice truth value math
regarding
these, e.g. we
>> could introduce Ellerman's "logical entropy"
which is
really a
>> partition entropy. There are also
connections with
some recent
>> theoretical work I've been doing on
"graphtropy" (using
"distinction
>> graphs" that generalize partitions), which
I'll post a
paper on
>> sometime in the next week or two.... But
that will be
another
email
>> for another day...
>
>
> Yeah graphical-entropy is something that I keep
trying
to work
on, except
> that every new urgent disaster of the day
distracts me
from it.
>
> --linas
>>
>>
>> -- Ben
>>
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http://goertzel.org
"I am God! I am nothing, I'm play, I am freedom, I
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I am the
boundary, I am the peak." -- Alexander Scriabin
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