# PCT and mathematical category theory

(Gavin Ritz 2011.05.13.16.32NZT)

This is an attempt at showing PCT as a mathematical
category Theory Construct.

I’m no expert at MCT and this needs
a lot of work but this is an attempt. I hope that there are some categorists in
the group so this can really be improved.

The diagram below shows three Control
systems (individuals) interacting. The objects V, O and D have interesting
mappings.

Object V (is the set of all possible
controlled variables)

Object I (is the set of all possible input
functions)

Object C (is the set of all possible comparators)

Object R (is the set of all possible Hierarchies)

Object O (is the set of all possible output
functions)

Object D (is the set of all possible
disturbances)

q,p,r,e,f,d, x are the maps.

So for example it reads, q is a map from V
to I, I and V are objects or sets with elements.

The mapping rules are:

1. q o (f o d), reads
q follows f follows d
2. p o q, p
follows q
3. e o (p o r), e
follows p follows r
4. f o e, f
follows e
5. x o e, x
follows e

What I think is that the set O is actually
a singleton set, because there’s is one possible element in the set that’s
to one mapping.

Why do this mapping at all, because the
set R and all its elements is the command set, this will have special logical
significance in terms of truth testing, like the subobject classifier and
Imperative logical calculus. On map objects D, V and O. Also MCT is a
good way to map dynamical systems.

If this is converted say to an algebraic expression
function C (error) = reference signal minus perceptual signal. (that is, equals
element in object O).

This may even help the economic argument because
the interaction of the maps with object D may be a place to start. PCT Economic
theory?

Regards

Gavin

···

(Gavin Ritz 2011.05.14.12.06NZT)

(Gavin
Ritz 2011.05.13.16.32NZT)

Has anyone ever mapped multiple Control
systems and the mapping relationships with disturbances and controlled variables.
How would this look?

If V is a set multiple controlled variables
and D is the set of disturbances and O is the set of multiple output functions what
type of mapping relationships would we see?

[From Rick Marken (2011.05.14.1500)]

Gavin Ritz (2011.05.14.12.06NZT)

Has anyone ever mapped multiple Control systems and the mapping
relationships with disturbances and controlled variables. How would this
look?

Well, I was hesitant to answer, not only because you think I'm such a
lousy teacher but also because I risk further confirming that opinion
by misunderstanding what you are asking here. But I believe that
multiple control systems have been "mapped" and simulated using
computer modeling (given my understanding of what "mapping" might
mean) so I'm posting this information in the hopes that it's helpful.
The "mappings" that I am posting all show the relationships between
the control systems, the variables controlled by these control systems
(controlled variables) and disturbances to the controlled variables.
I'll just mention "mappings" that are readily available on the net.

The earliest (and best) "mapping" of a hierarchy of control systems
can by found in the August issue of Bill Powers 1979 _Byte_ series on
the "Nature of Robots". A pdf of the article can be found at:

http://www.livingcontrolsystems.com/enclosures/byte_aug_1979.pdf

A working implementation of a three level control hierarchy with 6
systems at each level can be found in an Excel spreadsheet
implementation that is available at my MindReadings site:

The spreadsheet hierarchy above is pretty abstract. A more realistic
implementation of a hierarchy of control systems is Richard Kennaway's
model of a six legged walking insect named Archy.

http://www2.cmp.uea.ac.uk/~jrk/Robotics/Archy/Archy.html

A simpler (but also realistic) hierarchical control model, in this
case a model of bimanual coordinated movement, can be found at:

Best

Rick

···

--
Richard S. Marken PhD
rsmarken@gmail.com

(Gavin Ritz 2011.05.15.10.51NZT)

[From Rick Marken
(2011.05.14.1500)]

Gavin Ritz (2011.05.14.12.06NZT)

Has anyone ever mapped multiple Control systems
and the mapping

relationships with disturbances and controlled
variables. How would this

look?

Well, I was hesitant to answer, not only because you
think I’m such a

lousy teacher but also because I risk further
confirming that opinion

by misunderstanding what you are asking here.

Yes you do risk
confirming my opinion by misunderstanding what I’m asking but, however I
will respect you or anyone if the respect is mutual. And being honest here deserves
an honest fair response.

But I believe that

multiple control systems have been “mapped”
and simulated using

computer modeling (given my understanding of what
“mapping” might

mean) so I’m posting this information in the hopes

Thank you for responding,
this is not what I’m asking. These are not mathematical category mappings.

I’m looking for a
type of agent interaction mapping. Similar to the one I showed in the earlier
email.

Say we have 3 individuals
(they all have their own internal control systems) how would the mappings
looking between their disturbances, controlled variables and output functions.

For example individual is
a set (or object A) object A is the set of all control systems of object A.
Object B ditto, Object C ditto. How would their relationship mappings look?

I will be away on
business for some days so will be unable to respond.

Regards

Gavin

[From Rick Marken (2011.05.14.1640)]

Gavin Ritz (2011.05.15.10.51NZT)

Iï¿½m looking for a type of agent interaction mapping. Similar to the one I
showed in the earlier email.

I was afraid of that. I think you're probably on your own on this.

Best

Rick

···

Say we have 3 individuals (they all have their own internal control systems)
how would the mappings looking between their disturbances, controlled
variables and output functions.

For example individual is a set (or object A) object A is the set of all
control systems of object A. Object B ditto, Object C ditto. How would their
relationship mappings look?

I will be away on business for some days so will be unable to respond.

Regards

Gavin

--
Richard S. Marken PhD
rsmarken@gmail.com

(Gavin Ritz 2011.05.19.11.13NZT)

[From Rick Marken
(2011.05.14.1640)]

Gavin Ritz (2011.05.15.10.51NZT)

I’m looking for a type of agent interaction
mapping. Similar to the one I

showed in the earlier email.

I was afraid of that. I think you’re probably on your
own on this.

No worries then Rick, I
have not seen any work in this area of PCT. Huge amounts of very robust work
has been done with PCT as an internal-external “unit” but not much
with multiple inter-acting units.

An area that I’m interested
in.

Regards

Gavin

···

[From Rick Marken (2011.05.18.1800)]

Gavin Ritz (2011.05.19.11.13NZT)

Rick Marken (2011.05.14.1640)]

Gavin Ritz (2011.05.15.10.51NZT)

GR: Iï¿½m looking for a type of agent interaction mapping. Similar to the one I
showed in the earlier email.

RM: I was afraid of that. I think you're probably on your own on this.

GR: No worries then Rick, I have not seen any work in this area of PCT. Huge
amounts of very robust work has been done with PCT as an internal-external
ï¿½unitï¿½ but not much with multiple inter-acting units.

Actually, quite a bit has been done with multiple interacting control
units. I sent you some references earlier; I think they were mostly
stuff on the net. Here are some articles which are all about multiple
(at least two) interacting control units:

Marken, R. S. (2005) Optical Trajectories and the Informational Basis
of Fly Ball Catching, Journal of Experimental Psychology: Human
Perception & Performance, 31 (3), 630 - 634

Marken, R. S. (2001) Controlled Variables: Psychology As The Center
Fielder Views It, American Journal of Psychology, 114, 259-281

Marken, R. S. (1991) Degrees of Freedom in Behavior. Psychological
Science, 2, 92 - 100.

Marken, R. S. (1990) Spreadsheet Analysis of a Hierarchical Control
System Model of Behavior, Behavior Research Methods, Instruments, &
Computers, 22, 349 - 359.

Marken, R. S. (1986) Perceptual Organization of Behavior: A
Hierarchical Control Model of Coordinated Action. Journal of
Experimental Psychology: Human Perception & Performance, 12, 67 - 76.

All of the above research looks at the interaction between two or more
control systems that are all operating in the one person. Tom Bourbon
and his students have done quite a bit of work on the interaction
between control systems that are operating in two different people.
Also, Clark McPhail, Chuck Tucker and Bill Powers have collaborated on
some research on interacting control systems in separate agents. And
Kent McClelland has also developed models of multiple interacting
control systems.

An area that Iï¿½m interested in.

Then I suggest that you start with the work of the people I mention
above. We've been there and done a lot of that. But it would be great
if you could add to it. You don't need to start from scratch.

Best

Rick

···

--
Richard S. Marken PhD
rsmarken@gmail.com

(Gavin Ritz 2011.05.19.15.21NZT)

[From Rick Marken
(2011.05.18.1800)]

Gavin Ritz (2011.05.19.11.13NZT)

Rick Marken
(2011.05.14.1640)]

Gavin Ritz (2011.05.15.10.51NZT)

GR: I’m looking for a type of agent
interaction mapping. Similar to the one I

showed in the earlier email.

RM: I was afraid of that. I think you’re
probably on your own on this.

GR: No worries then Rick, I have not seen any work in this area
of PCT. Huge

amounts of very robust work has been done with
PCT as an internal-external

“unit” but not much with multiple
inter-acting units.

Actually, quite a bit has been done with multiple
interacting control

units. I sent you some references earlier; I think
they were mostly

stuff on the net. Here are some articles which are all

(at least two) interacting control units:

All of the above research looks at the interaction
between two or more

control systems that are all operating in the one person.

This is not what I’m
looking for, what I want is how the mapping looks between many people. Not an individual.
Not multiple control units but “multiple individuals with their multiple
control units”.

Regards

Gavin

[From Rick Marken (2011.05.18.2055)]

Gavin Ritz (2011.05.19.15.21NZT)

RM: All of the above research looks at the interaction between two or more
control systems that are all operating in the one person.

RM: This is not what Im looking for, what I want is how the mapping looks
between many people. Not an individual. Not multiple control units but
multiple individuals with their multiple control units.

Then what you want is the "Crowd" program, which is available at:

http://www.livingcontrolsystems.com/demos/tutor_pct.html

The program doesn't seem to want to run for me; it may not like my
version of windows (xp pro). But I have a version of the CROWD program
that does work if you want it. I'm attaching a description of the
program so you can see that it involves multiple individuals (up to
255) each with six control units.

Best

Rick

crowd_dos_demo.pdf (45.2 KB)

···

--
Richard S. Marken PhD
rsmarken@gmail.com

Content-Type: application/pdf; name="crowd_dos_demo.pdf"
Content-Disposition: attachment; filename="crowd_dos_demo.pdf"
X-Attachment-Id: f_gnv60s540

[From Rick Marken (2011.05.18.2100)]

Oops, I put my initials instead of Gavin's Here's what I should have
said (in case it wasn't obvious):

RM: All of the above research looks at the interaction between two or more
control systems that are all operating in the one person.

GR: This is not what Iï¿½m looking for, what I want is how the mapping looks
between many people. Not an individual. Not multiple control units but
ï¿½multiple individuals with their multiple control unitsï¿½.

RM: Then what you want is the "Crowd" program, which is available at:

http://www.livingcontrolsystems.com/demos/tutor_pct.html

Best

Rick

···

--
Richard S. Marken PhD
rsmarken@gmail.com

[From Bill Powers (2011.05.18.1045 MDT)]

8:55 PM 5/18/2011 -0700, Richard Marken wrote:

http://www.livingcontrolsystems.com/demos/tutor_pct.html

The program doesn't seem to want to run for me; it may not like my
version of windows (xp pro).

It doesn't? It works in my XP, both pro and home, and everyone else's that I know about. Please explain what you mean. Did you click on the Run button? I'll download it and check it out.

Best,

Bill P.

[From Bill Powers (2011.05.18.1045 MDT)]

OK, I should have checked this before. It's an old version, not the one in LCS3Programs, and it truly doesn't work right. Dag, can you take care of that?

8:55 PM 5/18/2011 -0700, Richard Marken wrote:

http://www.livingcontrolsystems.com/demos/tutor_pct.html

The program doesn't seem to want to run for me; it may not like my
version of windows (xp pro).

It doesn't? It works in my XP, both pro and home, and everyone else's that I know about. Please explain what you mean. Did you click on the Run button? I'll download it and check it out.

Best,

Bill P.

It works just fine, although this is not really what I'm looking for.

[From Bill Powers (2011.05.18.1045 MDT)]

OK, I should have checked this before. It's an old version, not the
one in LCS3Programs, and it truly doesn't work right. Dag, can you
take care of that?

8:55 PM 5/18/2011 -0700, Richard Marken wrote:

http://www.livingcontrolsystems.com/demos/tutor_pct.html

The program doesn't seem to want to run for me; it may not like my
version of windows (xp pro).

It doesn't? It works in my XP, both pro and home, and everyone else's
that I know about. Please explain what you mean. Did you click on the

Best,

Bill P.

[From Rick Marken (2011.05.19.0910)]

Bill Powers (2011.05.18.1045 MDT) re: the CROWD Program

BP: OK, I should have checked this before. It's an old version, not the
one in LCS3Programs, and it truly doesn't work right. Dag, can you
take care of that?

GR: It works just fine, although this is not really what I'm looking for.

Have you considered the possibility that what you are looking for has
nothing to do with PCT?

Of course not, what am I thinking.

Best

Rick

···

On Wed, May 18, 2011 at 11:39 PM, Gavin Ritz <garritz@xtra.co.nz> wrote:
--
Richard S. Marken PhD
rsmarken@gmail.com

(Gavin Ritz 2011.05.20.11.24NZT)

[From Rick Marken
(2011.05.19.0910)]

Bill Powers (2011.05.18.1045 MDT) re: the CROWD
Program

BP: OK, I should have checked this before. It’s
an old version, not the

one in LCS3Programs, and it truly doesn’t work
right. Dag, can you

take care of that?

GR: It works just fine, although this is not
really what I’m looking for.

Have you considered the possibility that what you are
looking for has

nothing to do with PCT?

Of course not, what am I thinking.

GR:Only you know what you are thinking.

Why I even bother conversing with you is a question
that I will not ponder on too long.

I really have no interest in
communication with you, for this reason.

Cheers this is it, no more I’ve

···

On Wed, May 18, 2011 at 11:39 PM, Gavin Ritz garritz@xtra.co.nz wrote:

[From Rick Marken (2011.05.19.1700)]

Gavin Ritz (2011.05.20.11.24NZT)--

GR: Why I even bother conversing with you is a question that I will not ponder
on too long.

You would not have to ponder on it for long if you learned PCT,
according to which you "bother" conversing with me because I am a
disturbance to some of the variables you control. In other words, you
respond to me for the same reason that I respond to you; to maintain
control.

I really have no interest in communication with you, for this reason.

That just means you are in a conflict; you don't want to reply (that's
why you say you have "no interest" in communicating) and you do want
to reply. Conflicts exist when you have two goals that are
incompatible in the sense that satisfaction of both goals requires
that the same lower level goal be in two different states at the same
time. This is also basic PCT.

Cheers this is it, no more Iï¿½ve had enough of your truly thoughtless

So the "no reply" side of the conflict is currently "winning". Let's
see how you do after this post;-)

Best

Rick

···

--
Richard S. Marken PhD
rsmarken@gmail.com