New Postulate formulation on AI KR

[Paola Di Maio]

Since posting  discussions about bridging the gap between symbolic and subsymbolic, (see related posts) I have formulated a simple postulate, and assigned a DOI so that anone following up on these discussions can cite the contribution accordingly. I ll be happy to review the postulate with comments and suggestions if any

To support AI explainability, learnability,verifiability and reproducibility, it is postulated that

for each MLA *machine learning algorithm, 

there should correspond a natural language expression or other type of symbolic knowledge representation

https://doi.org/10.6084/m9.figshare.9730268.v2  

https://figshare.com/articles/A_New_Postulate_for_Knowledge_Representation_in_AI/9730268  

[Alex Shkotin]

Paola,

I keep in mind a variation of your postulate:
Any algorithm (including MLA) has math representation.

This is all that we have been talking about since Leibniz: math knowledge representation:-)
The problem is that computer programs and DBs are not mathematical structures but they work:-)

Alex  

пн, 26 авг. 2019 г. в 05:27, Paola Di Maio <paola....@gmail.com>:

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[Paola Di Maio]

Thank you Alex

variations/elaborations  welcome (Vx)

There is a problem with math KR tho - which has been discussed forever

I ll summarise it briefly hoping to put things to rest:

Interdisciplinarity is fundamental to problem solving in most domains, and applied science and engineering are concerned primarily with problem solving, rather than purely theoretical formulations

Interdisciplinarity means not everyone uses maths (  apparently even mathematicians sometimes use different types of maths) Most domain experts are not mathematicians

Therefore, maths, like ml should also be translatable/mappable to to plain natural language, even better to some diagrammatic notation  or logical language that reproduces the logical schema ideally, problems and solutions should be represented in a multiplicity of ways

This particular postulate - luckily I dont wake up early morning to postulate very often - has been prompted but the need to fill a gap between symbolic an subsymbolic KR as per the posts, and having a citable reference 

I am sure it can be refined further

PDM

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[joseph simpson]

All:

The statement:

"Interdisciplinarity is fundamental to problem solving in most domains, and applied science and engineering are concerned primarily with problem solving, rather than purely theoretical formulations"

Appears to relate directly to General Systems Theory.  See:

https://www.researchgate.net/publication/323387658_A_General_Systems_Context_for_Structural_Modeling

https://www.panarchy.org/boulding/systems.1956.html

The proper alignment of natural language and formal language may be addressed in a range of view points.

Some issues and ideas are outlined at:

https://www.researchgate.net/publication/324138184_System_Concepts_and_Theories

Take care, be good to yourself and have fun,

Joe

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--

Joe Simpson

“Reasonable people adapt themselves to the world. 

Unreasonable people attempt to adapt the world to themselves. 

All progress, therefore, depends on unreasonable people.”

George Bernard Shaw

Git Hub link:

https://github.com/jjs0sbw

Research Gate link:

https://www.researchgate.net/profile/Joseph_Simpson3

YouTube Channel

https://www.youtube.com/user/jjs0sbw

Web Site:

https://systemsconcept.org/

[Paola Di Maio]

Thanks, yes!

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[Alex Shkotin]

Joe,

you are happy to work with systems and even in a general way. But who will work with not systems? (joke)

It's a pity Fig.1 in "System Concepts and Theories" does not have rectangles for formal mathematical models, as it's better to keep theory and models separately.

Alex 

вт, 27 авг. 2019 г. в 03:58, joseph simpson <jjs...@gmail.com>:

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[joseph simpson]

Alex:

Figure 1,  Scientific Method and Model Building is basically a process flow chart.

Take care, be good to yourself and have fun,

Joe

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[Alex Shkotin]

Joe,

is it possible to say that formal mathematical models you are talking about are finite?

Alex

вт, 27 авг. 2019 г. в 15:13, joseph simpson <jjs...@gmail.com>:

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[joseph simpson]

Alex:

Great question.

"is it possible to say that formal mathematical models you are talking about are finite?"

Some models (Formal Theories)  may be finite others may be infinite.

In this case, I assume you are using the terms "models" and "formal theory" interchangeably. 

To me the important ideas are associated with the model (theory) level of abstraction.

Abstract mathematical theories (models) can address both finite (matrix theory) and infinite (number theory) abstract models.

These abstract models (theories) may be formally applied to any specific domain of science.

When these abstract models (theories) are applied to a specific domain of science, the concrete, substantive data and information that populates the models transform the theory from "abstract" to "concrete specific."

General Systems Theory is viewed as occupying the space between abstract formal mathematical theories (models) and content specific theories (models) of specific scientific domains.

I tend to view this area of inquiry along a graded range starting from pure abstraction and flowing to specific concrete examples.

As Boulding wrote:

"General Systems Theory is a name which has come into use to describe a level of theoretical model-building which lies somewhere between the highly generalized constructions of pure mathematics and the specific theories of the specialized disciplines. "

Take care, be good to yourself and have fun,

Joe

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[Alex Shkotin]

Joe,

Let me say this way.
-math structure is a mathematical structure. (Matrix, set of natural numbers, labeled graph...)
-the theory is a special kind of text with axioms, definitions, theorems, and proofs.
So they exist separately.
Math structure is a model for theory if we have an interpretation of theory formulae such that all axioms are true.
So, a model is a role math structure may play or not play for theory.
And we came to the formula:

Ontology = formal theory + finite structure modeling it:-)

what do you think?  

Alex  

ср, 28 авг. 2019 г. в 01:57, joseph simpson <jjs...@gmail.com>:

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[joseph simpson]

Alex:

Interesting approach.

Your statement:

"Ontology = formal theory + finite structure modeling it:-)"

Includes two main parts:

1 - formal theory

2 - finite structure

We may be using slightly different definitions of the term theory, but I do not see this as a problem at this level of detail.

The finite structure that models the theory is an other area where we may be using slightly different definitions of structure, but again I see no issues at this time.

According to your statement, an ontology is a finite structure modeling a formal theory.

I have been addressing structures with different levels of abstraction, ranging from general (very abstract) to specific (very concrete.)

Please see the document, "Basic System Concepts," for an example of these levels of abstraction.  A general (abstract) concept cube is presented first in the paper which then is transformed into two specific examples, General System Cube and a Logic Cube.  The paper is available at:

https://www.researchgate.net/publication/331221935_Basic_System_Concepts

I think there is room to discuss the different types of finite structures  that are valuable in the production and utilization of and ontology. 

Take care, be good to yourself and have fun,

Joe

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[Alex Shkotin]

Joe,

I think I was wrong using =, + signs in my reply, as you did not understand me. You wrote, "According to your statement, an ontology is a finite structure modeling a formal theory." 

No, sir.

An ontology is a finite structure modeling a formal theory together with formal theory itself. 

Structure and theory in one bottle, if I may say this way.

Let's clarify that before move forward.

It seems you do not use "ontology" in your texts, but you have "model" and "theory." Is there any definition for ontology from your point of view?

Alex

  

ср, 28 авг. 2019 г. в 17:36, joseph simpson <jjs...@gmail.com>:

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[joseph simpson]

Alex:

Good plan, great questions.

It will take me some time to create a paper to address these questions.

My current plan is to have a paper ready for review around October 1st, 2019.

I have one other topic that is a higher priority, but I should be able to get that one handled in about a week or so.

I will send a message reply to this email chain when the paper is ready for review and discussion.

Take care, be good to yourself and have fun,

Joe

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[joseph simpson]

Alex:

My initial response to your question"

"Is there any definition for ontology from your point of view?"

is available in the document titled "Ontology as a System."

The document is available at:

https://www.researchgate.net/publication/336229633_Ontology_as_a_System

One of the key ideas in the document is the creation of a collection of simple (minimalist)  individual ontologies

that may be integrated upon demand to create a more robust ontology.

In any case, this is my initial attempt to more clearly connect structural modeling and ontology creation.

Take care and have fun,

Joe

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[Alex Shkotin]

Joe,

Thanks for the detailed answer. Give me some time for a careful response.

At first cut, it's a pity you don't mention my short ontology definition: formal theory plus finite model:-)

Alex

чт, 3 окт. 2019 г. в 04:29, joseph simpson <jjs...@gmail.com>:

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[joseph simpson]

Alex:

This is just our first rough cut at this topic.

Your ontology definition should play a major role in the next version of this report.

Your "ontology definition: formal theory plus finite model"

Is very abstract and open.  This definition provides a good basis for exploring different ontology types.

Questions your definition may help answer:

What is the most minimal form of an ontology?

Can a specification for a collocation of "composable"  ontology types be created?

What type of formal theories are acceptable?

What is the range of acceptable finite models?

What requirements are associated with a finite model?

The focus of the next version of this document is:

What constitutes a "minimal ontology?"

How to compose two or more minimal ontology systems into a more complete individual ontology.

This is just the start.

Take care and have fun,

Joe

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[John F. Sowa]

Alex,

>  my short ontology definition:   formal theory plus finite model :-)

Three problems with that definition:

1. There are infinitely many formal theories that have finite models, but nobody would consider them to be ontologies.  For example, the integers modulo N for any N>2.

2. Integers are useful components of almost all ontologies, and they require an infinite model.  The fact that there is no upper bound is a convenience, not a limitation.

3. And finally, many (most?) ontologies that need to interoperate with independently designed systems will have aspects that don't have a precise mapping to the other.  WordNet definitions, for example, are vague.  But WN is frequently used for mapping between ontologies.

John

[Alex Shkotin]

John, let me put the answer inside your letter.

Alex

пт, 4 окт. 2019 г. в 04:34, John F. Sowa <so...@bestweb.net>:

Alex,

>  my short ontology definition:   formal theory plus finite model :-)

Three problems with that definition:

1. There are infinitely many formal theories that have finite models, but nobody would consider them to be ontologies.  For example, the integers modulo N for any N>2.

AS: Why not? Maybe just because nobody needs this kind of ontology, but it may be good practice to train ontologists. For example, we'd like to set up theory T1 for natural numbers <4 with the axiom 

∀x:N x<3 → x<x+1

and the named structure:

S1{N{1 2 3} - set

<{(1 2) (1 3) (2 3)} - binary relation

+{(1 1):2 (1 2):3 (1 3):1 (2 1):3 (2 2):1 (2 3):2 (3 1):1 (3 2):2 (3 3):3} - mapping

}

where N, <, + just names.

Having FOL-processor or manually we may check that the structure S1 is a model for T1.

Note. I keep here Kolmogorov idea that 0 is not a natural number (as V. Arnold mentioned on TV). There is another school of science where it is:-) 

2. Integers are useful components of almost all ontologies, and they require an infinite model.  The fact that there is no upper bound is a convenience, not a limitation.

AS: We do not model natural and rational numbers - we use them:-)

3. And finally, many (most?) ontologies that need to interoperate with independently designed systems will have aspects that don't have a precise mapping to the other.  WordNet definitions, for example, are vague.  But WN is frequently used for mapping between ontologies.

AS: Formal ontology being a part of an information system (IS) is a formal theory kept by this system and the finite model of that theory and usually a model (in another sense) of some part of reality.
It is usual for a particular science or technology to have the same theory all around the world (national schools of science and technology understand each other). The formalization of this theory gives the possibility to put it into all IS.

And the alignment of formal theories defers dramatically from the alignment of finite models. 

∎

John

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[Alex Shkotin]

Joe,

I'll answer shortly inside your letter.

Alex

чт, 3 окт. 2019 г. в 23:54, joseph simpson <jjs...@gmail.com>:

Alex:

This is just our first rough cut at this topic.

Your ontology definition should play a major role in the next version of this report.

Your "ontology definition: formal theory plus finite model"

Is very abstract and open.  This definition provides a good basis for exploring different ontology types.

AS: The definition is only about the formal type of ontologies, and maybe it's better from the very beginning to consider  "formal ontology definition: formal theory plus finite model" otherwise we should consider "ontology definition: theory plus model" :-)

But I am about formal only i.e. about the idea of mathematical knowledge representation.

Questions your definition may help answer:

What is the most minimal form of an ontology?

AS: some minimal theory (one axiom) and minimal model...

Can a specification for a collocation of "composable"  ontology types be created?

AS: it seems like 2 questions for me: 

Can a specification for a collocation of "composable"  formal theory types be created?
Can a specification for a collocation of "composable"  finite structure types be created?   

What is it?

What type of formal theories are acceptable?

AS:  This is great question we need to see in practice.

What is the range of acceptable finite models?

AS:  This is great question #2 we need to see in practice. 

What requirements are associated with a finite model?

AS:  It should be computer-supported as well as formal theory. 

The focus of the next version of this document is:

What constitutes a "minimal ontology?"

AS: If we speak about minimal formal theory and minimal finite structure it looks like theoretic for me. 

Or maybe you are talking about minimal useful theory and structure. Then we should look at existing information systems (IS) as everything useful is in IS:-)

How to compose two or more minimal ontology systems into a more complete individual ontology.

AS: first we compose theories then models.

This is just the start.

AS: Is there any chance you focus on formal ontologies? As informal ontologies are the bread of philosophy.

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[John F. Sowa]

Alex,

If you use the word 'formal' in your definition, you're making a claim that your definition states the necessary and sufficient conditions for using the word 'ontology' correctly.  My examples show that the following definition is neither necessary nor sufficient.  Most things that people call ontologies do not conform to that definition:

> my short ontology definition: formal theory plus finite model :-)

If you like, you could state that as a recommendation.

John

[David Eddy]

John -

On Oct 03, 2019, at 9:34 PM, John F. Sowa <so...@bestweb.net> wrote:

 WordNet definitions, for example, are vague.  But WN is frequently used for mapping between ontologies.

And WordNet (or at least the v3.0 I have) does not even contain:

- VAX

- VMS

- DB2

- CICS

- JCL  (OAD at least has JCL)

At one point I spoke with current “owner” of WordNet about inclusion (or not) of un-natural language & technical jargon.  

Response was a look of total bafflement.

Vendors of all stripes & products spend big money to inject technobabble jargon into our brains… so we should pretend such terminology & concepts do not exist?

For certain we must keep technobabble far, far away from the purity of ontologies… right?

- David

[Alex Shkotin]

John,

From one point of view "formal theory plus finite model" is an informal definition of formal ontology mostly instead of "“An ontology is a specification of a conceptualization.” [KSL.Stanford.Edu, 2019]" see https://www.researchgate.net/publication/336229633_Ontology_as_a_System

From  another point of view, you are absolutely right "formal theory plus finite model" is the recommendation, for example for OWL2-ontology to look at any OWL2-axiom statement as part of a formal theory xor finite structure. 

Alex

пт, 4 окт. 2019 г. в 16:23, John F. Sowa <so...@bestweb.net>:

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[Adrian Walker]

David -

In Executable English, you can directly use previously unseen or jargon words such as

VAX

- VMS

- DB2

- CICS

and they take their meanings from context. 

                                                             -- Adrian

Adrian Walker
Executable English LLC  
San Jose, CA, USA
(USA) 860 830 2085 (California time)
www.executable-english.com

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[joseph simpson]

Alex:

It will take some time to create a response to your email note.

The next report version should be out between the middle of November and the first of December.

Take care and have fun,

Joe

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[Jon Awbrey]

Joe, All ...

The reason I've been maintaining an interdisciplinary perspective in my
postings to the Ontolog Forum, Structural Modeling, and Systems Science
groups is because each one stresses a distinct but necessary aspect of
a systems approach to scientific inquiry. I see much potential to be
had in integrating these views of the inquiry process, but it will
take a lot more thought and work to fully develop that potential.

The feature that jumps out at me as I scan the discussions on this thread --
and I've said this before about most of the discussions of systems I've
seen in these groups -- is the static nature of the pictures of systems
people are laying out. Whereas, for me, my whole reason for taking up
a systems approach to inference, information, inquiry, along with the
symbol systems we use to conduct their transactions, is to tease out
the shape and flow and dynamics of their transformations.

Regards,

Jon

[joseph simpson]

Jon:

You wrote:

"... is the static nature of the pictures of systems

people are laying out.  Whereas, for me, my whole reason for taking up
a systems approach to inference, information, inquiry, along with the
symbol systems we use to conduct their transactions, is to tease out
the shape and flow and dynamics of their transformations."

At this time I am exploring the creation of an initial structure associated with an unknown or poorly defined system.  The initial system structure is  static.

In the past, I have explored the dynamic transformations associated with a well-known system structure in a specific environment.  In this work I created the abstract relation type (ART) form.  The ART form concepts are used to evaluate large-scale systems in a range of contexts.  In many cases, the ART concepts employed evolutionary computation techniques in the evaluation process.

The next phase of exploration will combine the initial structural modeling techniques with the ART form approach.  Dynamic evaluation of system change produces a vast quantity of data.  A specific ontology that is used to organize, relate, and present this data is viewed as very valuable.  

Take care, be good to yourself and have fun,

Joe

[Alex Shkotin]

Joe and Mary,

Thank you for nice initial system analysis of the contemporary situation with ontology definitions and beings.

Please look at my notes to your sentences.

"In other cases, complex concepts, knowledge structures, and relationships must be arranged in a manner that interfaces among humans, software agents and other ontology processes and software agents."

=The best internal representation is a formal theory. The structure of formal theory is simple: primary constants, predicates, and functions; axioms; definitions of secondary constants, predicates, and functions; theorems and it's proofing.

By the way, the Reasoner like DL-reasoner should be mentioned somehow as it is a way to get new knowledge from ontology, beginning from inconsistency;-)

"At the most fundamental level, an ontology is a catalog of the types of things that exist in any specified domain."

=Let me say it's a formal theory.

"The lack of understanding in the domain space can be associated with the ontology types or the relationships between the types in the catalog."

=But tell me the domain where we do not have scientists and therefore informal theory;-)

"A more formal ontology may include formal concept descriptions, classification templates, class attributes, class relationships, domain constraints, rules, and axioms."

=Let me put formal theory terms after | to your text to compare:

A more formal ontology|theory may include: formal concept descriptions|definitions, classification templates|?, class attributes|functions, class relationships|predicates, domain constraints|axioms, rules|derivation? and axioms.

"Each ART configuration has the capability to represent an independent model of the domain knowledge."

=This is one more way to create ontology  - is there any example of ontology on ART?

"[Warfield, 1994] work in the area of the ‘science of generic design’ used binary matrices to encode domain knowledge. Charles Sanders Pierce's work in the area of the logic of relatives can be represented using binary matrices as shown by Jon Awbrey:

[ https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1 ]."

=A matrix is a form to represent finite structure — binary relation.

If you use a relation of relations then welcome to HOL:-)

"An ontology is a very loosely defined term."

=This is where we turn to the ontological definition: today ontology is a random mixture of sentences from formal theory together with sentences of the finite model of this theory:-)

The nearest approximation to math ontology may be microtheories of Cyc. But this should be studied. The problem with finite structure is mostly logical as technically any finite structure may be embedded in the theory if we assign the constant to every finite structure element:-) In my plan to show how to embed graph, the finest finite structure, to OWL2 ontology:-)

Alex

пт, 4 окт. 2019 г. в 19:53, joseph simpson <jjs...@gmail.com>:

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[joseph simpson]

Alex:

Thanks for your detailed comments.

Our next report, "Ontology as a System - Version 2.0" will address these and other comments associated the definition, design and use of any specific ontology.

Take care and have fun,

Joe

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[Alex Shkotin]

Joe,

Great! And small correction "A matrix is a form to represent finite structure — binary relation." ==> "A matrix is a form to represent finite structure — binary function."

:-)

вс, 6 окт. 2019 г. в 23:12, joseph simpson <jjs...@gmail.com>:

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