On Logical Triangulation

Purpose of this Site

[ATTENTION: all files are in the "files" sub-page. Google's "migration" destroyed all links.]

This site contains information on the theory of Logical Triangulation. It constitutes a novel approach to artificial intelligence and may aid other, more classical fields of AI in their development, as well as serve as an approach in and of itself. Its advantage is that it

It is based on the idea that thinking is a process of ordering information within an agent's "mind" according to "terms" or "atoms of thought", which are experienced by means of observation from the outer world. Each thought, each notion, everything the agent works with, is such an atom of thought. As opposed to other approaches, the relations between these atoms of thought are not to be pre-set by any human engineer (and so there is no ontology design or categorisation of whatever nature), but are to be established purely by the intelligent agent out of the perception of the atoms by itself. - From the formation of such relations, further relations can be concluded. The method of such conclusion of relations and evaluation of relations is referred to as Logical Triangulation and constitutes the central point of this theory.

Sample implementations are provided, as well as a first proposal of potential effects for feed-forward neural networks. (All implementations are either in Scheme or Fortran 90.)

A Short Summary of Logical Triangulation

For anyone interested what Logical Triangulatino is all about, the main reasoning mechanism is summarised below.

1. - The intelligent agent or system is perceiving through its sensors "elementary atoms of thought". These most primitive atoms of thought are each of axiomatic nature, they cannot be further decomposed or amended in any manner: they just "are". They can, however, be composed into more complex structues. It is up to the system designer to decide what kind of signals the sensors shall target, be it textual words, optical or acoustic input - the design is not focused on any certain "category" of information. The only requirement is that the information delivered by the sensors contains some form of regularity that can be learned and understood by the system.

2. - In putting these most primitive atoms of thought into relations, "higher atoms" (or super-atoms) can be formed. Each higher atom constitutes a relation between two other atoms. Each higher atom can be composed itself into an "even higher" atom together with another atom (so a new atom pair is formed). There is no theoretical limit as to the hierarchy level achievable by this. There is also no hierarchical limitation - even most elementary atoms can be combined with very high-level other atoms. Practical limits are imposed only by the fact that a too high level of abstraction will not be very useful and in a limited memory, it may be forgotten quickly. Atom may enter into any arbitrary number of combinations. (I.e. it is not "absorbed" nor "ceases to exist" once combined.) Atoms may also be "decomposed" - this occurs when the focus of the system is shifted from a higher and more abstract "super-atom" to its two "sub-atoms". This happens when the system focuses on "details" (such as those of an action that it wishes to undertake) or so that the system re-combines one or both of the sub-atoms into other atoms, effectively thereby "shifting its focus" to other high-level atoms.

3. - Each two atoms of thought can have two basic relational aspects - a relation of "vicinity" and a relation of "analogy". In the first case, it means that its sub-atoms are generally observed together (such as "branches and leaves" in a tree) or "next to each other" (hence the vicinity). In the second case, it means that its sub-atoms are generally observed alternatively and one in the place of the other (such as "night and day" or "sugar or honey" or "tea or coffee") - the notion of "analogy" is abstract, it merely means that the one term can be used with third terms in place of the other term. It means that "analogous" terms can either be very similar and can be used interchangeably, or they are so different that they are opposites. In any case, in an relation of "analogy", the terms are not used jointly but only the one in the place of the other is used. - These two relations are in truth aspects of one and the same underlying relation - the more often two atoms of are observed together, the more vicinal they appear, and the less they seem to be replacing one another, and vice versa. - The relation between two terms may and does change, and it is the task of the system to figure out the appropriate aspect - is it more of a vicinity connection or is it rather an analogy connection - as well as the strength of the respective connection. Operation on such connections is the central process in Logical Triangulation.

4. - Apart from expressing the world in terms in deliberate level of abstraction, the system also reasons with the terms. The basic idea is that it draws conclusions of the type "if the atoms X and Y are given, then the result must be an atom Z and cannot be an atom anti-Z". - However, as opposed to classical logic, the system cannot select which X and Y are in the "head-part" of the equation and which are in the "tail-part". This is why it cannot say that Z "follows from" some certain X and Y. Rather, it measures pure correlation: "Can X, Y and Z co-exist and be all true?" In this sense, it measures X according to Y and Z, Y according to X and Z, and Z according to X and Y. If in some case one atom is not known, e.g. Z is not known, it can be concluded from X and Y: "If X and Y co-exist, then that implies that an atom Z must exist". - If three atoms are found that can co-exist, each of their relations is "strengthened", as their co-existence constitutes congruent knowledge. If three atoms are found that cannot co-exist, the "trust" in them is shaken, and each relation is weakened (and at some point, one or more relations may be reversed). - It is this process of grouping together atoms and deciding whether their co-existence is reliable or unreliable that is the heart of Logical Triangulation.

5. - Logical Triangulation is based the idea that three super-atoms, where each super-atom has exactly one sub-atom uniquely in common with each other super-atom, can form exactly four groups of possible relation aspects. Two of these groups are congruent (and their relations would be strengthened by the system) and two of these groups are contradictory (and their relations would be weakened by the system as untrustworthy).

It shall be assumed that the following three super-atoms are involved:

X, consisting of the sub-atoms A and B;

Y, consisting of the sub-atoms A and C;

and Z, consisting of the sub-atoms B and C.

The question is what aspect of relation each of X, Y and Z expresses - vicinity or analogy - and each aspect of relation is measured against the other two aspects to see if they are congruent and their co-existence is possible.

The four possible relation groups are (whereby "-" denotes a vicinity connection and "=" denotes an analogy connection):

(i) A=B, A=C and B=C; this relation is congruent - if A is analogous to B and A is analogous to C, then B should also be analogous to C;

(ii) A-B, A-C and B=C; this relation is congruent - if A is vicinal to B and A is vicinal to C, then B and C are similar; or in another way: if A is vicinal to B and B is similar to C, then A should also be vicinal to C;

(iii) A-B, A=C and B=C; this relation is incongruent - if A is vicinal to B, then it cannot be that both A is similar to C and B is similar to C - one of the two should rather be vicinal to C than similar to C;

(iv) A-B, A-C and B-C; this relation is incongruent - if A is vicinal to B and A is vicinal to C, then B and C should rather be similar than vicinal.

Relations that participate in incongruent groups may still be supported by other relation groups with third atoms (any relation can participate in millions of triangles with varying outcomes). The system seeks no "absolute truths" in its reasoning and can easily operate with partly contradictory knowledge. It will just attempt to clarify knowledge in that area on which it presently focuses its attention.

6. - If the system knows two relations, it can conclude the third (obviously, forming only congruent relation groups):

If A=B and B=C is known, A=C is concluded;

if A-B and B=C is known, A-C is concluded;

and if A-B and B-C is known, again A=C is concluded.

7. - The system operates through dynamically forming super-atoms of arbitrary hierarchy and again decomposing super-atoms as needed, and evaluating connections. It does not seek "absolute truths", but "truths in the current situation" that shall guide its output-reaction to an input that is evaluated by means of the systems knowledge of relations between atoms and their aspects.

For further information, refer to the full description of the theory.

The Theory

The full text may be found here:

ontriangulation20140521.pdf

(The previous version has been:

ontriangulation20140510.pdf

- this has been revised in order to improve readability and precision. Kind thanks to comp.ai.philosophy for the suggestions.)

The source code of the four presented systems together with detailed instructions on their use is contained in the above pdf. They are once more shown below for your convenience:

LarCom A and its support files:

larcom-a.scm

simpledefinitions.txt

simpleinput.txt

LarCom B and its support files:

larcom-b.scm

atomdefinitions.txt

userinput.txt

LarCom C and its support files:

larcom-c.f95

aiinput.txt

listofallatoms.txt

unusedatomsymbol.txt

LarCom D:

larcom-d.f95

After over twelve and half years of consideration, I have published this theory here for the first time on 10th May 2014.

If you have further questions, you may contact me under: "k e d a l i o n [dot] d a i m o n [at-sign] g m a i l [dot] c o m".

For a public discussion, kindly go to: weblog

- Nino Ivanov

Hyper- and Hypo-Triangulation

7th April 2017

I have discovered a possibility to extend logical triangulation. - In the above theory, it is described how to reason "between any three atoms" who have the necessary sub-atoms in common. But I have said nothing about their SUPER-ATOMS and their SUB-ATOMS. However, If the atoms X, Y and Z triangulate by means of their sub-atoms A, B and C, then it is actually possible to draw infinitely many conclusions from this. - I believe this discovery to be of importance, as it opens the door to "rules" (which are nothing else than "SUPER-ATOMS" to large amounts of sub-atoms, and such (up to infinitely) large super-atoms can now be concluded).

hyper-hypo20170329.pdf

cross20170330.pdf

You may find implementations here:

https://github.com/KedalionDaimon

and specifically under:

https://github.com/KedalionDaimon/maptri

- Nino Ivanov