Final Report on CCG Generalization
TelArt Inc., Chantilly Virginia
December 31, 2000
Dr. Paul S. Prueitt
In May 2000, Zenkin and Prueitt discussed the possibility that there might be a common scheme in the use to CCG technology in new knowledge generation. This discussion occurred after Dr. Prueitt had developed an appreciation of the Computer Cognitive Graphics (CCG) technology as it is applied to number theory. Drs Art Murray, Alex Citkin, Peter Kugler, Bob Shaw, Michael Turvey, and Kevin Johnson have assisted him in this evaluation. Our first project report to the ARL delineates this understanding.
Our final report to ARL generalizes the CCG technology. From the generalized CCG it becomes clear that the re-application of the original CCG technology to other objects of investigation can be achieved if and only if all real aspects of the syntactic and semantic representational problems are addressed.
The claim made in this ARL Report is that the CCG technology, as applied to number theory, has addressed all real aspects of the syntactic and semantic representational problems. A specific object of investigation, this being the theorems of classical number theory, shapes how these aspects are addressed. No semantic issues exist, except as noted in the problem of induction. Thus the mass of the CCG technology is merely formal and syntactic. The core, however, is essential and this core is about how one manages mental induction.
We hold that induction is not and cannot be considered algorithmic in nature. We cite Western Scholarship Robert Rosen and Roger Penrose work as well as the work of J. J. Gibson and Karl Pribram. Thus the interface between algorithm computers and human mental activity is necessary in any generation of new knowledge. It appears that much of Russian Applied Semiotics is based on this Peircean concept that an interpretant is actually required during the generation of new knowledge.
Section 1: A review of the CCG application to number theory and its generalization
Specifically, elementary number theory is a formal construct that is built on the Peano axiom, the additive and multiplicative operators, and on the use of a principle of mathematical induction. The CCG technology can then be seen to have the following parts:
A representational schema
A visualization schema
An induction schema
In number theory, the representation allows color codes to represent division properties. In classical number theory, division does not induce non-integers, but rather truth evaluation of whether a remainder is zero, (or any other integer - depending on the way the theorem is stated). Now it is important to state that Dr. Prueitt has some background in number theory and as a consequence of seeing number theory in a new light, there appears to be new theorems regarding invariances of residues. Perhaps others can see these new theorems also. There is additional work that would have to be achieved in order to tease a the specific statement about these theorems.
The possibility of new mathematics is pointed out only because there is a mental effort required to think about how color representation fits into the CCG methodology. That this effort has lead to an intuition about new mathematics is a statement that must be taken on a plausibility argument. This plausibility is ultimately how the CCG methodology should be judged. Although positive results of various types exist and can be shown, the conceptual grounding for CCG remains foreign, not only due to Russian origin but primarily due to the misrepresentations made by modern Artificial Intelligence regarding the nature of human induction.
There is a suggestion that there are some new theorems that are delineated by Prueitt's mental intuition when the cognitive effort is made to describe the CCG techniques. Prueitt is willing to discuss this issue at the proper time. However, the existence of new theorems in number theory is not of immediate interest. Prueitt claims that the theory of algebraic residues is not completely developed, and that pure mathematicians who know this field well will be able to quickly see the same intuition. This intuition comes immediately from the realization of how color-coding is used in the CCG applied to number theory.
How would CCG assist in our experiencing intuitions?
An induction is to be established by the physical representation (by colors in this case) and the subsequent representation of truth/falseness evaluation of specific properties of a generated sequence of numbers using the 2 dimensional grid. Zenkin has many examples of how this has worked for him. Prueitt sees a different class of theorems because he has a different mathematical training and internal percepts about the Peano axiom and the additive and multiplicative operators. Any other pure mathematician would see theorems that are new, depending on the nature of the intuitions that are resident in the mind of the pure mathematician. CCG would be useful in the completion of mathematical reasoning from whatever experience the mathematician might have.
Prueitt holds that any formal system can be vetted using slight modifications of the CCG representation and visualization demonstrated by Zenkin. If this claim is correct, then areas of abstract algebra would fall under the technique. The requirement is that a human has deep intuitions about an object of investigation and that the representation and visualization setup a route to induction regarding truth / falseness of theorems (see the work of Russian father of quasi axiomatic theory, Victor Finn, on routs to induction).
This means that someone who is deeply involved with algebra and who studied the CCG applications to number theory would likely begin to (immediately) see how to represent and visualize relationships such as the property of being a generator of a semi group. Once this new mental intuition is established, then a principle of induction is required that allows the validation, or falsification, of intuitions. In the past application of CCG to number theory, this validation of intuition is equivalent to a formal proof, and yet is made using a proxy that is visual in nature. The proxy is established via the notion of a super-induction where the visual observation of a property transfers to a formal declaration of fact. This transfer is the core of the CCG technology and is not dependent on the specific representation or the visualization, as long as the visualization schema matches (completely) all syntactic and semantic representational problems.
In formal systems, the problem of syntactic and semantic representation is not only simple; but is also complete. There is little or no semantic dimension. Only the truth evaluation is semantic and this semantic evaluation is incompletely represented in the iterated folding of syntactic structure (via rules of deductive inference). In essence, one can almost claim that the only semantic aspect about number theory is that someone who sees the elegance of it can experience it as beautiful. The caveat is captured by the Godel theories on completeness and consistency, and on related notions communicated by Cantor and others (including Robert Rosen’s work on category theory). Of course, Zenkin is one of those who have advanced a disproof of Cantor's argument regarding the categorical non-correspondence between the whole numbers and the real numbers.
Prueitt reads this disproof in a certain way. The argument that Canton's diagonalization theory is flawed is really a comment on the nature of common mathematical induction. As Kevin Johnson has pointed out, there are many many ways to perform an induction. The common mathematical induction simply depends on an ordering of theorems in such a way that the tail of this sequence of theorems has invariance with respect to the truth evaluation. The CCG representation and visualization simply allows a pure mathematician a by-pass of all orderings except one that results in visualization of the targeted invariance of a tail of a sequence of theorems. This by-pass is non-algorithmic and thus must be managed by a human.
One can see this as a search space problem. In many cases modern computer science has identified what are called NP-complete problems. The NP-complete problem can be proved not to be computably solved with the iterative application of the folding (application) of the fundamental axioms and properties in the set up of the formal system. However, visual acuity by a human might see a route to a solution. In fact, Prueitt has made the argument that biological systems have evolved in such a way as to by-pass NP complete problems. He claims that the capacity for seeing a solution that cannot be computed is fundamental to biological intelligence.
In formal systems, the by-pass is simply a lifting away from and a replacement into the formal construct. This there is still no semantic dimension to the solution. This concept of lifting is consistent with Brower’s notion of intuition (Bob Shaw – private communication). This means that the solution, once found, to NP-complete problems can then be proved using common inference and common induction. It is just a question of skipping and reordering.
Possible application to EEG and stock market data analysis
Zenkin and Prueitt were hoping that EEG data could be easily found with expert opinions about differential meaning of data patterns in context. Due to the uncertainty of how we might precede Prueitt did not pursue a collaborative relationship with EEG experts in Karl Pribram's lab or in any other lab. Such collaboration requires that the method we have devised for visualization be well developed and that our collaborative project with the Russians be well funded.
As we worked on this issue, it became clear that we could describe such a method only if the communication between Russia and the United States was better. We need to involve neuroscientists both in St. Petersburg (Juri Kropotov) and in the USA (Karl Pribram).
Given our limited resources, Prueitt decided to attempt to generalize the CCG methodology and then project this generalization back onto some object of investigation. The idea was that the generalization and separation of parts of the CCG techniques would show us how to proceed.
What we needed to figure out first was how to characterize the CCG method in such a way that aspects of the method could be separated into functional parts. Then each part might be generalized and then projected into a new use case.
We were open to possible investment directed at using indices in the analysis of stock market performance. This possibility still exists. However, it is felt that this application is unwise and not directed at a scientific or mathematical objective.
However, our thinking about the markets allowed us to see, for the first time, that we needed to have an Image Library. We needed a repository for the consequences of the evocation of knowledge about, or an intuition about, the past or future performance of the market. At this point, the work of other Russian applied semioticians (Pospelov and Finn) come into play. The Library becomes a repository for a system of tokens, each token deriving token meaning from intuitions vetted by the CCG representation and visualization, and confirmed by an induction. The system is then a formalism that is open to human manipulation as well as formal computations. The formalism has both a first order and a second order (control or tensor) system.
We have come face to face with the core difference between a formal system, like number theory or algebra, and a natural object of investigation, like the stock market. It is this difference that is ignored by most Western mathematicians and computer scientists. It is also this difference that illuminates the nature of Russian applied semiotics. The case of this assertion will not be full made here; as to some extent the assertion is ultimately a statement of belief.
In any case Zenkin and Prueitt both agreed that an Image Library might be built as a type of Artifact Warehouse, where the artifacts were the consequences of a super-induction mediated by some representational and visualization schema.
The problems are then defined as
How does one represent the object of investigation
How does one visualize the accrual of invariance
How does one establish conditions of induction
Prueitt has some experience with scatter gather methods used in the standard methods for vectorization of text. Thus he chose a collection of 312 Aesop fables to be his target of investigation. This choice was a secondary choice, since TelArt Incorporated continued to hope that situations in Russian might allow Alex Zenkin the time required to make a paper on his own attempted application of CCG to scientific data of some sort. As the deadline for our Final Report neared, it became clear that Prueitt would have to write the Final Report without additional original work from Russia.
In the next sections, results of the generalization of CCG are applied to the problem of parsing text. Text parsing ultimately is to be applied to a routing of information or a retrieval of information. Prueitt is designing a system for a worldwide evaluating of Indexing Routing and Retrieval (IRR) technologies, and thus the use of Prueitt's background was capitalized on for the purpose of completing our contractual obligations.
The URL announcing this IRR evaluation is at: link
Section 2: The Application of CCG generalization to the problem of text parsing
ARL has expressed an interest in text parsing. Thus it is important to specify how we see test parsing fitting into an architecture that deploys a generalization of Russian CCG technology.
The diagram for this architecture has the following parts:
1) a preprocessor for data streams
2) an image library
3) an visualization interface and control system
4) indexing engine,
5) routing and retrieval engines
6) a viewer or reader interface with feedback loop to 2, 3, 4 and 5
Component 6 is where we might have a decision support interface. Components 2, 3 , 4, and 5 are really the components of a knowledge warehouse. These components are built up over time. In the data mining terminology, component 1 is called a data-cleaning component.
The preprocessor simply must do what is necessary to put the incoming information into a regular data structure. The cleaned data structure can have many different types of forms, however in the text-parsing task all of these forms have deficits. Again, this is due to the indirect relationship that the textual information has to the experience of awareness or knowledge. We seek to transfer the interpretability of natural text into an image framework.
The viewer interface is used in real time on problems of some consequence. Thus there is a systems requirement that a feed forward loop be established from one decision-making event to the next. The feed forward loop must touch components 2, 3, 4 and 5.
This architecture allows at least three places where human perception can make differences in the systems computations. The first is in the development of the image library (perhaps a token library is a more general concept). The second is in the indexing of the library components and perhaps some data streams. The third is in the decision support system (component 6).
In Section 1, we made several claims. One of these is that the flow of information from an object of investigation can be vetted by a human computer system if and only if all real aspects of the syntactic and semantic representational problems are addressed. The flow of information from an object of investigation is likely to suffer from the data source being somewhat indirect, as in EEG data and linguistic data. Data sources such as astronomic data are more direct and thus more like the formal data sources such as number theory.
The partial success of statistical methods on word frequencies attests to the fact that a partial solution to these problems leads to an imperfect result. The glass is either half empty or half full. We do not know how to make this judgment, because today there is no completely satisfactory automated text parsing system.
The indirect nature of the data source would seem to imply that a human interpretant is necessary before there can be really successful text parsing. Thus the notion of vetting is proper, since this notion implies causation on a process that is mediated by a knowledgeable source and human judgment. The goal of a CCG system for text parsing is to transfer the interpretive degrees of freedom of text into an image framework. Once in the framework certain algorithm paths can produce suggestive consequences in new context.
It is not yet known if new methodology, entirely separate from the existing routing and retrieval technologies, will give rise to new and more successful results. We have suggested that much of the statically work on word frequencies is hard limited by the nature of anticipation. The statistical sciences can tell perhaps everything about the past, but cannot always predict the future. Moreover, we have the problem of false sense making. The meaning of words is enabled with ambiguity just for the purpose of predicting the meaning of words in contexts that are bound in a perceptional loop. This loop involves both memory and anticipation.
One can revisit the TREC and TIPSTER literatures, as we at TelArt will be doing over the next three months. In this review, we find not only statistical approaches, such as those made by David Lewis at AT&T and Susan Dumas at BellCore, but also a few linguistic and semantic methods. These methods are being reviewed as part of the Indexing Routing and Retrieval (IRR) evaluation by conducted by TelArt for a commercial client.
An understanding of routing and retrieval techniques might assist the generalization of the CCG technology. This generalization was done to establish some broad principles that might be formative to a proper text parsing system. The CCG technology can then be seen to have the following parts:
1) A representational schema
2) A visualization schema
3) An induction schema
New thinking on indexing, sorting or arrangements of data atoms may also provide value to our task. As we look for tools and methods we will of course be somewhat hampered by the now proprietary nature of new technologies. However, this is just part of the task we set for ourselves.
The application of CCG technology to the problem of text parsing requires that our group have a command of all existing IRR technologies and theory.
Section 3: Strategy for organizing text data using CCG plus other techniques
Computer Cognitive Graphics (CCG) is a technique developed by Alexander Zenkin. The technique uses a transformation of sequence values into a rectangular grid. Visual acuity is then used to identify theorems in the domain of number theory.
Our purpose has been to generalize this work and to integrate the components of CCG technique with knowledge discovery methods more widely used for automating the discovery of semantically rich interpretations of text.
An application of knowledge discovery in text is underway. As a general capability a parse of text is to be made in the context of a number of text understanding tasks.
Test Corpus: We have several test collections, however only one is non-proprietary. This corpus is a collection of 312 Aesop Fables. We will demonstrate some work on identification of the semantic linkages between the 312 elements of this collection in the last section (section 4).
In a text parsing system, the interpretation of text is oriented towards retrieval of similar text units, the routing of text units into categories and the projection of inference about the consequences of meaning.
As part of a commercial activity, TelArt Inc. is developing behavioral criterion for commercial clients regarding proper evaluation of text vectorization, Latent Semantic Indexing, CCG – derived, and other informational indexing systems. Most of the direct work on criterion is proprietary in nature. However, the movement from one domain to another domain of interest will produce a variation of behavioral criterion. Thus client criterion can be generalized to fit many other domains outside the domain of content evaluation. This moves us towards the notion of ultrastructure as developed by Jeff long at the Department of Energy.
Thus we envision the development of generalized behavioral criterion for evaluation of Indexing Routing and retrieval technologies. Behavioral criterion for intelligence vetting is an interest to TelArt Inc.
TelArt is, of course, interested in a full-scale program involving principals at New Mexico State University and universities in Russia on situational logics and text parsing systems.
Use case type analysis of a process of text parsing
A.1: Convert a word sequence into a sequence of stemmed words.
A.1.1: Take the words in each document and filter out words such as “the” and “a”.
A.1.2: Make replacements of words into a stemmed class:
running à run
storage à store
A.1.3: The set of stemmed classes will be referred to as C.
A.2: Translate the stems to positive integers randomly distributed on a line segment between 0 and 1,000,000,000.
A.2.1: This produces a one-dimensional semantically pure representation of the word stem classes.
A.2.2: The topology of the line segment is to be disregarded.
A.2.3: A new topology is to be generated by adaptively specifying the distance d(x,y) for all x, y in C.
A.3: Introduce a pair wise semantics
A.3.1: Following a training algorithm, points are moved closer together or further apart based on either algorithmic computation or human inspection
A.3.2: Topologies can be produced by various methods.
A.3.2.1: Generalized Kohonen feature extraction, evolutional computing techniques.
A.3.2.2: From one point topological compactification of the line segment, and cluster using the Prueitt feature extraction algorithm.
A.4: Pair wise semantics can drive a clustering process to produce a new semantically endowed topology on the circle.
A.4.1: Features of the new space are identified by the nearness of representational points
A.4.2: Prueitt feature extraction is seen to be equivenant to Kohonen feature extraction
A.4.3: Prueitt feature extraction produces a specific distribution of points on the unit circle.
A.4.3.1: The Prueitt distribution is defined as a limiting distribution which has semantically relevant clusters
A.4.3.2: As in associative neural networks, and other feature extraction algorithms, the Prueitt distribution is driven by a randomization of selection (of the order of pairing evaluation), and thus is not unique to that data set.
A.4.3.3: Iterated feature extraction produces a quasi- complete enumeration of features, encoded within a topology.
A.4.3.4: The limiting distribution has a endowed semantics.
A.5: The clustering process can be mediated by human perceptual acuity
A.5.1: Phrase to Phrase association by human introspection as coded in a relational matrix.
A.5.2: Latent Semantic Indexing for co-occurrence, correlation and associative matrices.
A.6: The limiting distribution is viewed via CCG methodology
A.6.1: The CCG methodology is linear and assumes a regular limiting distribution.
A.6.1.1: The CCG technique finds a coloring patterning and modules (related in the number of columns in the grid), such that a principle of Super distribution can be used to prove a specific theorem using the limiting distribution.
A.6.1.2: Coloring pattern and modules is specific to the theorem in question
A.6.1.3: Given a limiting distribution that is metastable (a pattern is repeated) or stable (one condition is show to exist always beyond a certain point), then a theorem exists.
A.6.2: A non-linear CCG technique might have a grid of dimension higher than 2, as well as transformation operators that are statistical.
Locally, the clustered tokens on a number line can be rough partitioned using neighborhoods (see Figure 1)
Figure 1: Topology induced by clustering
The circles in Figure 1 indicate boundaries of categories. These categories overlap producing areas a ambiguity. However rough sets and the Prueitt voting procedure anticipates the ambiguous boundaries and addresses this issue in a non-fuzzy fashion.
We can map the categories to an n dimensional space. In this case n = 8.
x --> (0,1,0,0,0,0,0,0) or (0,0.7,0,0,0,0,0,0)
y --> (0,0,0,0,1,0,0,0) or (0,0,0,0,0.6,0,0,0)
z --> (0,0,0,0,0,1,1,0) or (0,0,0,0,0,0.3,0.2,0)
Where the first representation is a Boolean representation of inclusion and the second representation measures show a degree of closeness to the center of the distribution.
The question of how to assign category memberships to new text elements is addressed in the tri-level routing and retrieval architecture and voting procedure as well as in Russian Quasi Axiomatic Theory.
There are some important and general problems in automated recognition of patterns in data streams. When the stream is a stream of text, then the text can be converted or transformed into tokens that have been shaped by linguistic and semantic considerations. The discovery of proper shaping methodology is what is at issue. Shaping is an induction.
For example, an illness might be recognized from a written description of symptoms. The text of the descriptions can be converted to number sequences. Shaping might come from relevance feedback, or some type of feed forward mechanism such as thought to account for the shaping of action during motor control, see the neuropsychological and neural architecture work by James Houk and Andy Barto.
The development of an information base from a text database is at least a three step process:
1) the transformation from text to tokens,
2) the shaping of token identification, and
3) the development of relationships between tokens
The first step, in text vectorization scatter / gather methods, is to randomize the location of numbers by a uniform projection into a line segment. The projected numbers become tokens for the text units. The nearness of tokens on the number line is discarded. Some transformations might then be applied to produce clusters of values around prototypes of concepts that are expressed in the text (in theory).
Our hope is that CCG images can be involved in a visualization of these prototypes. The clusters, once identified can be used in the definition of a tri-level category policy to be used to place next text elements within an assignment relative to these categories.
The objective of our study is to see if we can find a representational schema, transformation of topologies, and some relationship that becomes clear as a process of analysis moves our attention deeper into a sequence of numbers or generalizations of the notion of number. As we neared the completion of our contract time (December 2000) it became clear to Prueitt that the generalization of number that we needed is the cluster space itself. The clustering process itself produces states, and each state in generated from the previous state using a specific iterated axiom (like the Peano axiom). .
The cluster iteration is dependent on a scatter / gather process. This process is well known in the text vectorization community. We see no way (at least not now) to start with anything other than a scatter of representational points into a line segment – or perhaps a n-dimensional sphere (see discussion regarding Pospelov oppositional scales).
Obtaining the even distribution of numbers is a critical process. However, there are degrees of uncertainties that actually play in favor of eventual positive utility. No matter how the distribution occurs, visual acuity made about distributions in cluster space can reinforce interpretations of the meaning of cluster patterns that are made by the human.
Reinforcement learning within an image space?
At the beginning of our current project (February, 2000), we did not know how we would handle the CCG visualization task. We felt that the visualization can come in any of at least three points.
1) In the clustering process itself
2) In the provision of semantic linkages between clusters
3) In the reinforcement of positive and negative learning
Our thinking is that the identification of tokens should be followed by the identification of a relational linkage between tokens. However, how does one impose such a relational linkage given the nature of human concepts? One answer is a reinforcement architecture.
A mapping could be developed between CCG images, derived in some way (see for example, as specified in the use cases in the last section), and objective physiological, psychological, and other medical parameters. The mapping might be complex, in the sense envisioned by the tri-level voting procedure, Russian Quasi Axiomatic Theory, or it could depend simply on visual acuity. The CCG – technique separates and then organizes regular patterns using a two-dimensional grid and color-coding. We looked for a separation and organization process that makes sense of non-regular patterns – such as those found in word distributions. These are visualized as point distributions on a circle.
It might be that normal distance metrics on CCG – type images might produce an ability to automatically parse new text for concepts – simply using a distance in image space. Any correspondence between image distances and conceptual linkages would be of extreme utility. In any case, we need a means to reinforce learning. We need an interface between user and computational systems. This interface must be used in critical situations where judgments are made and consequences determined. The interface must preserve the notion that human interpretant and judgment is an essential aspect of the experience of knowledge.
An interpretation of the data invariance in data from astronomical sources might be made. In this case a relationship would be found between various types of invariance in the data, as identified by a CCG type image, and scientific interpretation of that invariance. A reference system could be developed that extrapolated between visualization.
This domain appears simpler than the domain of text parsing and a Russian application of CCG visualization to scientific data streams is expected.
EEG data has been considered, however, the problem has been the acquisition of data as well as the development of a methodology for elucidating knowledge from knowledgeable persons in the field of EEG analysis.
Content evaluation seems to be the most difficult and to have the most potential for a true scientific breakthrough and for aggressive commercialization. However, we are well aware of the considerations that come from any proper understanding of the deep issues.
Content evaluation for full text data bases might follow a specific approach:
1) a inventory of CCG type images is recognized by a human as having value
2) the images are used as retrieval profiles against CCG representations of text not yet recognized
3) an associative neural network would take reinforcement into account in the modification of semantic interpretation
Several theorists have identified a problem. This problem is stated well by Alex Zenkin when he talks about expert knowledge and the role of well-defined goals to the CCG based vetting process. We hold that this problem can be solved using ecological systems theory based in part of the work of Robert Shaw, J. J. Gibson and A. N. Whitehead.
Comments from Alex Zenkin
VISAD works with any set of objects that is described by any set of tokens (signs, properties, etc.).
The set of tokens must include semantics on the set of objects.
The required semantics is defined by common goals of the problem under consideration (purchase, sale, advertisement, production, market, etc.). It is obvious, that objects will have different sets of tokens depending on the goal.
The requirement for goal-oriented information means that it is impossible to describe all objects of a large patent base by an unified set of tokens.
Not knowing terminal aims and semantic features of objects, we can construct a set of only formal syntactical, grammatical, and statistical tokens.
However, Zenkin sees the following possibilities.
First of all, we may formulate the terminal goals
1) Why we need to analyze the data?
2) What kind of problems will the new knowledge help us solve?
We may extract from the patent base a class of related objects (say, by means of key words).
A set of tokens related to terminal goals must be constructed by professionals in the problem domain defined by the terminal goals.
We may use VISAD in order to:
1) Visualize the given class of objects.
a. a. The class will have a distinctive CCG – type image that is used as an icon
b. b. The icon will have possible dynamics. Zenkin has suggested that Lefebvre type gestured images be used as a control language
c. c. Prueitt has developed some notion on state / gestures response interfaces to contextualized text databases.
2) Define more accurately the initial set of tokens
d. a. It is an interactive process of experts with the problem and of learning their professional knowledge and intuition.
e. b. Our experience shows that professionals frequently find out new ideas and solutions during VISAD usage.
f. c. Vladimir Lefebvre's has useful ideas on reflexive dynamic interaction "expert – an experts group", "expert – visualized problem"
g. d. Vladimir Lefebvre's "faces representation" for qualitative analysis of situations might be useful here.
3) Automate (with a visual CCG-support) classification of the given class of objects
4) Automate (with a visual CCG-support) creation of notions on classes
5) Creating a mathematical (logical) model of the kind "What will be IF …" or "reason - consequence", "cause - effect", etc
6) Automated (with a visual CCG-support) recognition (purposeful search) of objects in image-structured information base.
Additional comments on VISAD by Alex Zenkin
VISAD usage allowed us to make clear well-known information on logical and mathematical paradoxes (about 20 objects described by about 20 tokens). IN recent work in Russia, Zenkin discovered some new tokens, made a new classification of some paradoxes, and formulated notions on necessary and sufficient conditions of paradoxicality as a whole. These notions disclose the nature of paradoxicality.
Main stages of this work are described in Zenkin’s paper "New Approach to Paradoxes Problem Analysis" published in "Voprosy Filosofii" (Problems of Philosophy), 2000, no. 10, pp. 81-93 (see Annotation, today so far in Russian.
The VISAD and its usage to analyze large bases of text unstructured DATA is a large independent project.
Note from Paul Prueitt to Alex Zenkin November 17, 2000
CCG analysis of EEG data is a complex task. The data set that Prueitt received from Pribram’s lab has what is an unknown format of the numbers. Figuring out this format has stopped his attempt to put EEG data together.
If this data format problem is figured out, we still have the difficult issue of how to make sense of the syntactic structure that might be imposed
For this reason, Prueitt has turned to the unstructured text domain, since he has already solved the conversion of text to numbers and categories problem.
Prueitt’s hope was that Zenkin would be able to find both a source of data and the expertise to make a small experiment in Russia. We hope that Zenkin is able to solve both these pragmatic problems and thus produce an prototype application of CCG to unstructured scientific data.
Criterion for knowledge creation
We have developed a metaphoric linkage between Zenkin’s CCG application to number theory and parsing and interpretation applications to other data sets. The linkage is related to the notion of induction, Zeno’s paradox, Cantor’s diagonalization conjecture, and Godel’s complimentarity / consistency arguments.
In our view, knowledge creation occurs when a jump occurs from one level of organization to another level of organization. This jump is referred to as an “induction”. The scholarship on this notion of induction is concentrated in the area called ecological physics (J. J. Gibson, Robert Shaw) and perceptual measurement (Peter Kugler). We hold that inductive jumps require the existence of scales of organization in physical systems (see the work of Howard Pattee).
In the case of mathematical finite induction, the jump occurs when one assumes an actual infinity, and uses this assumption in the characterization of the positive integers. In the case of Zenkin’s principle of super induction, the jump allows the truth of the presence of one condition to be transferred to the proof of a theorem.
Zenkin’s principle of super induction is still an induction that occurs about formal structures that are regular and linear, even through perhaps intricate.
One requirement for CCG application to number theory is the existence of an infinite sequence of positive integers – generated by the Peano axiom and the imposed property of addition and multiplication. This sequence serves as an index on sequences of theorems. The objective of both induction and super induction is to find a way to demonstrate that the truth of an intricate relationship transfers. The transfer is between secession (Peano axiom), addition and multiplication properties of integers. This transfer will exist beginning at some point and then continue being true from that point in the sequence of theorems.
With any representation of concepts in text, we have a different situation. There is no relevant truth-value that can be assigned to some relationship between numbers, or so it seems at first glance.
Section 4: Study of cluster iterations and visualization
In reviewing the first three sections of this report, I find the concepts difficult. We must cover so much. First, we must cover what we have discovered about the knowledge creating methodology that is applied by Alexander Zenkin to the knowledge domain of number theory. Then we have to address what we think are the philosophical issues related to the notions of induction and super induction. Then we must develop a scientific foundation and a plan for applying the CCG techniques and concepts to natural language parsing.
The core issues are:
1) How do we replace the concept of number with some concept that has something to do with semantic tokens and linkages between these tokens?
2) How do we replace the original notion of super-induction so that the replacement has the same rigor as that discovered by Alexander Zenkin?
We have only begun preliminary work on the first core issue. The second core issue remains largely a proper statement of something that needs to be done, but that has not yet been done.
As we neared the completion of our contract time (December 2000) it became clear that the generalization of the concept of number that we needed is the cluster space itself.
The clustering process itself produces states, and each state in generated from the previous state using a specific iterated set of rules. These rules collectively can be thought of as an axiom, which, like the Peano axiom, is iterated. The visualization of a sequence of cluster spaces requires considerable work with computer programmers. However the literature has many examples of visualization of cluster spaces (specifically systems like Pathfinder and the Spires).
The first elements of the sequences should show a random scatter of the points. Figure one shows a random scatter of points to a circle.
Random scatter of 312 points on a circle
This is the scatter part of the scatter / gather methods. The gather part is generally either:
1) take two points randomly and evaluate the “nearness’ between the two corresponding documents and use this metric to move the two points a little closer or further apart on the circle.
2) take a one and a cluster at random and computer the “nearness” between the document and the cluster and move the point closer or further apart from the cluster on the circle.
The identification of a cluster is a critical issue, and perhaps this is where we might use some corollary to the CCG visualization. Again, it is difficult to be specific because there are many different ways to accomplish something. For example, whereas the first part of a gather process might be purely algorithmic (using the Prueitt voting procedure or the standard cosine measure); the second part might allow the use of visual acuity to select the boundaries of a cluster, and to make a judgment about the topics and themes that are central to that cluster. Once clusters are identified, then one might ask a user to review a cluster’s conceptual graph and the conceptual graph related to the point.
A question can be posed to the user regarding the “nearness” of the two conceptual graphs. The answer can then drive the gather process.
The question of nearness is problematic unless handled in a grounded fashion. The grounding that we suggest is Pospelov’s notion of an oppositional scale. Pospelov has made the conjecture that there are 117 categories of semantic linkage. The enumeration of a theory of semantic type by Pospelov is conjectured to be language independent, and to be related to the actual linkages that formed when the natural language systems go through the early formative process. The work of Stu Kauffman on autocatalytic sets establishes one way to justify the notion that semantic linkage in natural language in fact has a specific structure. Regardless of how one justified an enumerated set of dimensions, semantic linkage must be expressed through degrees of freedom required for each of these dimensions.
The analysis of the previous paragraph implied that the gather part of a scatter / gather method be made on a 117 dimensional sphere. The notion of nearness is thus specific to those oppositional scales that are seen to be relevant to a comparison such as between two cognitive graphs.
Such comparison is both situational and relative to a point of view. Thus the indexing of semantic linkage in text collections needs to be vetted by human introspection, and the consequences of this vetting process encoded into some type of compression. The gather process is exactly such a compression.
No matter how the limiting distribution occurs, visual acuity made about distributions in cluster space can reinforce interpretations of the meaning of cluster patterns that are made by the human. The limiting distribution reveals clusters that need to be validated as a relevant knowledge artifact in context. Knowledge validation is where we expect to seen formal notion related to Zenkin’s notion of super induction.
One requirement for CCG application to number theory is the existence of an infinite sequence of positive integers – generated by the Peano axiom and the imposed property of addition and multiplication. This sequence serves as an index on sequences of theorems. The objective of both induction and super induction is to find a way to demonstrate that the truth of an intricate relationship transfers. The transfer is between secession (Peano axiom), addition and multiplication properties of integers onto an evaluation of truth.
This transfer of evaluation will exist beginning at some point and then continue from that point in the sequence of theorems. The validated limiting distribution of a gather process has exactly the property we need. At some point the gather process does not change the clustered patterns. From this point on the clusters remain invariant.