Functional Categorization of Knowledge

Presented at AAAI - SSS 92 Symposium:
Propositional Knowledge Representation)
FUNCTIONAL CATEGORIZATION OF KNOWLEDGE

Dr Sakir Kocabas
ITU, Maslak, Turkey


Abstract
The continuous increase of human knowledge rendered the classification of knowledge an important task, from very early ages, for philosophers like Aristotle down to the modern age, to Wittgenstein. These classifications were necessitated by the difficulties in understanding, memorization and transmission of knowledge. An analogous task now is faced in knowledge based artificial intelligence systems as the needs arise to build larger and more versatile systems. In this paper we introduce a method for organising knowledge into several linguistic categories. We describe how this categorization introduces clarity in representing different types of knowledge, how it facilitates the analysis of complex propositions into their simple constituents, and how these in turn can be assembled into complex constructs such as frames and schemata.


1 Introduction
Large knowledge systems of the future, especially those that will represent scientific theories in physics, chemistry, biology astronomy, etc., cannot be confined to narrow domains of expertise. However, as the amount of knowledge given to or acquired by a system increases, two main problems arise: _knowledge acquision bottleneck_ (Lenat, Prakash, & Shepherd, 1986) and _brittleness_ (McCarthy, 1983; Holland, 1986). Knowledge acquisition bottleneck is related to the acquisition of new knowledge by instruction or various other methods of learning. Minsky (1984) and Lenat et al. (1986) point out that, in acquiring new knowledge, the human mind overcomes this problem by recalling similar concepts it already knows about and by recording the exceptions to the case in consideration. Brittleness on the other hand, is related to the difficulties in having the knowledge system expand beyond the scope originally contemplated by their designers.

One solution to the brittleness problem, proposed by McCarthy (1983), is to provide a system with the ability to acquire commonsense knowledge and reasoning. Another solution, which emphasizes inductive learning and is based on general purpose learning algorithms, has been proposed by Holland (1986). His general purpose "classifier systems", are essentially reactive rule based systems relying on genetic algorithms.

Holland (1986) proposes induction as the basic - and perhaps the only - way of making large advances in overcoming brittleness. In considering the specific problems induction faces in this context, Holland identifies the creation of useful ways of categorizing input as the primary task. He suggests that categories must be incorporated into rules that "point" both to actions and to an aura of associated categories. That is, as they are induced, they must be arranged in a "tangled hierarchy", enabling the system to model its environment appropriately.

Holland seems to have an Aristotelian notion of concept-based categorization(1) in mind here as opposed to functional categorization of knowledge. By "category", he understands common abstract features between objects or frames, so that when these are captured, one set of feature can be used in developing a similar frame. (Frames can be regarded as complex propositions linked together by inheritance features.)

Lenat (1983) proposes to carry along multiple representations simultaneously and to shift from one representation to another to enable the knowledge system to carry out the most frequent operations more quickly. He says that this has not been much studied or attempted in artificial intelligence, except in very small worlds. Lenat's CYC system (see, Lenat & Guha, 1990) makes use of this idea.

Porter and Kibler (1986, p. 283) state that in machine learning research, most systems have been built on a small number of rules (heuristics) without having to address the problem of organizing learned knowledge into a coherent, efficiently accessible whole. Also, Cheng and Fu (1984) emphasize that, compared with knowledge representation or the formalization of concepts, little work has been done in the area of knowledge organisation.

Woods (1986) offers a simple classification of knowledge. He states that the "knowledge of the world" consists of two kinds of things - facts about what is or has been true (the known world state) and rules for predicting changes over time. He mentions the need for taxonomic organisation. Woods also argues that the standard notations and semantics of the predicate calculus are insufficient by themselves - they need to be supplemented with additional mechanisms e.g., for non-monotonic reasoning, and metalogical reasoning.

A "functional" knowledge system described by Brachman and Levesque (1983), distinguishes between "definitional" and "factual" information. Their system contains two "boxes" of knowledge; one for maintaining analytical knowledge, and the other to build descriptive domain theories. They also use two languages for their representation, a frame language for analytical knowledge and an "assertional langage" for the descriptive domain theories.

Langley (1986) states that although concept learning has been a basic mainstay of the machine learning community, most research in this area has ignored a number of well-established psychological phenomena. He says that basic-level categories appear to have a special status in human memory, being retreived more quickly and being acquired earlier than other concepts, and suggests that more work is needed on concept formation, for such research would yield a better understanding of human concepts and their acquisition, and it should also lead to improved methods of nonhuman concept learning (also see, Gennari, Langley & Fisher (1989).

His notion of category referred here, is also concept based, in which clusters of concepts are regarded as categories, which in turn, are organised in hierarchies. A concept based approach addresses the issue of conceptual organisation of knowledge. Whereas, it can be argued that a considerable proportion of human cognitive activity is propositional. Therefore a functional organisation of knowledge needs to be developed at least in equal priority and depth.

The categorization intoduced in this paper is based on fundamental methodological and linguistic criteria: methods of verification, meaning (use), and the function of expressions. It is not aimed at the classification of concepts, but of simple propositions. In philosophical terms, our concept of category is based on the deep grammar of propositions and therefore, is quite different from the concept-based notions.

2. Theoretical and Philosophical Background: Piaget and Wittgenstein The future of artificial intelligence, to a certain extent, depends on the studies on cognitive development. Because of its better tractability by means of natural language, human cognitive development is still the best source for developing knowled ge based models in artificial intelligence. The foundations of cognitive science was laid by Piaget's work on human cognitive development in the early 1920s.

Piaget (1971) made extensive studies on human cognitive development and the development of language. There are several reasons why his work is of interest: It is related with 1) the linguistic methods of knowledge acquisition (questions and their classification), 2) the order of knowledge acquisition according to the types of knowledge acquired, 3) the methods of relating the acquired knowledge, and 4) the theoretical foundations for the organisation of knowledge.

Piaget (1971, p. 30) draws our attention to the importance of questions in cognitive development. From the standpoint of cognitive development, a question is a spontaneous search for information. He studies questions asked by the child between the ages of six and seven, and classifies them as questions 1) for causal explanation, 2) about reality, 3) on actions and intentions, 4) rules, 5) about classification, and 6) arithmetical questions.

It may be worthwhile to consider the child's questions from the standpoint of the organisation of acquired knowledge. Explanations seem to play a critical role in these activities. It is conceivable that the child's mind is actively involved in organising its knowledge during the knowledge acquisition processes. Analogously, intelligent knowledge systems may have to be given the ability to ask as many meaningful questions as possible during knowledge acquisition and learning, particularly in the development stage of such systems. The difficulties encountered in knowledge acquisition can be avoided by the use of such strategies of learning frequently used by the child. Lenat et al. (1986) use a similar strategy in developing the knowledge base of their CYC system, but their knowledge acquisition methods are not automated.

Piaget's observations on the "why" questions of the child can be viewed as the manifestation of the operations of an effective organizational capability for acquired knowledge. The child's lack of interest in logical justification of explanations can also be explained within this perspective: Confronted with a vast amount of knowledge to be learned, the primary cognitive problem must naturally be how to organise the acquired knowledge rather than learning how any proposition can be theoretically, logically or mathematically justified. Piaget's classification of questions and explanations, albeit for psychological purposes, constitutes an important step in understanding how expressions are used in language and how they might be classified according to their function and methods of verification.

Having seen some of Piaget's views on human cognitive development, we can now take a brief look at Wittgenstein's philosophical work on the grammatical distinctions between various types of expressions. Wittgenstein considered the functional classification of expressions as one of the most important tasks in philosophy. In one of his earlier works (Wittgenstein, 1965, pp. 44-45), he provides an illustrative analogy between the task of organising a heap of books into a library and the classification of knowledge. Wittgenstein pronounces that some of the greatest achievements in philosophy could only be compared with taking up some books which seemed to belong together and putting them on different shelves; nothing more being final about their positions than that they no longer lie side by side.

Wittgenstein's use of the methods of distinction between various types of expressions are at times implicit and scattered in his books. At Cambridge in 1939 (see, Wittgenstein, 1974), he devoted a whole series of lectures on how to distinguish mathematical sentences from non-mathematical statements. For example, in (Wittgenstein, 1974, p. 34) he says: "'Professor Hardy believes that x1 > x0' is not a mathematical statement. It is no more a mathematical statement than "William said that 7x8=54' is a mathematical statement." Here, Witggenstein is trying to draw a distinction between a factual statement and a mathematical sentence. Indeed, later (Wittgenstein, 1974, p. 111), he clearly states that in his discussions he aims to show the essential difference between the uses of mathematical propositions and non-mathematical propositions which seem to be "exactly analogous" to the former. One of the criteria that he proposes for distinguishing between mathematical and non-mathematical statements is the prefix "by definition ...", which is applicable to logico-mathematical and formal statements, but not to factual statements. In his various studies, Wittgenstein distinguishes between philosophical statements and theoretical statements; first-person psychological statements and theoretical statements; and basic belief statements and theoretical statements (e.g., see, Wittgenstein, 1953; Wittgenstein, 1970; Wittgenstein, 1978).


The Method of Categorization
The development of large and reliable knowledge systems requires an effective knowledge organisation. It seems that one way of achieving this is to categorize simple propositions into functionaly different classes and then assemble them into frames as necessary. Here there are two questions to be asked: 1) How many grammatically different categories can be identified in language? 2) What can be the criteria to be used in distinguishing between these categories? Before we attempt to answer these questions, the problems of consistency and completeness in knowledge systems need to be reconsidered.

3.1 Consistency, Completeness, Validity and Truth
Since Goedel's (1962) famous theorem, it is widely accepted that an overall completeness and consistency may not be achievable in large bodies of knowledge. However, a domain dependent consistency and completeness can and needs to be maintained. It was Wittgenstein (1981, p. 118) who pointed out that a contradiction is not the end of the world, but something that sets a limit to a particular "language game" or "grammar". His remarks opened the way to a context dependent concept of consistency in the philosophy of language. More recently, from the viewpoint of artificial intelligence, de Kleer (1986) argues that there is no necessity that the overall database be consistent in a knowledge system. He suggests that a context dependent concept of consistency provides a better way of achieving control in problem solving. Similar arguments can be found in recent artificial intelligence literature (see, e.g. Lenat & Feigenbaum, 1987; 1991.)

It may even be meaningless to try to achieve an overall consistency and completeness in complex bodies of knowledge. The simple reason is that "truth" and "verification" have different meanings in different "grammars". In other words, quite different methods and criteria are employed in establishing the "truth" of propositions in different categories. To illustrate, let us consider the expressions:

(1) 1^3 + ... + n^3 = (n(n+1)/2)^2
(2) Jupiter is a planet.
(3) In a nuclear reaction, the amount of energy released is equal to the rest-mass difference multiplied by the square of the velocity of light (E=mc^2).
(4) Calculus was invented by Newton and Leibniz independently.

The criteria and methods that are used in establishing the "validity" of each of these expressions are quite different:.The first expression is a mathematical statement which can be proved by the axioms of arithmetic using mathematical induction, and no factual investigation is needed to prove it. The second is a formal statement the "truth" of which is implied by the order of the concepts "planet" and "Jupiter" in language. Unlike theoretical, hypothetical, empirical statements, formal statements are not compared with facts for their validity. The third one is a theoretical statement which is used as part of a model to understand and explain certain physical phenomena. Finally, the last sentence is a historical statement, the "validity" of which is established by the methods used in historical study.

These four statements can be considered as belonging to four different categories, namely the categories of logico-mathematical, formal, theoretical, and historical statements. Propositions belonging to these categories can be found in the body of almost any scientific work. Therefore, knowledge systems should take into account such categorical distinctions. A system of criteria has been developed for functional classification of propositions, based on the works of Piaget (1971) and Wittgenstein (1974) outlined above. The principles for the functional categorization of propositions will now be described.

3.2 The Categorization Criteria
Propositions can be categorized according to their functions (uses) in language. The criteria that can be used in such categorizations can initially be divided into three groups:

1) Methodological (epistemic) criteria. The differences between the methods of verification (or falsification) of propositions may help to determine their categories. Verification methods of theoretical statements are different from those of logico-mathematical statements. In addition, verification may not apply to some propositions at all (e.g., allegorical and fictional sentences). A theoretical, hypothetical or empirical statement must be testable, or else must be derivable from testable theoretical statements, while logico- mathematical and formal statements are not verified by testing (i.e. by observation and/or experimentation).

Verification of formal statements is embedded in the logical structure of their constitutent concepts in language. In this way, formal statements reflect the concept structure in language. For example the sentences, "A cat is an animal," and "Calcium is an element" are formal statements and no one would ask for their verification by the verification methods of theoretical, hypothetical or empirical statements. Definitions also belong to the category of formal statements. On the other hand, logico- mathematical statements (e.g., theorems, non-theorems, lemmas and corollaries) are subject to the criterion of provability in a formal or informal calculus.

2) Grammatical labels used in language in the form of prefixes and postfixes., which play a role in the categorization of expressions. Some examples are: "By definition, ...", "According to the theory, ...", "According to the hypothesis, ...", "According to our experiences, ...", "I believe that, ...", "I do not believe that, ...", "I feel that, ...", "Probably, ...", "Possibly, ...", etc. It seems that at times we use these prefixes like flags or labels for classes of expressions, to signify their places in our knowledge.

Each of these prefixes are meaningful with the expressions of a certain category, while with others they are not. For example, it is meaningful to say: "By definition, 2+2 is 4," while it is not meaningful to say: "By definition, the gravitational force between two bodies is proportional to their masses and inversely proportional to the square of the distance between them." Similarly, it is not meaningful to say: "According to the hypothesis, 2+2 is 4," while it is meaningful to use the prefix with a theoretical or hypothetical statement.

3) Comparisons and contrasts of propositions with the ones already identified as belonging to a category. Comparisons can be very useful in cases where other criteria may fail to identify the category of a statement. Consider Fermat's "last theorem" for example. One might think that it is a hypothesis like those which physicists use to hypothesize certain subatomic particles (e.g. quarks). Fermat's last theorem is easily comparable with some other familiar arithmetical theorems, whereas physical hypotheses are more readily comparable with other similar physical hypotheses.

By using these criteria, it is possible to build a system of categorization which can be used in distinguishing the following categories: i) logical propositions, ii) mathematical statements, iii) formal statements, iv) grammatical (or meta) statements, v) theoretical-hypothetical-empirical statements, vi) historical sentences, vii) factual statements. In addition to these, there are other identifiable categories which can be listed as: a) basic belief statements, b) sensory and intentional statements, c) metaphorical and allegorical statements, d) fictional sentences. However, these will not be considered here, because the seven categories above seem to be sufficient to represent scientific knowledge to a large extent.

A list of questions that can be used in determining the category of a given statement S, is as follows:

- Does the statement S describe an event currently in effect?
- Does the statement S describe a past event or state of affairs?
- Does the statement S define a concept?
- Is the statement S testable against facts?
- Is the statement S varifiable by observations/experiments?
- Is the statement S provable/deducible?
- Is the statement S about another statement?
- Is the statement S a comment on a property/relation?
- Is the statement S a comment on a state of affairs?
- Is the statement, "Possibly, S" meaningful?
- Is the statement, "Probably, S" meaningful?
- Is the statement, "By definition, S" meaningful?
- Is the statement, "According to the hypothesis, S" meaningful?
- Is the statement, "According to the theory, S" meaningful?
- Is the statement, "According to the experiences, S" meaningful?
- Is the statement, "According to the rules, S" meaningful?
- Is the statement, "According to (so and so), S" meaningful?

The determination of the category of a given statement is a classification problem, in which the parameters are the questions listed.

3.3 Examples of Categorized Propositions
Some examples of statements belonging to the seven categories named above are as follows:

Logical Statements:
- A proposition and its negation imply all other propositions.
- A proposition P is derivable from propositions P and Q.

Mathematical Statements:
- The sum of the inner angles of a triangle is 180 degrees.
- There are natural numbers x, y, z, such that x^3 + y^3 = z^3.

Formal Statements:
- Jupiter is a planet.
- Cu is the chemical symbol of copper.

Grammatical Statements:
- Superconductivity is an important property.
- The statement, "Aluminium substitution reduces superconductivity," is consistent (with the knowledge base).
- Why BaPbBiO3 is a superconductor is not explainable (by the existing knowledge).

Theoretical-hypothetical-empirical Statements:
- The decay products of neutron are proton, electron and antineutrino.
- Electronegativity of nickel is 1.9.
- In metallic superconductors, superconductivity and electron density are positively related.

Historical Propositions:
- Superconductivity was discovered by H.K. Onnes in 1911.
- Partial substitution of sulfur for oxygen in LaNiO3 has been tried in superconductivity experiments.

Factual Statements:
- The price of Aluminium is 1.75 US dollars.
- The price of Scandium is over 50,000 US dollars.

Simple and complex propositional knowledge can be further organised within each category into levels of expressions. The levels can be based on Russell's logical theory of types (see, Whitehead & Russell, 1970). The categorization has been applied to varying extents in four different computational models of scientific reasoning and discovery in particle physics (see, Kocabas, 1989a; 1989b; and 1991), and is currently implemented in the development of a computational model of discovery in oxide superconductivity. These will not be described here, but a discussion based on our observations about the implementations will be given instead.

4. Discussion
The importance of the categorization of knowledge lies in the following possible advantages it may provide in the acquisition, representation, refinement and effective use of knowledge: 1) simple and complex propositions of different categories can be represented, accessed and used separately, 2) automatic transformation of knowledge from one form of representation into another one, especially from predicate statements into frames, can be made easier, which can be very useful in knowledge acqu isition during the development of large knowledge systems like CYC (Lenat et al., 1986), 3) some useful general formal and informal rules applicable to each category can be found, 4) logic mistakes in designing knowledge systems can be minimized, 5) search procedures can be made more effective, for categorization eliminates unnecessary search activities in the system, (e.g., formal questions can be answered by conducting search only in formal knowledge, theoretical questions in theoretical knowledge and so on), 6) more detailed and effective identification and resolution of conflicts and theory revision (or truth maintenance) is possible, and 7) dynamic reorganisation of knowledge can be made easier.

Categorization of knowledge facilitates truth maintenance, as in such systems "truths" of propositions of different categories are established differently.(A more detailed description of conflict resolution based on the categorization is given in Kocabas, 1989c.) Furthermore, some priorities of validity can be given to knowledge belonging to certain categories. For example, if a factual statement is in contradiction with a hypothesis, the problem is resolved by simply giving priority to factual knowledge over the theoretical, rather than removing the hypothesis.

In a categorized knowledge system, hypothesis generation is more systematic. New hypotheses are generated from factual and theoretical knowledge by induction and other forms of generalization. Similarly, new hypotheses can be generated from theoretical knowledge by specialization, abstraction and abduction. Certain forms of reasoning do not apply to certain categories of knowledge (e.g., specialization is not applicable to factual knowledge).

Combined uses of frame and logic representation provides a more structured representation of knowledge. Frame representation by itself is less efficient in making full use of logical and extralogical inference such as deduction, abstraction and abduction (see, Lenat & Guha, 1990). On the other hand, logic representation alone provides a less efficient basis for certain kinds of reasoning (e.g. taxonomic and analogical reasoning). The integration of frame and logic representations allows the copying and transformation of knowledge from frames into predicate statements. Brewka (1987) describes a method for translating knowledge from frames into predicate statements. However, he makes no mention of any theoretical work on translating predicate statements into frames in a general and systematic way. Categorization provides this opportunity, as it allows the transfer or transformation of predicate statements into frames.

In a categorized system, knowledge acquisition does not have to be in the form of frames. Knowledge can be entered in simple predicate statements to be categorized by means of a small set of transformation functions. They can then be automaticall y structured into frames by means of a set of knowledge assembly functions. Naturally, the frames must reflect the categorical distinctions in their structure. For example, every frame can have slots corresponding to the category names such as "logical", "formal", "the" and "factual".

Categorization facilitates the analysis of descriptive knowledge. Other methods of expressing complex propositions in terms of their atomic constitutents had been developed. E.g. Kowalski (1979, p. 33) describes the method of expressing n-ary relationships as a conjunction of n+1 binary relationships. Complex descriptions can also be represented in this way. This method is directly applicable in frame systems, but it decomposes propositions into decriptions instead of atomic sentences, which may not always be desirable for reasons of hindering the effective use of logical and extralogical inference.

Complex propositions can be analyzed to and represented in simple, categorized predicate statements. Consider, for example the proposition: "The pulsar, which was discovered by radio astronomers is a rapidly spinning neutron star whose radio sign al regularly turns off and on." This sentence can be split into simpler propositions:

formal: A pulsar is a neutron star.
the: A pulsar spins rapidly.
the: The radio signal of a pulsar rapidly turns on and off.
historical: The pulsar was discovered by radio astronomers.

As is seen, the constituent propositions are not only representationally, but also categorically distinguishable. Feigenbaum has recently redefined problem solving in terms of "knowledge assemby" rather than search (see, Engelmore & Morgan, 1988, vi i). In this new definition the emphasis is on "finding the right piece of knowledge to build into the right place in the emerging solution structure". In a categorized knowledge system simple propositions can be assembled together to build more complex constructs such as complex propositions, frames and schemata.

Inevitably, the categorization has its own drawbacks, beside introducing hopes in resolving some important knowledge level problems in the development of large knowledge systems. First of all, it imposes a structure on descriptive kowledge, which is based on a set of criteria. The validity of these criteria can be argued, but if humans utilize such criteria in organising their knowledge, it is worth considering to utilize them in computational models.

Another problem can be with certain propositions whose category may not be easy to decide. In such cases, the proposition can be maintained in more than one category. However, such cases are rare, and the duplications do not pose a serious difficulty. On the other hand, categorization errors can be identified and resolved by a set of "knowledge administration functions" which can be built to supervise the "distribution", "maintenance" and the "assembly" of knowledge in the categories of the system.

Holland's (1986) suggestion of using general purpose learning algorithms and giving emphasis on inductive learning as a solution to the brittleness problem has appeal. However, systems based on such methods use complex input and output units and their complexity grows as the system is given knowledge of different kinds and levels. In the end, the brittlenes problem transforms into two problems: Input and output management. To a certain extent this can be avoided in a multi layered general purpose system, but the theoretical basis for such systems has not been sufficiently developed. Additionally, Holland's proposal underestimates the role of the deductive methods in learning.

The categorization scheme introduced, is much more detailed than the classification suggested by Woods (1986) in which he divides knowledge into "facts" and "rules". His "facts" include what we call factual statements, simple logical, formal, mathematical, theoretical, and historical propositions. His "rules" include complex logical, formal, mathematical, and theoretical rules, which can be called rules of inference, as well as action rules (or production rules). The categorization meets some of the requirements proposed by Aiello et al., (1986), as it allows knowledge and metaknowledge to be represented in the same form. In this way, it allows the inference mechanisms to be accessible at both levels.

5. Conclusions
In this paper we described a methodology for organising descriptive knowledge into several functional categories, and discussed the ways in which it can integrate different methods of representation such as predicate logic and frame representations. The categorization helps to resolve some of the major problems of developing and maintaining large knowledge systems. These problems are, brittleness, knowledge acquisition bottleneck, and the identification and resolution of conflicts. Clarity and simplicity are essential in building complex knowledge systems, because as the system grows, it becomes more and more difficult to keep track of the relationships between domain concepts. (Lenat et al., 1986; 1990) provide dramatic examples of the complexities of adding more knowledge to a large knowledge system.) The categorization scheme introduced, provides clarity and simplicity, as it allows different kinds of knowledge to be represented in an organised way. Commonsense, as well as scientific knowledge can be represented in the categories described. The categorization has been implemented in several computational systems which model reasoning and discovery in astronomy, particle physics, and high-temperature superconductivity, incorporating various methods of learning and conflict resolution. More detailed implementations are being carried out.

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