Knowledge Representation Tibor Vámos
3. TECHNIQUES, PRACTICES, AND METHODOLOGIES 1 C ONCEPTUAL R EPRESENTATION
3.4. O PEN W ORLD R EPRESENTATION
Relational representation of traditional logic is occluded by the Closed World Assumption, which includes only a limited number of concepts and relations, and supports the hypothesis that the whole world is explorable in a well-defined way. Uncertainty methods open some windows to the real world of unexplored, unexpected phenomena. This is especially true for the nontraditional uncertainty methods which ignore the hypothesis of excluded middle and of independence of basic events. The price of a more permissive method is the increased softness of their basic model and consequently the inefficiency of reasoning capabilities. From the point of view of logical and mathematical rigor, they are less and less welcome. This used to be the history of all methods beyond the classical probabilistic theory. One of the compromises is that all methods, the fuzzy representation included, can be attached at any instance of an acceptable hypothesis to the relational representation of classical logic. This is the intention of practically all methods concerning an open world: restricting the part of soft representation to the unavoidable limits. The other way towards the open world of reality is the development of modern logic. It grew out of the
modalities of classical logic, i.e. the limitations of the validity to certain subjects, interpretation fields. This is done by the different interpretations of the quantifiers (Figure 8). The next, far leading step is a deeper intrusion into the meaning. The quantifiers define an external frame for the interpretation, the meaning of the variables, analyzed and interpreted by intensional logic, opens every possible interpretation due to the specific usage of the words in different cultures, different ages, different disciplines, different mood, different social environment (Figure 9).
Some readings of Modal Operators
[open box] A [open diamond] A
It is necessarily true that A
It is possibly true that A Alethic logic
It will always be true that A
It will be sometimes that A Temporal logic
It ought to be that A
It can be that A Deontic logic
It is known that A The opposite of A is not known Logics of knowledge It is believed that
that A
The opposite of A is not believed Logics of belief After every
terminating execution of the program, A is true
There is an execution of the program that terminates with A true
Dynamics logic
FIGURE 8 Modal representation. (After A. Thayse, D. Snyers: Languages and logics, in Thayse (Ed.): From Modal Logic to Deductive Databases, Chichester: John Wiley, 1989.)
Peter looks for his car
-- In the world of a certain parking place where automobiles are parking * type of parking automobiles
-- In the world of a railway station where railway cars are assigned to the ticket owners * type of railway cars
-- In the world of desires where somebody has a lottery ticket, can win a car, and is just looking at the
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* type of lottery prizes
FIGURE 9 Intentional representation.
The open world is full of contradictions. Because not all cases, relations, phenomena are known, one meets new facts in everyday practice, information which is not in strict logical relation with the previous knowledge, cannot be included into the framework of previous definitions. Three ways are used in common practice. The first preserves the earlier knowledge, its frames, structures and recognizes those which could not be fit in as exceptions. The second tries to detach this world of different phenomena by creating other new worlds. The third tries to create a unified new world of renewed concepts and relational hypotheses. The first can be illustrated by the favorite example of nonmonotonic logic, the exception of the birds (like the penguin or the oyster) which cannot fly, like the "regular" birds of the northern hemisphere. The second is the case of discovering the viral and bacterial variants of pneumonia or the difference between hepatitis A and B. The third is the case of the Kuhnian new paradigms in science, quantummechanics unifying the theories of the micro and the macroworld, or relativity theory, the Newtonian and the cosmological world. All these have representational methods in modern logic.
Nonmonotonic logic is the conceptual frame of logical relations where some new information is in discord with earlier information; the flow of information does not corroborate the consequence in a monotonic way. Two obvious procedures are at hand: narrowing down the conceptual frame to the extent where no contradiction exists, or the extension of the concept with new rules valid for the exceptions. The first is the
circumscription of McCarthy, the second is the use of default rules of Reiter.
In a system of an open world, not all statements corroborate the previous ones, or those hypotheses or conclusions that are logical antecedents or consequences of these statements. Monotonic logic is created in a closed world where all further statements, data, and facts add to the validity of the previous ones. Nonmonotonic logic looks at these contradictory sets of statements and attempts to find a consistent way of resolving the situation by adding further new conditions or canceling some old ones.
In classical logic, theorems are results of valid inferences; in nonmonotonic logic, one starts with all inferred statements, whether they are consistent or not. Then in most cases, one starts to find a fixed point, i.e., a minimal set where all statements get a consistent context. A typical paradigm is Tweety the bird, Tweety the penguin. Either we should exclude the penguin from the class of birds or flying should not be a characteristic attribute.
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Several methods were developed for restoring consistency, e.g., circumscription and default logic. Circumscription: "We know some objects in a given class and we have some ways of generating more. We jump to the conclusion that this gives all the objects in the class. Thus, we circumscribe the class to the objects we know how to generate."
"It is only a conjecture, because there might be an object such that" a predicate on the object "is not generated in this way." The heuristics of circumscription -- when one can plausibly conjecture that the objects generated in known ways are all there are -- are completely studied.
Circumscription is not deduction in disguise, because every form of deduction has two properties that circumscription lacks -- transitivity and what we may call monotonicity. Transitivity says that ‘if a is a consequence of b and c is a consequence of b, then c is a consequence of a. (If snowing, the road is slippery, if slippery then driving is dangerous, consequently if snowing then driving is dangerous.) In realistic cases (circumpscription), driving on snow by a car equipped with tire chains is not dangerous.’ Monotonicity says that within a class of statements all further sentences corroborate the statement further." No exception or contradiction occurs. This is not the case in circumscription and generally in
nonmonotonic logic because we find some exception cases, e.g., our penguin Tweety is a bird that does not fly. The way of circumventing this difficulty is to find a minimum model where all sentences are true (e.g., European birds, cars with standard, low mileage tires, etc.). Nevertheless, "It is not always true that a sentence true in all minimal models can be proved by circumscription. Indeed, the minimal model of Peano's axiom is the standard model of arithmetic and Gödel's theorem is the assertion that not all true sentences are theorems. Minimal models don't always exist, and when they exist, they are not always unique."1
1The quotations are from McCarthy (1977); the logical formulae are substituted by verbal explanation and examples.
Default logic: Default logic is a method for treating nonmonotonic problems. The basic idea is a distinction between general "hard" rules (facts) and their defaults, i.e., those rules which extend the world of the fact by exemptions and irregularities. (All birds fly but penguins do not.) The method is similar to
circumscription, the main difference is the theoretically well-formed idea of circumscription's minimal set, related to the fixed point theorems. A weakness of default logic is the arbitrary and occasional nature of the default rules. They cannot be inferred within the system, they can be used for further inference with much caution, and they can yield trivial contradictions.
The general class of default theories is mathematically intractable and that many desirable features of the logic can be obtained only for the class, so-called normal default theories, namely, theories in which all defaults have the form:
Such defaults are extremely common; for example, "Typically, birds fly" would be written:
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without explicating specific conditions [beta](x) under which the inference should be blocked. While most naturally occurring defaults are normal, the interactions among such defaults lead to
anomalous conclusions. For example, suppose we use normal defaults to express the facts that Tweety is a penguin, that penguins are birds, and that birds typically fly but penguins typically do not. We have:
The theory has two extensions:
and
depending on which default rule is applied first.
In both cases an automatic operating check system is needed for the control of the flow of new information if it really fits into the limited frame of circumscription or can be included into one of the existing default rules. This automatic record keeping and checking system is the Truth Maintenance System of Doyle.
A remarkable fact is the application of first-order classical logic for all these purposes of nonmonotonic logic. The necessary logical test of soundness is not different from the usual program test procedures; it can be completed only if the test program covers the whole possible logical graph of inference or better by the resolution principle executing the proof for the impossibility of the negative statement of the system. (All men are mortal -- we cannot find a man who was immortal). The complexity of these test procedures is another hint at the advice: do not try to develop specific tools for a real large scale system but use the ones which are developed by professionals of great experience and tested in real-life environments by several well referenced similar applications!
The second mainstream for creating coherent knowledge chunks is the definition of agents. Agent can be anything that operates on certain chunks of the "total" knowledge, a person, an enzyme, any process of natural sciences and industry, even time which operates on everything. An agent is in a narrow sense a modality, an external definition how the logic of the system should be interpreted and how it should work, defined by this interpretation. The same case is viewed in a different way by a legal expert agent, who uses the deontic logic modality, and by somebody who has some different cultural beliefs. By this way, different possible worlds are created, a separate world for each agent. In a cooperative process, such as manufacturing a car or building a house, the specialists of each production phase or the machines used for different tasks, the agents in this sense, have own worlds of their own professional knowledge, their task-executing programs. Usually, not only the strongly restricted task-oriented execution program is different, but the entire view. The artist of the interior design has a view about the roles of bricks completely different from a bricklayer.
If the worlds of the different agents can be well separated and well interfaced, no further problem exists, all can be reduced to the methods and their problems discussed earlier. A special type of agent is the system which coordinates different agents; one of the typical representations are the Blackboard systems
which concentrate on different input agents. They convey various information from various resources, sometimes in various styles. The system's task is to organize the information into a usable structure. The blackboard metaphor is valid only for the nonintelligent part of this task: having a device which can receive many kinds of records. The real problem is the interpretation of the information and creation of
rules for putting them into the right place of the object representation, frame, net, etc.
The agents can be supplied with goals, and these goals can create plans. An intelligent robot has a goal of reaching and catching an object in an environment where other objects, even other moving robots are present, and the task of the robot is to plan a trajectory which avoids all obstacles and gets to the goal. The goal is usually defined by logical statements on the final situation. The plan is a graph, represented by rules of logic, constraints, i.e., the obstacles are some frame-like representations of their instant locations. The robot task is a general metaphor of any goal-oriented actions satisfying some constraints on the environment. The nature of real environments suggests the understanding of all kinds of open world conditions, nonmonotonicity.
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A special kind of method, autoepistemic logic was drafted by Stalnaker and Moore for creating different closed worlds for different agents, based on their views, beliefs. The intriguing problem is a belief on the belief of the other agent. The metaphor of the autoepistemic logic procedure is the game with many distorting mirrors, each reflecting the deformed picture of the previous one, and performing some operation on the real object but seen on the last mirror!
The real difficulty is, like in real life, the possible and unforeseeable conflict among the agents resp. their worlds. This is where the knowledge representation joins the disciplines of conflict resolution: decision support, social choice, voting and polling, theories and methods of uncertainty and games. This relation of uncertainty calculations and decision was mentioned before. The decision support system which does or advises a choice, a preference of possible conflict resolution should be an integral knowledge part of the expert system. Which conflict resolution procedure should be used, what the previously definable preferences are, these are obvious prerequisite knowledge constituents as this was exemplified by the different possible strategies of medical diagnosis and therapy. An important result of Arrow was the theoretical proof for the non-existence of any finally rational choice in the presence of several rational preferences. This fact is experienced in our everyday life, and it suggests the application of a
sophisticated decision support system for seeing pros and cons of alternatives but the final decision is mostly a human responsibility. Knowledge representation should know this, the simplest solution is in any dubious situation a return to a man-machine dialog with the user or with the domain expert.
The third way of creating coherence in knowledge is the creation of new frames, new structures, which is mostly a human learning and ingenuity task, but can be supported also by machine learning.
The first-order logic structures behind the various systems permit not only the use of these systems for knowledge acquisition in a conducted domain expert-knowledge engineer dialog but can be a background for learning if information can be fed into the system in a suitable way. In the learning mode of operation not only the previously defined slots are filled in with the new information and according to a prescribed way but a search can be initiated for new structures, such as new relations for a relational database. The open world is fundamentally represented in natural language; this is the reason language is still, and remains for ever, a subject of investigations in human relations. The bridge between natural language and computer representation is, therefore, a basic problem and -- as far as it can be approximated -- a basic tool for knowledge representation. This is the real blackboard! Syntactical parsing, semantic analysis
belong to these tools, all achievements of general computer linguistics. These are, therefore, ingredients of every more sophisticated knowledge representation system.
Case-based reasoning can use all these methods of representation. The essential problem is the same as that of human expertize: remembering to a certain similar and some only likely similar cases the expert should decide if the similar decision is applicable or not. The practice of the Anglo-Saxon precedence- based Common Law or of a medical consultant are the best examples. The cases are represented in frames, scripts; the distances to other schemes are mostly defined by fuzzy-like estimations, Bayesian- types of conditional probabilities, distance measures used in clustering problems like the Nearest Neighbor methods. The representation of the singular cases and case prototypes like a given diagnostic pattern or a textbook malady description is the simpler part of the task; the invention of the pertinent similarity measure, definition of its relation to viewpoints (very special modalities) is the real human challenge.