![]() |
|||
![]()
|
![]() |
![]() |
![]() |
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
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: 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.
|
![]() |
|
Use of this site is subject certain Terms & Conditions. Copyright (c) 1996-1999 EarthWeb, Inc.. All rights reserved. Reproduction in whole or in part in any form or medium without express written permission of EarthWeb is prohibited. Please read our privacy policy for details. |