Reasoning in Practical Situations Pei Wang Department of Computer and Information Sciences Temple University pei.wang@temple.edu http://www.cis.temple.edu/∼pwang/ 1 Problems in Practical Situations An automatic reasoning system usually consists of the following major components: (1) a formal language that represents knowledge, (2) a semantics that defines meaning and truth value in the language, (3) a set of inference rules that derives new knowledge from existing knowledge, (4) a memory that stores knowledge, and (5) a control mechanism that chooses premises and rules in each inference step. The first three components are usually referred to as a logic, or the logical part of the reasoning system, and the last two as an implementation of, or the control part of the system. The most influential theory for the logic part of reasoning systems is the modern symbolic logic, especially, first-order predicate logic. For the control part, it is the theory of computability and computational complexity. Though these theories have been very successful in many domains, their application to reasoning in practical situations shows fundamental differences from human reasoning in these situations. Traditional theories of reasoning are certain in several aspects, whereas actual human reasoning in practical situations is often uncertain in these aspects. – The meaning of a term in traditional logic is determined according to an interpretation, therefore it does not change as the system runs. On the contrary, the meaning of a term in human mind often changes according to experience and context. Example: What is “game”? – In traditional logic, the meaning of a compound term is completely determined by its “definition”, which reduces its meaning into the meaning of its components and the operator (connector) that joins the components. On the contrary, the meaning of a term in human mind often cannot be fully reduced to that of its components, though is still related to them. Example: Is a “blackboard” exactly a black board? – In traditional logic, a statement is either true or false, but people often take truth value of certain statements as between true and false. Example: Is “Tomorrow will be cloudy” true or false? – In traditional logic, the truth value of a statement does not change over time. However, people often revise their beliefs after getting new information. Example: After learning that Tweety is a penguin, you may change some of your beliefs formed when you only know that it is a bird. M.Gh. Negoita et al. (Eds.): KES 2004, LNAI 3215, pp. 285–292, 2004. c Springer-Verlag Berlin Heidelberg 2004
286 P. Wang – In traditional logic, a contradiction leads to the “proof” of any arbitrary conclusion. However, the existence of a contradiction in a human mind will not make the person to do so. Example: Have you ever had a contradiction in your mind? Do you believe 1 + 1 = 3 at that time? – In traditional reasoning systems, inference processes follow algorithms, therefore are predictable. On the other hand, human reasoning processes are often unpredictable, in the sense that sometimes a inference process “jumps” in an unanticipated direction. Example: Have you ever waited for “inspiration” for your writing assignment? – In traditional reasoning systems, how a conclusion is derived can be accurately explained. On the contrary, human mind often generate conclusions whose source cannot be backtracked. Example: Have you ever said “I don’t know why I believe that. It’s just my intuition.”? – In traditional reasoning systems, every inference process has a pre-specified goal, and the process stops whenever its goal is achieved. However, though human reasoning processes are also guided by various goals, they often cannot be completely achieved. Example: Have you ever tried to find the goal of your life? When can you stop thinking about it? Furthermore, all the inference rules of traditional logic are deduction rules, where the truth of the premises is supposed to guarantee the truth of the conclusion. In a sense, in deduction the information in a conclusion is already in the premises, and the inference rule just reveals what is previously implicit. For example, from “Robins are birds” and “Birds have feather,” it is valid to derive “Robins have feather.” In everyday reasoning, however, there are other inference rules, where the conclusions seem to contain information not available in the premises: Induction produces generalizations from special cases. Example: from “Robins are birds” and “Robins have feather” to derive “Birds have feather.” Abduction produces explanations for given cases. Example: from “Birds have feather” and “Robins have feather” to derive “Robins are birds.” Analogy produces similarity-based results. Example: from “Swallows are similar to robins” and “Robins have feather” to derive “Swallows have feather.” None of the above inference rules guarantee the truth of the conclusion even when the truth of the premises can be supposed. Therefore, they are not valid rules in traditional logic. On the other hand, these kinds of inference seem to play important roles in learning and creative thinking. If they are not valid according to traditional theories, then in what sense they are better than arbitrary guesses? Finally, traditional logic often generates conclusions that are different from what people usually do. Sorites Paradox: No one grain of wheat can be identified as making the difference between being a heap and not being a heap. Given then that one grain of wheat does not make a heap, it would seem to follow that two do not, thus three do not, and so on. In the end it would appear that no amount of wheat can make a heap. Reasoning in Practical Situations 287 Implication Paradox: Traditional logic uses “P → Q” to represent “If P , then Q”. By definition the implication proposition is true if P is false or if Q is true, but “If 1+1 = 3, then the Moon is made of cheese” and “If life exists on Mars, then robins have feather” don’t sound right. Confirmation Paradox: Black ravens are usually taken as positive evidence for “Ravens are black.” For the same reason, non-black non-ravens should be taken as positive evidence for “Non-black things are not ravens.” Since the two statements are equivalent in traditional logic, white sacks are also positive evidence for “Ravens are black,” which is counter-intuitive. Wason’s Selection Task: Suppose that I show you four cards, showing A, B, 4, and 7, respectively, and give you the following rule to test: “If a card has a vowel on one side, then it has an even number on the other side.” Which cards should you turn over in order to decide the truth value of the rule? According to logic, the answer is A and 7, but people often pick A and 4. 2 Assumptions of Reasoning Systems None of the problems listed in the previous section is new. Actually, each of them have obtained many proposed solutions, in the form of various non-classical logics and reasoning systems. However, few of these solutions try to treat the problems altogether, but see them as separate issues. A careful analysis reveals a common nature of these problems: they exist in “practical situations”, not in mathematics, which is about idealized situations. At the time of Aristotle, the goal of logic was to find abstract patterns in valid inference that apply to all domains of human thinking. It remained to be the case until the time of Frege, Russell, and Whitehead, whose major interest was to set up a solid logic foundation for mathematics. For this reason, they developed a new logic to model valid inference in mathematics, typically the binary deduction processes that derives theorems from axioms and postulations. What is the deference between “practical situations” as in everyday life and “idealized situations” as in mathematics? A key difference is their assumption on whether their knowledge and resources are sufficient to solve the problems they face. On this aspect, we can distinguish three types of reasoning systems: Pure-Axiomatic Systems. These systems are designed under the assumption that both knowledge and resources are sufficient. A typical example is the notion of “formal system” suggested by Hilbert (and many others), in which all answers are deduced from a set of axioms by a deterministic algorithm, and which is applied to some domain using model-theoretical semantics. Such a system is built on the idea of sufficient knowledge and resources, because all relevant knowledge is assumed to be fully embedded in the axioms, and because questions have no time constraints, as long as they are answered in finite time. If a question requires information beyond the scope of the axioms, it is not the system’s fault but the questioner’s, so no attempt is made to allow the system to improve its capacities and to adapt to its environment. 288 P. Wang Semi-axiomatic Systems. These systems are designed under the assumption that knowledge and resources are insufficient in some, but not all, aspects. Consequently, adaptation is necessary. Most current AI approaches fall into this category. For example, non-monotonic logics draw tentative conclusions (such as “Tweety can fly”) from defaults (such as “Birds normally can fly”) and facts (such as “Tweety is a bird”), and revise such conclusions when new facts (such as “Tweety is a penguin”) arrive. However, in these systems, defaults and facts are usually unchangeable, and time pressure is not taken into account [Reiter, 1987]. Many learning systems attempt to improve their behavior, but still work solely with binary logic where everything is black-and-white, and persist in always seeking optimal solutions of problems [Michalski, 1993]. Although some heuristicsearch systems look for less-than-optimal solutions when working within time limits, they usually do not attempt to learn from experience, and do not consider possible variations of time pressure. Non-axiomatic Systems. In this kind of system, the insufficiency of knowledge and resources is built in as the ground floor (as explained in the following). Pure-axiomatic systems are very useful in mathematics, where the aim of study is to idealize knowledge and questions to such an extent that the revision of knowledge and the deadlines of questions can be ignored. In such situations, questions can be answered in a manner so accurate and reliable that the procedure can be reproduced by an algorithm. We need intelligence only when no such pure-axiomatic method can be used, due to the insufficiency of knowledge and resources. Many arguments against logicist AI [Birnbaum, 1991, McDermott, 1987], symbolic AI [Dreyfus, 1992], or AI as a whole [Searle, 1980, Penrose, 1994], are actually arguments against a more restricted target: pure-axiomatic systems. These arguments are valid when they reveal many aspects of intelligence that cannot be produced by a pureaxiomatic system (though these authors do not use this term), but some of the arguments seriously mislead by taking the limitations of these systems as restricting all possible AI systems. Unlike in mathematics, in practical situations a system has to work with insufficient knowledge and resources. By that, I mean the system works under the following restrictions: Finite: The system has a constant information-processing capacity. Real-Time: All tasks have time constraints attached to them. Open: No constraint is put on the content of the experience that the system may have, as long as they are representable in the interface language. 3 NARS Overview NARS (Non-Axiomatic Reasoning System) is an intelligent reasoning system designed to be adaptive and works under the assumption of insufficient knowledge and resources [Wang, 1995]. Here the major components of the system are briefly introduced. For detailed technical discussion, please visit the author’s website. Reasoning in Practical Situations 289 When a system has to work with insufficient knowledge and resources, what is the criteria of validity or rationality? This issue needs to be addressed, because the aim of NARS is to provide a normative model for intelligence in general, not a descriptive model of human intelligence. It means that what the system does should be “the right thing to do,” that is, can be justified against certain simple and intuitively attractive principles of validity or rationality. In traditional logic, a “valid” or “sound” inference rule is one that never derives a false conclusion (that is, it will by contradicted by the future experience of the system) from true premises. However, such a standard cannot be used in NARS, which has no way to guarantee the infallibility of its conclusions. However, this does not mean that every conclusion is equally valid. Since NARS is an adaptive system whose behavior is determined by the assumption that future situations are similar to past situations, in NARS a “valid inference rule” is one whose conclusions are supported by evidence provided by the premises used to derive them. Furthermore, restricted by insufficient resources, NARS cannot exhaustively check every possible conclusion to find the best conclusion for every given task. Instead, it has to settle down with the best it can find with available resources. Model-theoretic semantics is the dominant theory in the semantics of formal languages. For a language L, a model M consists of the relevant part of some domain described in another language ML, and an interpretation I that maps the items in L onto the objects in the domain (labeled by words in ML). ML is referred to as a “meta-language,” which can be either a natural language, like English, or another formal language. The meaning of a term in L is defined as its image in M under I, and whether a sentence in L is true is determined by whether it is mapped by I onto a “state of affairs” that holds in M. With insufficient knowledge and resources, what relates the language L, used by a system R, to the environment is not a model, but the system’s experience. For a reasoning system like NARS, the experience of the system is a stream of sentences in L, provided by a human user or another computer. In such a situation, the basic semantic notions of “meaning” and “truth” still make sense. The system may treat terms and sentences in L, not solely according to their syntax (shape), but in addition taking into account their relations to the environment. Therefore, What we need is an experience-grounded semantics. NARS does not (and cannot) use “true” and “false” as the only truth values of sentences. To handle conflicts in experience properly, the system needs to determine what counts as positive evidence in support of a sentence, and what counts as negative evidence against it, and in addition we need some way to measure the amount of evidence in terms of some fixed unit. In this way, a truth value will simply be a numerical summary of available evidence. Similarly, the meaning of a term (or word) in L is defined by the role it plays in the experience of the system, that is, by its relations with other terms, according to the experience of the system. The “experience” in NARS is represented in L, too. Therefore, in L the truth value of a sentence, or the meaning of a word, is defined by a set of sentences, 290 P. Wang also in L, with their own truth values and meanings — which seems to have led us into a circular definition or an infinite regress. The way out of this seeming circularity in NARS is “bootstrapping.” That is, a very simple subset of L is defined first, with its semantics. Then, it is used to define the semantics of the whole L. As a result, the truth value of statements in NAL uniformly represents various types of uncertainty, such as randomness, fuzziness, and ignorance. The semantics specifies how to understand sentences in L, and provides justifications for the various inference rules. As said above, NARS needs a formal language in which the meaning of a term is represented by its relationship with other terms, and the truth value of a sentence is determined by available evidence. For these purposes, the concept of (positive or negative) evidence should be naturally introduced into the language. Unfortunately, the most popular formal language used in first-order predicate logic does not satisfy the requirement, as revealed by the “Confirmation Paradox” [Hempel, 1943]. A traditional rival to predicate logic is known as term logic. Such logics, exemplified by Aristotle’s Syllogistic, have the following features: [Bocheński, 1970, Englebretsen, 1981] 1. Each sentence is categorical, in the sense that it consists of a subject term and a predicate term, related by a copula intuitively interpreted as “to be.” 2. Each inference rule is syllogistic, in the sense that it takes two sentences that share a common term as premises, and from them derives a conclusion in which the other two (unshared) terms are related by a copula. Traditional term logic has been criticized for its poor expressive power. In NARS, this problem is solved by introducing various types of compound terms into the language, to represent set, intersection and difference, product and image, statement, and so on. The inference rules in term logic correspond to inheritance-based inference. Basically, each of them indicates how to use one item as another one, according to the experience of the system. Different rules correspond to different combinations of premises, and use different truth-value functions to calculate the truth value from those of the premises, justified according to the semantics of the system. The inference rules in NAL uniformly carry out choice, revision, deduction, abduction, induction, exemplification, comparison, analogy, compound term formation and transformation, and so on. NARS cannot guarantee to process every task optimally — with insufficient knowledge, the best way to carry out a task is unknown; with insufficient resources, the system cannot exhaustively try all possibilities. Since NARS still needs to try its best in this situation, the solution used in NARS is to let the items and activities in the system compete for the limited resources. Again, the validity of the resource allocation policy is justified according to the past experience of the system (rather than its future experience), and the aim is to satisfy the goals of the system as much as possible. In the system, different data items (tasks, beliefs, and concepts) have different priority values attached, according to which the system’s resources are Reasoning in Practical Situations 291 distributed. These values are determined according to the past experience of the system, and are adjusted according to the change of situation. A special data structure is developed to implement a probabilistic priority queue with a limited storage. Using it, each access to an item takes roughly a constant time, and the accessibility of an item depends on its priority value. When no space is left, items with low priority will be removed. The memory of the system contains a collection of concepts, each of which is identified by a term in the formal language. Within the concept, all the tasks and beliefs that have the term as subject or predicate are collected together. The running of NARS consists of individual inference steps. In each step, a concept is selected probabilistically (according to its priority), then a task and a belief are selected (also probabilistically), and some inference rules take the task and the belief as premises to derive new tasks and beliefs, which are added into the memory. The system runs continuously, and interacts with its environment all the time, without stopping at the beginning and ending of each task. The processing of a task is interwoven with the processing of other existing tasks, so as to give the system a dynamic and context-sensitive nature. 4 Conclusion To work in practical situations, a reasoning system should adapt to its environment, and works with insufficient knowledge and resources. Since traditional theories do not make such an assumption, new theories are needed. The practice of NARS shows that it is possible to build an intelligent reasoning system according to the above requirement, and such a system provides unified solutions to many problems. References [Birnbaum, 1991] Birnbaum, L. (1991). Rigor mortis: a response to Nilsson’s “Logic and artificial intelligence”. Artificial Intelligence, 47:57-77. [Bocheński, 1970] Bocheński, I. (1970). A History of Formal Logic. Chelsea Publishing Company, New York. Translated and edited by I. Thomas. [Dreyfus, 1992] Dreyfus, H. (1992). What Computers Still Can’t Do. MIT Press, Cambridge, Massachusetts. 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