, Last updated: December 30, 2009

Publications of Pei Wang

My later publications and demonstrations have been moved to

Publications Finished at Indiana University

Aspects of NARS

Reasoning under Uncertainty

Psychological Issues


Non-Axiomatic Reasoning System --- Exploring the essence of intelligence

Dissertation for the Doctor of Philosophy degree in the Computer Science Department and the Cognitive Science Program of Indiana University, 1995.

The dissertation has 181 pages. It is available as *.ps file(5M), * file(460K), and *.ps.gz file (430K).


Every artificial-intelligence research project needs a working definition of ``intelligence'', on which the deepest goals and assumptions of the research are based. In the project described in the following chapters, ``intelligence'' is defined as the capacity to adapt under insufficient knowledge and resources. Concretely, an intelligent system should be finite and open, and should work in real time.

If these criteria are used in the design of a reasoning system, the result is NARS, a non-axiomatic reasoning system.

NARS uses a term-oriented formal language, characterized by the use of subject--predicate sentences. The language has an experience-grounded semantics, according to which the truth value of a judgment is determined by previous experience, and the meaning of a term is determined by its relations with other terms. Several different types of uncertainty, such as randomness, fuzziness, and ignorance, can be represented in the language in a single way.

The inference rules of NARS are based on three inheritance relations between terms. With different combinations of premises, revision, deduction, induction, abduction, exemplification, comparison, and analogy can all be carried out in a uniform format, the major difference between these types of inference being that different functions are used to calculate the truth value of the conclusion from the truth values of the premises.

Since it has insufficient space--time resources, the system needs to distribute them among its tasks very carefully, and to dynamically adjust the distribution as the situation changes. This leads to a ``controlled concurrency'' control mechanism, and a ``bag-based'' memory organization.

A recent implementation of the NARS model, with examples, is discussed. The system has many interesting properties that are shared by human cognition, but are absent from conventional computational models of reasoning.

This research sheds light on several notions in artificial intelligence and cognitive science, including symbol-grounding, induction, categorization, logic, and computation. These are discussed to show the implications of the new theory of intelligence.

Finally, the major results of the research are summarized, a preliminary evaluation of the working definition of intelligence is given, and the limitations and future extensions of the research are discussed.

A New Approach for Induction: From a Non-Axiomatic Logical Point of View

Technical Report No. 102 of CRCC. A .ps file of the paper is here.

Abstract: Non-Axiomatic Reasoning System (NARS) is designed to be a general-purpose intelligent reasoning system, which is adaptive and works under insufficient knowledge and resources. This paper focuses on the components of NARS that contribute to the system's induction capacity, and shows how the traditional problems in induction are addressed by the system. The NARS approach of induction uses an term-oriented formal language with an experience-grounded semantics that consistently interprets various types of uncertainty. An induction rule generates conclusions from common instance of terms, and a revision rule combines evidence from different sources. In NARS, induction and other types of inference, such as deduction and abduction, are based on the same semantic foundation, and they cooperate in inference activities of the system. The system's control mechanism makes knowledge-driven, context-dependent inference possible.

Problem-Solving under Insufficient Resources

Technical Report No. 101 of CRCC. A .ps file of the paper is here. A revised version appears in the Working Notes of the AAAI Fall Symposium on Flexible Computation, Cambridge, Massachusetts, November 1996.

Abstract: This paper discusses the problem of resources-limited information processing. After a review of relevant approaches in several fields, a new approach, controlled concurrency, is described and analyzed. This method is proposed for adaptive systems working under insufficient knowledge and resources. According to this method, a problem-solving activity consists of a sequence of steps which behaves like an anytime algorithm --- it is interruptible, and the quality of the result is improved incrementally. The system carries out many such activities in parallel, distributes its resources among the them in a time-sharing manner, and dynamically adjusts the distribution according to the feedback of each step. The step sequence for a given problem is formed at run time, according to the system's knowledge structure, which is also dynamically formed and adjusted. Finally, this approach is compared with other approaches, and several of its properties and implications are discussed.

Grounded on Experience: Semantics for intelligence

Technical Report No. 96 of CRCC. A .ps file of the paper is here.

Abstract: Model-theoretic semantics is inappropriate for adaptive systems working with insufficient knowledge and resources. An experience-grounded semantics is introduced in this paper, using NARS, an intelligent reasoning system, as a concrete example. In NARS, the truth value of a sentence indicates the amount of available evidence, and the meaning of a term indicates its experienced relationship with other terms. Accordingly, both truth value and meaning are dynamic and subjective. This approach provides new ideas to the solution of some important problems in artificial intelligence.

On the Working Definition of Intelligence

Technical Report No. 94 of CRCC. A .ps file of the paper is here.

Abstract: This paper is about the philosophical and methodological foundation of artificial intelligence (AI). After discussing what is a good ``working definition'', ``intelligence'' is defined as ``the ability for an information processing system to adapt to its environment with insufficient knowledge and resources''. Applying the definition to a reasoning system, we get the major components of Non-Axiomatic Reasoning System (NARS), which is a symbolic logic implemented in a computer system, and has many interesting properties that closely related to intelligence. The definition also clarifies the difference and relationship between AI and other disciplines, such as computer science. Finally, the definition is compared with other popular definitions of intelligence, and its advantages are argued.

Confidence as Higher Order Uncertainty

Technical Report No. 93 of CRCC. A .ps file of the paper is here.

Abstract: With insufficient knowledge, the conclusions made by a reasoning system are usually uncertain. If the system is open to new knowledge, it also suffers from a higher order uncertainty, because the first order uncertainty evaluations are uncertain themselves --- they can be changed by future evidence. Several approaches have been proposed for handling higher order uncertainty, including the Bayesian approach, higher-order probability, and so on. Though each of them has its advantages, none of them is satisfactory, for various reasons. A new measurement, confidence, is defined to indicate higher order uncertainty, which is understood as relative stability of first order uncertainty evaluation, and is processed accordingly.

Heuristics and Normative Models

Technical Report No. 92 of CRCC. A .ps file of the paper is here. A revised version appears in International Journal of Approximate Reasoning, 14(4), 221--235, 1996.

Abstract: Evidence has be collected that probability theory is not a proper descriptive model of intuitive human judgment. Some heuristics have been proposed as such a descriptive model. This paper argues that probability theory has limitations even as a normative model. A new normative model of judgment under uncertainty is introduced, by which some heuristics can be justified from a set of basic assumptions about intelligence.

Reference Classes and Multiple Inheritances

Technical Report No. 91 of CRCC. A .ps file of the paper is here. A revised version appears in International Journal of Uncertainty, Fuzziness and Knowledge-based Systems, 3(1), 79-91, 1995.

Abstract: The reference class problem in probability theory and the multiple inheritances (extensions) problem in non-monotonic logics can be referred to as special cases of conflicting beliefs. The current solution accepted in the two domains is the specificity priority principle. By analyzing an example, several factors (ignored by the principle) are found to be relevant to the priority of a reference class. A new approach, Non-Axiomatic Reasoning System (NARS), is discussed, where these factors are all taken into account. It is argued that the solution provided by NARS is better than the solutions provided by probability theory and non-monotonic logics.

A Unified Treatment of Uncertainties

Technical Report No. 87 of CRCC. A .ps file of the paper is here. A revised version appears in Proceedings of the Fourth International Conference for Young Computer Scientists, 462-467, Beijing, July 1995.

Abstract: This paper begins by defining an environment of reasoning under uncertainty, then it is shown why some existing approaches cannot be used in such a situation. A new approach, Non-Axiomatic Reasoning System, is introduced, which is built under the assumption that the system's knowledge and resources are usually insufficient to handle the tasks imposed by the environment. The system can consistently represent several types of uncertainty, and can carry out multiple operations on these uncertainties. Finally, the new approach is compared with the previous approaches in terms of uncertainty representation and interpretation.

The Interpretation of Fuzziness

Technical Report No. 86 of CRCC. A .ps file of the paper is here. A revised version appears in IEEE Transactions on Systems, Man, and Cybernetics, 26(2), 321--326, 1996.

Abstract: By analyzing related issues in psychology and linguistics, two basic types of fuzziness can be attributed to similarity and relativity, respectively. In both cases, it is possible to interpret grade of membership as the proportion of positive evidence, so as to treat fuzziness and randomness uniformly.

A Defect in Dempster-Shafer Theory

Technical Report No. 85 of CRCC. A revised version appears in Proceedings of the Tenth Conference of Uncertainty in Artificial Intelligence . Eds. R. Lopez de Mantaras and D. Poole. Seattle, WA, 1994. The .ps file of a new version the paper is here. The posters presented in UAI-94 are also available.

Abstract: By analyzing the relationships among chance, weight of evidence and degree of belief, we show that the assertion ``probability functions are special cases of belief functions'' and the assertion ``Dempster's rule can be used to combine belief functions based on distinct bodies of evidence'' together lead to an inconsistency in Dempster-Shafer theory. To solve this problem, we must reject some fundamental postulates of the theory. We introduce a new approach for uncertainty management that shares many intuitive ideas with D-S theory, while avoiding this problem.

From Inheritance Relation to Non-Axiomatic Logic

Technical Report No. 84 of CRCC. A .ps file of the paper is here. A revised version appears in the International Journal of Approximate Reasoning, 11(4), 281-319, 1994.

Abstract: Non-Axiomatic Reasoning System is an adaptive system that works with insufficient knowledge and resources. At the beginning of the paper, three binary term logics are defined. The first is based only on an inheritance relation. The second and the third suggest a novel way to process extension and intension, and they also have interesting relations with Aristotle's syllogistic logic. Based on the three simple systems, a Non-Axiomatic Logic is defined. It has a term-oriented language and an experience-grounded semantics. It can uniformly represents and processes randomness, fuzziness, and ignorance. It can also uniformly carries out deduction, abduction, induction, and revision.

Comparing Categorization Models: A psychological experiment

Technical Report No. 79 of CRCC. A .ps file of the paper is here.

Abstract: In this paper, a psychological experiment is proposed to compare different theories on the internal structure of categories: classical theory, prototype theory, exemplar theory, and NARS.

Non-Axiomatic Reasoning System (Version 2.2)

Technical Report No. 75 of CRCC. A .ps file of the paper is here.

Abstract: Non-Axiomatic Reasoning System (NARS) is an intelligent reasoning system, where intelligence means working and adapting with insufficient knowledge and resources. NARS uses a new form of term logic, or an extended syllogism, in which several types of uncertainties can be represented and processed, and in which deduction, induction, abduction, and revision are carried out in a unified format. The system works in an asynchronously parallel way. The memory of the system is dynamically organized, and can also be interpreted as a network. After present the major components of the system, its implementation is briefly described. An example is used to show how the system works. The limitations of the system are also discussed.

Belief Revision in Probability Theory

Technical Report No. 74 of CRCC. A revised version appears in Proceedings of the Ninth Conference of Uncertainty in Artificial Intelligence , 519-526. Eds. David Heckerman and Abe Mamdani. (San Mateo, CA: Morgan Kaufmann), 1993. A .ps file of the paper is here.

Abstract: In a probability-based reasoning system, Bayes' theorem and its variations are often used to revise the system's beliefs. However, if the explicit conditions and the implicit conditions of probability assignments are properly distinguished, it follows that Bayes' theorem is not a generally applicable revision rule. Upon properly distinguishing belief revision from belief updating, we see that Jeffrey's rule and its variations are not revision rules, either. Without these distinctions, the limitation of the Bayesian approach is often ignored or underestimated. Revision, in its general form, cannot be done in the Bayesian approach, because a probability distribution function alone does not contain the information needed by the operation.