Many AI problems, when formalized, reduce to evaluating the probability that a propositional expression is true. In this paper we show that this problem is computationally intractable even in surprisingly restricted cases and even if we settle for an approximation to this probability. We consider various methods used in approximate reasoning such as computing degree of belief and Bayesian belief networks, as well as reasoning techniques such as constraint satisfaction and knowledge compilation, that use approximation to avoid computational difficulties, and reduce them to model-counting problems over a propositional domain. We prove that counting satisfying assignments of propositional languages is intractable even for Horn and monotone formulae, and even when the size of clauses and number of occurrences of the variables are extremely limited. This should be contrasted with the case of deductive reasoning, where Horn theories and theories with binary clauses are distinguished by the e...

: Belief functions are mathematical objects defined to satisfy three axioms that look somewhat similar to the Kolmogorov axioms defining probability functions. We argue that there are (at least) two useful and quite different ways of understanding belief functions. The first is as a generalized probability function (which technically corresponds to the inner measure induced by a probability function). The second is as a way of representing evidence. Evidence, in turn, can be understood as a mapping from probability functions to probability functions. It makes sense to think of updating a belief if we think of it as a generalized probability. On the other hand, it makes sense to combine two beliefs (using, say, Dempster's rule of combination) only if we think of the belief functions as representing evidence. Many previous papers have pointed out problems with the belief function approach; the claim of this paper is that these problems can be explained as a consequence of confounding the...

The term Soft Computing (SC) represents the combination of emerging problem-solving technologies such as Fuzzy Logic (FL), Probabilistic Reasoning (PR), Neural Networks (NNs), and Genetic Algorithms (GAs). Each of these technologies provide us with complementary reasoning and searching methods to solve complex, real-world problems. After a brief description of each of these technologies, we will analyze some of their most useful combinations, such as the use of FL to control GAs and NNs parameters; the application of GAs to evolve NNs (topologies or weights) or to tune FL controllers; and the implementation of FL controllers as NNs tuned by backpropagation-type algorithms.

Non-Axiomatic Reasoning System is an adaptive system that works with insu cient knowledge and resources. At the beginning of the paper, three binary term logics are de ned. The rst 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 de ned. 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.

When simultaneously reasoning with evidences about several different events it is necessary to separate the evidence according to event. These events should then be handled independently. However, when propositions of evidences are weakly specified in the sense that it may not be certain to which event they are referring, this may not be directly possible. In this paper a criterion for partitioning evidences into subsets representing events is established. This criterion, derived from the conflict within each subset, involves minimising a criterion function for the overall conflict of the partition. An algorithm based on characteristics of the criterion function and an iterative optimisation among partitionings of evidences is proposed.

In this paper we present an application where we put together two methods for clustering and classification into a force aggregation method. Both methods are based on conflicts between elements. These methods work with different type of elements (intelligence reports, vehicles, military units) on different hierarchical levels using specific conflict assessment methods on each level. We use Dempster-Shafer theory for conflict calculation between elements, Dempster-Shafer clustering for clustering these elements, and templates for classification. The result of these processes is a complete force aggregation on all levels handled.

For the case of evidence ordered in a complete directed acyclic graph this paper presents a new algorithm with lower computational complexity for Dempster's rule than that of step-by-step application of Dempster's rule. In this problem, every original pair of evidences, has a corresponding evidence against the simultaneous belief in both propositions. In this case, it is uncertain whether the propositions of any two evidences are in logical conflict. The original evidences are associated with the vertices and the additional evidences are associated with the edges. The original evidences are ordered, i.e., for every pair of evidences it is determinable which of the two evidences is the earlier one. We are interested in finding the most probable completely specified path through the graph, where transitions are possible only from lower- to higher-ranked vertices. The path is here a representation for a sequence of states, for instance a sequence of snapshots of a physical object's track. A completely specified path means that the path includes no other vertices than those stated in the path representation, as opposed to an incompletely specified path that may also include other vertices than those stated. In a hierarchical network of all subsets of the frame, i.e., of all incompletely specified paths, the original and additional evidences support subsets that are not disjoint, thus it is not possible to prune the network to a tree. Instead of propagating belief, the new algorithm reasons about the logical conditions of a completely specified path through the graph. The new algorithm is O(|Θ| log |Θ|), compared to O(|Θ|^log |Θ|) of the classic brute force algorithm. After a detailed presentation of the reasoning behind the new algorithm we conclude that it is feasible to reason without approximation about completely specified paths through a complete directed acyclic graph.

In this article we investigate a problem within Dempster-Shafer theory where 2^q - 1 pieces of evidence are clustered into q clusters by minimizing a metaconflict function, or equivalently, by minimizing the sum of weight of conflict over all dusters. Previously one of us developed a method based on a Hopfield and Tank model. However, for very large problems we need a method with lower computational complexity. We demonstrate that the weight of conflict of evidence can, as an approximation, be linearized and mapped to an antiferromagnetic Potts spin model. This facilitates efficient numerical solution, even for large problem sizes. Optimal or nearly optimal solutions are found for Dempster-Shafer clustering benchmark tests with a time complexity of approximately O(N^2 log^2 N). Furthermore, an isomorphism between the antiferromagnetic Potts spin model and a graph optimization problem is shown. The graph model has dynamic variables living on the links, which have a priori probabilities that are directly related to the pairwise conflict between pieces of evidence. Hence, the relations between three different models are shown.

By analyzing the relationships among chance, weight of evidence and degree ofbelief, it is shown that the assertion \chances 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, some fundamental postulates of the theory must be rejected. A new approach for uncertainty management is introduced, which shares many intuitive ideas with D-S theory, while avoiding this problem. 1

The notion of interval sets is introduced as a new kind of sets, represented by a pair of sets, namely, the lower and upper bounds. The interval-set algebra may be regarded as a counterpart of the interval-number algebra. It provides a useful tool to represent qualitative information. Operations on interval sets are also defined, based on the corresponding set-theoretic operations on their members. In addition, basic properties of interval-set algebra are examined, and the relationships between interval sets, rough sets and fuzzy sets are analyzed. 1