Among the many approaches for reasoning about degrees of belief in the presence of noisy sensing and acting, the logical account proposed by Bacchus, Halpern, and Levesque is perhaps the most expressive. While their formalism is quite general, it is restricted to fluents whose values are drawn from discrete finite domains, as opposed to the continuous domains seen in many robotic applications. In this work, we show how this limitation in that approach can be lifted. By dealing seamlessly with both discrete distributions and continuous densities within a rich theory of action, we provide a very general logical specification of how belief should change after acting and sensing in complex noisy domains.
In the situation calculus, a monotonic solution to the frame problem was provided by Reiter @cite_0 . The situation calculus was extended to reason about knowledge whilst incorporating this solution in @cite_45 , and to reason about noisy effectors and sensors by BHL. Throughout, we operated under the setting of knowledge expansion, that is, observations are assumed to resolve the agent's uncertainty and never contradict what is believed. The topic of lifts this assumption @cite_14 , but it is not considered here. See @cite_43 for an account of belief revision in the situation calculus. Other action logics have enjoyed similar extensions. For example, @cite_69 proposes an extension to dynamic logic for reasoning about degrees of belief and noisy actions, and @cite_29 provides a computational framework for probabilistic reasoning using the stable model semantics, but they are propositional. In @cite_82 @cite_65 , the fluent calculus was extended for probabilistic beliefs and noisy actions. None of these admit continuous probability distributions.
@cite_69 @cite_14 @cite_29 @cite_65 @cite_0 @cite_43 @cite_45 @cite_82
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These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions. This paper proposes a method for handling the frame problem for knowledge-producing actions. An example of a knowledge-producing action is a sensing operation performed by a robot to determine whether or not there is an object of a particular shape within its grasp. The work is an extension of Reiter's approach to the frame problem for ordinary actions and Moore's work on knowledge and action. The properties of our specification are that knowledge-producing actions do not affect fluents other than the knowledge fluent, and actions that are not knowledge-producing only affect the knowledge fluent as appropriate. In addition, memory emerges as a side-effect: if something is known in a certain situation, it remains known at successor situations, unless something relevant has changed. Also, it will be shown that a form of regression examined by Reiter for reducing reasoning about future situations to reasoning about the initial situation now also applies to knowledge-producing actions. We consider a modified version of the situation calculus built using a two-variable fragment of the first-order logic extended with counting quantifiers. We mention several additional groups of axioms that can be introduced to capture taxonomic reasoning. We show that the regression operator in this framework can be defined similarly to regression in the Reiter's version of the situation calculus. Using this new regression operator, we show that the projection and executability problems are decidable in the modified version even if an initial knowledge base is incomplete and open. For an incomplete knowledge base and for context-dependent actions, we consider a type of progression that is sound with respect to the classical progression. We show that the new knowledge base resulting after our progression is definable in our modified situation calculus if one allows actions with local effects only. We mention possible applications to formalization of Semantic Web services. John McCarthy's situation calculus has left an enduring mark on artificial intelligence research. This simple yet elegant formalism for modelling and reasoning about dynamic systems is still in common use more than forty years since it was first proposed. The ability to reason about action and change has long been considered a necessary component for any intelligent system. The situation calculus and its numerous extensions as well as the many competing proposals that it has inspired deal with this problem to some extent. In this paper, we offer a new approach to belief change associated with performing actions that addresses some of the shortcomings of these approaches. In particular, our approach is based on a well-developed theory of action in the situation calculus extended to deal with belief. Moreover, by augmenting this approach with a notion of plausibility over situations, our account handles nested belief, belief introspection, mistaken belief, and handles belief revision and belief update together with iterated belief change.