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Logics for Quantified Partial Belief |
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Agents operating in the real world, whether natural or artificial, are constantly faced with information which is imperfect, in that it may be incomplete, approximate, unreliable, and/or uncertain. However, if they are to be successful, agents must be able to form beliefs based on this information, and use these beliefs to reason and act.
One of the most popular approaches to modelling knowledge and belief in AI is the use of epistemic logics. Traditional epistemic logics only consider categorical belief: agents either believe a proposition completely, or they don't believe it at all. However, beliefs originating from inperfect information are, in general, partial. We have extended the framework of epistemic logics to capture notions of partial belief, and to model weak forms of reasoning. These logics give us formal tools to build intelligent systems that operate under conditions of uncertainty and incompleteness. Technically, we use belief functions to quantify strength of belief on a numerical scale.
Our first step in this direction has been the development of a logic, called "Belief Function Logic" (BFL), in which partial belief is represented by formulae of the form "F:[a,b]", where F is a sentence in standard first-order logic, and a and b are numbers in [0,1]. We read "F:[a,b]" as "I believe with strength at least a that F is true; and with strength at least (1-b) that it is false." BFL has been given semantics in terms of belief functions in a possible-world framework and has been shown to behave well as a "hybrid" logic, in that it preserves the properties of both first-order logic and of belief functions. In fact, BFL is an instance of the Hybrid Belief Systems presented in the "HBS" leaflet.
BFL is not a full-fledged modal logic: for example, BFL cannot express combinations of belief formulae (eg, "I believe P or I believe Q") or nested belief (eg, "I believe that I don't believe P"). Our next step has been to represent partial belief by modal operators of the form "Ba", where a is a number in [0,1]. The formula "BaP" is then read "I believe at the degree a that P is true." We have defined a family of modal logics based on these operators. As for BFL, these logics have been given semantics in terms of belief functions over a set of possible worlds. We are also investigating the use of these logics to model some aspects of the dynamics of belief, including belief combination and update. These dynamics play a crucial role in applications where the agent is subject to a continuous flow of information, and/or the environment changes overtime.
Non-monotonic forms of reasoning occur when some previous conclusions --which typically was inferred by using default assumptions-- must be withdrawn in the light of new, contradicting information. A major source of difficulties in modeling non-monotonic reasoning is that there are in general multiple ways to revise a belief set in order to recover from a contradiction. An impressive amount of AI literature is devoted to the problem of how to define "reasonable" non-monotonic inference operators.
Thanks to their ability to reason about degrees of belief in a state of partial inconsistency, the logics above provide a framework to formalize preferences between alternative revisions. We have shown that this framework is able to capture most of the approaches to non-monotonic reasoning currently developed in AI. We have also used this framework to develop a new non-monotonic inference operator that solves many of the drawbacks of the existing approaches.
Logic is one of the main formal tools used to develop artificial intelligence products. It provides a formal basis to most of the current work in knowledge representation and reasoning. However, classical logics only deal with the idealized case where knowledge is completely accurate. Our work on logics for quantified partial belief aims at extending the applicability of logic-based techniques to formalize domains where knowledge is incomplete, uncertain, and may contain errors and default assumptions. Most real-world domains are of this type.
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