Nina Gierasimczuk, Believing in Formal Learning Theory

This talk brings together two types of learning. The first kind, getting to know about facts, is formalized and analyzed in the domain of belief revision and the diverse frameworks of epistemic and doxastic logics. The main aim here is to formalize the elementary dynamics of knowledge and epistemic attitudes towards incoming information. The second kind, learning as a process, is studied within the framework of formal learning theory. In this framework a general concept (language, grammar, theory) gets to be identified by an agent on the basis of some elementary data (sentences, results of experiments) over a long period of time. The learning agent is allowed to change his mind on the way, and the process is successful if it results in convergence to an appropriate hypothesis. In a sense this kind of learning is built on top of the first kind, it consists of an iteration of simple getting-to-know events.
In this talk we propose a way to use the framework of learning theory to evaluate belief-revision policies. On the inductive inference side, we are interested in the paradigm of language learning. As possible concepts that are inferred we take sets of atomic propositions. Therefore, receiving new data corresponds to getting to know about facts. On the side of belief revision we follow the lines of dynamic epistemic logic (see van Benthem, 2007). Hence, we interpret current beliefs of the agent (hypothesis) as the content of those possible worlds that he considers most plausible. The revision does not only result in the change of the current hypothesis, but can also induce modification of the agent’s plausibility order. We are mainly concerned with identifiability in the limit (Gold, 1967). In the first part we restrict ourselves to learning from sound and complete streams of positive data. We show that learning methods based on belief revision via conditioning (update) and lexicographic revision are universal, i.e., provided certain prior conditions, those methods are as powerful as identification in the limit. Those prior conditions, the agent’s prior dispositions for belief revision, play a crucial role here. We show that in some cases, these priors cannot be modeled using standard belief-revision models (as based on well-founded preorders), but only using generalized models (as simple preorders). Furthermore, we draw conclusions about the existence of tension between conservatism and learning power by showing that the very popular, most ‘conservative’ belief-revision methods fail to be universal. Then we turn to the case of learning from both positive and negative data. Here, along with information about facts the agent receives negative data about things that do not hold of the actual world. We again assume these streams to be truthful and we draw conclusions about iterated belief revision governed by such streams. This enriched framework allows us to consider the occurrence of erroneous information. Provided that errors occur finitely often and are always eventually corrected we show that the lexicographic revision method is still reliable, but more conservative methods fail.
Our results can be interpreted as showing that applying certain types of rules in certain contexts can be analyzed in terms of whether they can be relied upon in the ‘quest for the truth’ (the analysis of inductive inference in terms of reliability has been for the first time provided by Kelly, 1996). In our framework we can naturally treat the procedural aspect of iterated belief revision, address some intermediate stages of such iterations and relate them to the ultimate success of a belief-revision policy.
The results presented in this talk come from a joint work with Alexandru Baltag and Sonja Smets.

van Benthem, J. (2007). Dynamic logic for belief revision. Journal of Applied Non-Classical Logics, 2:129–155.
Gold, E. (1967). Language identification in the limit. Information and Control, 10:447–474.
Kelly, K. (1996). The Logic of Reliable Inquiry. Oxford University Press, Oxford.