A dynamic logic for learning theory (original) (raw)
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Bridging learning theory and dynamic epistemic logic
Synthese, 2009
This paper discusses the possibility of modelling inductive inference in dynamic epistemic logic (see e.g. ). The general purpose is to propose a semantic basis for designing a modal logic for learning in the limit. First, we analyze a variety of epistemological notions involved in identification in the limit and match it with traditional epistemic and doxastic logic approaches. Then, we provide a comparison of learning by erasing and iterated epistemic update (Baltag and Moss 2004) as analyzed in dynamic epistemic logic. We show that finite identification can be modelled in dynamic epistemic logic, and that the elimination process of learning by erasing can be seen as iterated belief-revision modelled in dynamic doxastic logic. Finally, we propose viewing hypothesis spaces as temporal frames and discuss possible advantages of that perspective.
Logic and Topology for Knowledge, Knowability, and Belief
The Review of Symbolic Logic, 2019
In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge [30]. Building on Stalnaker's core insights, and using frameworks developed in [11] and [4], we employ topological tools to refine and, we argue, improve on this analysis. The structure of topological subset spaces allows for a natural distinction between what is known and (roughly speaking) what is knowable; we argue that the foundational axioms of Stalnaker's system rely intuitively on both of these notions. More precisely, we argue that the plausibility of the principles Stalnaker proposes relating knowledge and belief relies on a subtle equivocation between an " evidence-in-hand " conception of knowledge and a weaker " evidence-out-there " notion of what could come to be known. Our analysis leads to a trimodal logic of knowledge, knowability, and belief interpreted in topological subset spaces in which belief is definable in terms of knowledge and knowability. We provide a sound and complete axiomatization for this logic as well as its uni-modal belief fragment. We then consider weaker logics that preserve suitable translations of Stalnaker's postulates, yet do not allow for any reduction of belief. We propose novel topological semantics for these irreducible notions of belief, generalizing our previous semantics, and provide sound and complete axiomatizations for the corresponding logics.
Outstanding Contributions to Logic, 2014
Dynamic epistemic logic, broadly conceived, is the study of rational social interaction in context, the study, that is, of how agents update their knowledge and change their beliefs on the basis of pieces of information they exchange in various ways. The information that gets exchanged can be about what is the case in the world, about what changes in the world, and about what agents know or believe about the world and about what others know or believe. This chapter gives an overview of dynamic epistemic logics, and traces some connections with propositional dynamic logic, with planning and with probabilistic updating.
Learning, from a Logical Point of View
Research Notes in Neural Computing, 1993
Learning is a pervasive topic in Artificial Intelligence (AI). It was already a well expressed concern in the very first works in the field and has since continued to be present in AI activity, sometimes as a subsidiary part of an area of research in AI, other times as a subfield on its own. Although one can distinguish several schools of thought, most of these would agree that (artificial) learning is at least "any process by which a system improves its performance". This conception allows for many approaches; differences stemming from methodological, technical and teleological commitments. In this paper, I will concentrate on the approach taken by logically inspired works. In this approach, the "improvement" that would account for learning is interpreted, either by extending or refining the class of sentences deducible in a formal theory, or by achieving better efficiency in the deduction process. I will present some characteristic devices used by logicians to obtain these improvements and will illustrate how purely formal devices are complemented by metalogical constructs to represent learning-theoretical tenants.
On C-Learnability in Description Logics
Lecture Notes in Computer Science, 2012
We prove that any concept in any description logic that extends ALC with some features amongst I (inverse), Q k (quantified number restrictions with numbers bounded by a constant k), Self (local reflexivity of a role) can be learnt if the training information system is good enough. That is, there exists a learning algorithm such that, for every concept C of those logics, there exists a training information system consistent with C such that applying the learning algorithm to the system results in a concept equivalent to C.
Topological Subset Space Models for Belief
ArXiv, 2016
In recent work, Robert Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge [28]. Building on Stalnaker’s core insights, we employ topological tools to refine and, we argue, improve on this analysis. The structure of topological subset spaces allows for a natural distinction between what is known and (roughly speaking) what is knowable; we argue that the foundational axioms of Stalnaker’s system rely intuitively on both of these notions. More precisely, we argue that the plausibility of the principles Stalnaker proposes relating knowledge and belief relies on a subtle equivocation between an “evidence-in-hand” conception of knowledge and a weaker “evidence-out-there” notion of what could come to be known. Our analysis leads to a trimodal logic of knowledge, knowability, and belief interpreted in topological subset spaces in which belief is definable in terms of knowledge and knowability. We provide a sound and complete axiomatization for...
Finite identification from the viewpoint of epistemic update
Information and Computation, 2011
Finite identifiability of sets from positive data Dynamic epistemic logic Epistemic temporal logic Epistemic update Formal learning theory constitutes an attempt to describe and explain the phenomenon of learning, in particular of language acquisition. The considerations in this domain are also applicable in philosophy of science, where it can be interpreted as a description of the process of scientific inquiry. The theory focuses on various properties of the process of hypothesis change over time. Treating conjectures as informational states, we link the process of conjecture-change to epistemic update. We reconstruct and analyze the temporal aspect of learning in the context of dynamic and temporal logics of epistemic change. We first introduce the basic formal notions of learning theory and basic epistemic logic. We provide a translation of the components of learning scenarios into the domain of epistemic logic. Then, we propose a characterization of finite identifiability in an epistemic temporal language. In the end we discuss consequences and possible extensions of our work.
The Topological Theory of Belief
2015
Stalnaker introduced a combined epistemic-doxastic logic that can formally express a strong concept of belief, a concept which captures the 'epistemic possibility of knowledge'. In this paper we first provide the most general extensional semantics for this concept of 'strong belief', which validates the principles of Stalnaker's epistemic-doxastic logic. We show that this general extensional semantics is a topological semantics, based on so-called extremally disconnected topological spaces. It extends the standard topological interpretation of knowledge (as the interior operator) with a new topological semantics for belief. Formally, our belief modality is interpreted as the 'closure of the interior'. We further prove that in this semantics the logic KD45 is sound and complete with respect to the class of extremally disconnected spaces and we compare our approach to a different topological setting in which belief is interpreted in terms of the derived set operator. In the second part of the paper we study (static) belief revision as well as belief dynamics by providing a topological semantics for conditional belief and belief update modalities, respectively. Our investigation of dynamic belief change, is based on hereditarily extremally disconnected spaces. The logic of belief KD45 is sound and complete with respect to the class of hereditarily extremally disconnected spaces (under our proposed semantics), while the logic of knowledge is required to be S4.3. Finally, we provide a complete axiomatization of the logic of conditional belief and knowledge, as well as a complete axiomatization of the corresponding dynamic logic.