DS-2020-07: On the Path to the Truth: Logical & Computational Aspects of Learning

DS-2020-07: Sandoval, Ana Lucia Vargas (2020) On the Path to the Truth: Logical & Computational Aspects of Learning. Doctoral thesis, University of Amsterdam.

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In this dissertation we study various perspectives on learning and its relation to knowledge and belief within a formal approach. We mostly focus on inductive inference (or, inductive learning), the process of inferring general conclusions from incoming information. Our work is based in two areas that, independently, study dynamics of information, Dynamic Epistemic Logic (DEL) and Formal Learning Theory (FLT).

In Part I we investigate information dynamics from, on the one hand, incoming observations and, on the other, from incoming truthful announcements.

In Chapter 3 we present two dynamic modal logics to reason about learning from incoming observations in the spirit of FLT. Our first logic uses subset space semantics and the standard notion of a learning function to model learning in the limit. Our second logic extends the first framework in order to model learning in the limit from partial observations with a fully rational learner in the style of AGM belief revision theory. We present expressivity, soundness and completeness results for both logics.

In Chapter 4, we shift our focus to information gathering via public announcements and arbitrary public announcements in scenarios with multiple learners. We resolve problematic issues encountered in the work of Balbiani et al. (2008) concerning the unsound finitary rule proposed for the original Arbitrary Public Announcement Logic (APAL). This leads us also to solving the long standing open question of finding a recursive axiomatization for a strong version of APAL (and its variant Group Announcement Logic (GAL)).

In Part II, we focus completely on the learning model of finite identification in FLT. We obtain a more fine-grained theoretical analysis of the distinction between finite identification with positive information (pfi) and with complete (positive and negative) information (cfi). We show that the difference between pfi and cfi, if not as huge as in learning in the limit, is considerable not only in power but also in character.

In Chapter 5, we focus purely on the structural differences between families that are pfi and families that are cfi, ignoring computational aspects. We investigate whether any finitely identifiable family is contained in a maximal finitely identifiable one. We get a positive answer in the setting of positive data for families containing only finite languages. We provide a strong negative result in the setting of complete data showing that any finitely identifiable family can be extended to a larger one which is also finitely identifiable. We also study how many maximal extensions a positively identifiable family has. Our leading conjecture, which we partially resolve, is that any positively identifiable family of finite languages either has only finitely many maximal pfi extensions or uncountably many.

In Chapter 6 we study the computational properties of a family of languages. In particular, we analyze infinite anti-chains of finite languages. We provide negative answers to the questions: is every anti-chain of finite languages that is cfi also pfi? Is every maximal anti-chain of finite languages pfi (or cfi)? We also investigate a variation of finite identification that considers a learner who identifies a language as soon as it is objectively certain which language it is and explore the connection between pfi and cfi in this setting. We then study a variation of cfi which considers a learner that can ask queries to the teacher.

Overall, this dissertation, on the one hand brings closer together Dynamic Epistemic Logic and Formal Learning Theory, resulting in novel logics of information dynamics that formalize various learning theoretic notions. On the other hand, it uses tools in combinatorics and recursion theory to provide a detailed analysis of the differences between finite identification with positive data and finite identification with complete data.

Item Type: Thesis (Doctoral)
Report Nr: DS-2020-07
Series Name: ILLC Dissertation (DS) Series
Year: 2020
Subjects: Computation
Depositing User: Dr Marco Vervoort
Date Deposited: 14 Jun 2022 15:17
Last Modified: 14 Jun 2022 15:17
URI: https://eprints.illc.uva.nl/id/eprint/2176

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