Topological Modelling of Knowledge Change Lecture 2 Nina - PowerPoint PPT Presentation

Topological Modelling of Knowledge Change Lecture 2 Nina Gierasimczuk Department of Applied Mathematics and Computer Science Technical University of Denmark PhDs in Logic VIII Darmstadt, May 9th-10th, 2016

Outline Introduction Learning and Solving in the Limit Topological Toolbox Characterization of Learnability and Solvability Constructive Order-driven Learning Towards Epistemic Logic of Learnability Topological semantics Doxastic interpretations Intermediate (but Interesting ) Conclusions

Outline Introduction Learning and Solving in the Limit Topological Toolbox Characterization of Learnability and Solvability Constructive Order-driven Learning Towards Epistemic Logic of Learnability Topological semantics Doxastic interpretations Intermediate (but Interesting ) Conclusions

Marrying Topology and Epistemology (via Learnability) 1. Learnability and solvability in terms of topological separation properties. 2. Constructive, order-based learning by updating. 3. Consequences for logic of belief (doxastic logic).

Epistemic Spaces and Observables Definition An epistemic space is a pair S = ( S , O ) consisting of a state space (a set of possible worlds) S and a countable set of observable properties O ⊆ P ( S ). V U s u w t

Learning: Streams of Observables Definition Let S = ( S , O ) be an epistemic space. ◮ A data stream is an infinite sequence � O = ( O 0 , O 1 , . . . ) of data from O . ◮ A data sequence is a finite sequence σ = ( σ 0 , . . . , σ n ). Definition Take S = ( S , O ) and s ∈ S . A data stream � O is: ◮ sound with respect to s iff every element listed in � O is true in s . ◮ complete with respect to s iff every observable true in s is listed in � O . We assume that data streams are sound and complete.

Learning: Learners and Conjectures Definition Let S = ( S , O ) be an epistemic space and let σ 0 , . . . , σ n ∈ O . A learner is a function L that on the input of S and data sequence ( σ 0 , . . . , σ n ) outputs some set of worlds L ( S , ( σ 0 , . . . , σ n )) ⊆ S , called a conjecture . Definition S = ( S , O ) is learnable by L if for every state s ∈ S we have that for every sound and complete data stream � O for s , there is n ∈ N s.t.: L ( S , ( O 0 , . . . , O k )) = { s } for all k ≥ n . An epistemic space S is learnable if it is learnable by a learner L .

Example of a Learnable Space Let S = ( S , O ) such that S = { s n | n ∈ N } , O = { p i | i ∈ N } , and for any k ∈ N , p k = { s i | 0 ≤ i ≤ k } . S is learnable. p 0 p 1 p 2 p 3 p 4 . . . s 0 s 1 s 2 s 3 s 4

Example of a Learnable Space Let S = ( S , O ) such that S = { s n | n ∈ N } , O = { p i | i ∈ N } , and for any k ∈ N , p k = { s i | 0 ≤ i ≤ k } . S is learnable. p 0 p 1 p 2 p 3 p 4 . . . s 0 s 1 s 2 s 3 s 4

Example of a Learnable Space Let S = ( S , O ) such that S = { s n | n ∈ N } , O = { p i | i ∈ N } , and for any k ∈ N , p k = { s i | 0 ≤ i ≤ k } . S is learnable. p 0 p 1 p 2 p 3 p 4 . . . s 0 s 1 s 2 s 3 s 4

Example of a Learnable Space Let S = ( S , O ) such that S = { s n | n ∈ N } , O = { p i | i ∈ N } , and for any k ∈ N , p k = { s i | 0 ≤ i ≤ k } . S is learnable. p 0 p 1 p 2 p 3 p 4 . . . s 0 s 1 s 2 s 3 s 4

Example of a Non-Learnable Space Consider S = ( S , O ), where S := { s n | n ∈ N } ∪ { s ∞ } , and O = { p i | i ∈ N } , and for any k ∈ N , p k := { s k , s k +1 , . . . } ∪ { s ∞ } . S is not learnable. p 0 p 1 p 2 p 3 . . . s 0 s 1 s 2 s 3 s ∞

Example of a Non-Learnable Space Consider S = ( S , O ), where S := { s n | n ∈ N } ∪ { s ∞ } , and O = { p i | i ∈ N } , and for any k ∈ N , p k := { s k , s k +1 , . . . } ∪ { s ∞ } . S is not learnable. p 0 p 1 p 2 p 3 . . . s 0 s 1 s 2 s 3 s ∞

Example of a Non-Learnable Space Consider S = ( S , O ), where S := { s n | n ∈ N } ∪ { s ∞ } , and O = { p i | i ∈ N } , and for any k ∈ N , p k := { s k , s k +1 , . . . } ∪ { s ∞ } . S is not learnable. p 0 p 1 p 2 p 3 . . . s 0 s 1 s 2 s 3 s ∞

Example of a Non-Learnable Space Consider S = ( S , O ), where S := { s n | n ∈ N } ∪ { s ∞ } , and O = { p i | i ∈ N } , and for any k ∈ N , p k := { s k , s k +1 , . . . } ∪ { s ∞ } . S is not learnable. p 0 p 1 p 2 p 3 . . . s 0 s 1 s 2 s 3 s ∞

Example of a Non-Learnable Space Consider S = ( S , O ), where S := { s n | n ∈ N } ∪ { s ∞ } , and O = { p i | i ∈ N } , and for any k ∈ N , p k := { s k , s k +1 , . . . } ∪ { s ∞ } . S is not learnable. p 0 p 1 p 2 p 3 . . . s 0 s 1 s 2 s 3 s ∞

Questions, Answers, and Problems Definition A question Q is a partition of S , whose cells A i are called answers to Q . Given s ∈ A ⊆ S , A ∈ Q is called the answer to Q at s , denoted A s . Definition Q ′ is a refinement of Q if all answers of Q is a disjoint union of answers of Q ′ . Definition A problem P is a pair ( S , Q ) consisting of S = ( S , O ) and Q over S . P ′ = ( S , Q ′ ) is a refinement of P = ( S , Q ) if Q ′ is a refinement of Q .

Illustration U V s t P Q u v

Illustration U V s t P Q u v

Solving in the Limit Definition A learning method L solves a problem P = ( S , Q ) in the limit iff for every state s ∈ S and every data stream � O for s , there exists some k ∈ N such that: L ( S , � O [ n ]) ⊆ A s for all n ≥ k . A problem is solvable in the limit if there is a learner that solves it in the limit.

General Topology Definition A topology τ over a set S is a collection of subsets of S (open sets) s.t.: 1. ∅ ∈ τ , 2. S ∈ τ , 3. for any X ⊆ τ , � X ∈ τ , and 4. for any finite X ⊆ τ we have � X ∈ τ. Definition Take a set X ⊆ S . 1. The interior of X : Int ( X ) = � { U ∈ τ | U ⊆ X } . 2. A subset Y ⊆ S is closed if an only if its complement, Y c is open. 3. The closure of X : X = ( Int ( X c )) c = � { Y | X ⊆ Y and Y is closed } .

Separability of States Definition A topology τ over S is T 1 (strongly separated) just if every state s ∈ S is separable from every other state in S , i.e., for all s , t ∈ S , if s � = t then there is an U ∈ τ such that s ∈ U and t �∈ U . Definition A topology τ over S is T 0 (weakly separated) just if all distinct states are separable one way or another, i.e., for all s , t ∈ S if s � = t then there is and U ∈ τ such that either s ∈ U , t �∈ U or s �∈ U , t ∈ U .

Illustration V U s u w t Figure: t and u are not separable U V s t Figure: weakly separated space, T 0 U V s t Figure: strongly separated space, T 1

Locally Closed and Constructible Sets Definition A set A is locally closed if A = U ∩ C , where U is open and C is closed. A set is constructible if it is a finite disjoint union of locally closet sets. Definition A topology τ is T d iff for every s ∈ S there is a U ∈ τ such that U \ { s } ∈ τ , i.e., for every s ∈ S there is a U ∈ τ such that { s } = U ∩ { s } . T d is a separation property between T 0 and T 1.

ω -Constructible Sets Definition An ω -constructible set is a countable union of locally closed sets. Proposition Every ω -constructible set is a disjoint countable union of locally closed sets.

Outline Introduction Learning and Solving in the Limit Topological Toolbox Characterization of Learnability and Solvability Constructive Order-driven Learning Towards Epistemic Logic of Learnability Topological semantics Doxastic interpretations Intermediate (but Interesting ) Conclusions

The Topology Associated with an Epistemic Space Definition The topology τ S associated with an epistemic space S = ( S , O ) is a collection of subsets of S of the following properties: 1. for any O ∈ O it is the case that O ∈ τ S 2. ∅ ∈ τ S , 3. S ∈ τ S , 4. for any U ⊆ τ S , � U ∈ τ S , and 5. for any x , y ∈ τ S we have x ∩ y ∈ τ S . V U s u w t

The Topology Associated with an Epistemic Space Definition The topology τ S associated with an epistemic space S = ( S , O ) is a collection of subsets of S of the following properties: 1. for any O ∈ O it is the case that O ∈ τ S 2. ∅ ∈ τ S , 3. S ∈ τ S , 4. for any U ⊆ τ S , � U ∈ τ S , and 5. for any x , y ∈ τ S we have x ∩ y ∈ τ S . V U s u w t

Characterization of Solvability in the Limit Theorem A problem P = ( S , Q ) is solvable in the limit iff Q has a countable locally closed refinement. Corollary An epistemic space S = ( S , O ) is learnable in the limit iff it is countable and satisfies the T d separation axiom.

Outline Introduction Learning and Solving in the Limit Topological Toolbox Characterization of Learnability and Solvability Constructive Order-driven Learning Towards Epistemic Logic of Learnability Topological semantics Doxastic interpretations Intermediate (but Interesting ) Conclusions

Order-driven learning: Motivation ◮ Belief Revision: minimal states give beliefs. ◮ Computational Learning Theory: co-learning, learning by erasing. ◮ Philosophy of Science: Ockham’s razor.

Conditioning Definition Conditioning wrt a prior ≤ on S , is defined in the following way: � n � � L ≤ ( O 1 , . . . , O n ) := Min ≤ O i i =1 whenever � i O i has any minimal elements; and otherwise: n � L ≤ ( O 1 , . . . , O n ) := O i . i =1 Definition Conditioning is said to be standard if the prior ≤ is well-founded .