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). s t u w U V

Learning: Streams of Observables

Definition

Let S = (S, O) be an epistemic space.

◮ A data stream is an infinite sequence

O = (O0, O1, . . .) of data from O.

◮ A data sequence is a finite sequence σ = (σ0, . . . , σn).

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 σ0, . . . , σn ∈ O. A learner is a function L that on the input of S and data sequence (σ0, . . . , σn) outputs some set of worlds L(S, (σ0, . . . , σn)) ⊆ 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, (O0, . . . , Ok)) = {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 = {sn | n ∈ N}, O = {pi | i ∈ N}, and for any k ∈ N, pk = {si | 0 ≤ i ≤ k}. S is learnable. s0 s1 s2 s3 s4 p0 p1 p2 p3 p4 . . .

Example of a Learnable Space

Let S = (S, O) such that S = {sn | n ∈ N}, O = {pi | i ∈ N}, and for any k ∈ N, pk = {si | 0 ≤ i ≤ k}. S is learnable. s0 s1 s2 s3 s4 p0 p1 p2 p3 p4 . . .

Example of a Learnable Space

Let S = (S, O) such that S = {sn | n ∈ N}, O = {pi | i ∈ N}, and for any k ∈ N, pk = {si | 0 ≤ i ≤ k}. S is learnable. s1 s2 s3 s4 p0 p1 p2 p3 p4 . . . s0

Example of a Learnable Space

Let S = (S, O) such that S = {sn | n ∈ N}, O = {pi | i ∈ N}, and for any k ∈ N, pk = {si | 0 ≤ i ≤ k}. S is learnable. s0 s1 s3 s4 p0 p1 p2 p3 p4 . . . s2

Example of a Non-Learnable Space

Consider S = (S, O), where S := {sn | n ∈ N} ∪ {s∞}, and O = {pi | i ∈ N}, and for any k ∈ N, pk := {sk, sk+1, . . .} ∪ {s∞}. S is not learnable. s0 s1 s2 s3 s∞ p0 p1 p2 p3 . . .

Example of a Non-Learnable Space

Consider S = (S, O), where S := {sn | n ∈ N} ∪ {s∞}, and O = {pi | i ∈ N}, and for any k ∈ N, pk := {sk, sk+1, . . .} ∪ {s∞}. S is not learnable. s0 s1 s2 s3 s∞ p0 p1 p2 p3 . . .

Example of a Non-Learnable Space

Consider S = (S, O), where S := {sn | n ∈ N} ∪ {s∞}, and O = {pi | i ∈ N}, and for any k ∈ N, pk := {sk, sk+1, . . .} ∪ {s∞}. S is not learnable. s0 s2 s3 s∞ p0 p1 p2 p3 . . . s1

Example of a Non-Learnable Space

Consider S = (S, O), where S := {sn | n ∈ N} ∪ {s∞}, and O = {pi | i ∈ N}, and for any k ∈ N, pk := {sk, sk+1, . . .} ∪ {s∞}. S is not learnable. s0 s1 s2 s3 s∞ p0 p1 p2 p3 . . .

Example of a Non-Learnable Space

Consider S = (S, O), where S := {sn | n ∈ N} ∪ {s∞}, and O = {pi | i ∈ N}, and for any k ∈ N, pk := {sk, sk+1, . . .} ∪ {s∞}. S is not learnable. s0 s1 s3 s∞ p0 p1 p2 p3 . . . s2

Questions, Answers, and Problems

Definition

A question Q is a partition of S, whose cells Ai are called answers to Q. Given s ∈ A ⊆ S, A ∈ Q is called the answer to Q at s, denoted As.

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

t s u v V U P Q

Illustration

t s u v V U P Q

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]) ⊆ As 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 T1 (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 T0 (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

s t u w U V

Figure: t and u are not separable

t s V U

Figure: weakly separated space, T0

V t U s

Figure: strongly separated space, T1

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 Td 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}. Td is a separation property between T0 and T1.

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 τS associated with an epistemic space S = (S, O) is a collection

• f subsets of S of the following properties:
• 1. for any O ∈ O it is the case that O ∈ τS
• 2. ∅ ∈ τS,
• 3. S ∈ τS,
• 4. for any U ⊆ τS, U ∈ τS, and
• 5. for any x, y ∈ τS we have x ∩ y ∈ τS.

s t u w U V

The Topology Associated with an Epistemic Space

Definition

The topology τS associated with an epistemic space S = (S, O) is a collection

• f subsets of S of the following properties:
• 1. for any O ∈ O it is the case that O ∈ τS
• 2. ∅ ∈ τS,
• 3. S ∈ τS,
• 4. for any U ⊆ τS, U ∈ τS, and
• 5. for any x, y ∈ τS we have x ∩ y ∈ τS.

s t u w U V

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 Td 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: L≤(O1, . . . , On) := Min≤ n

• i=1

Oi

• whenever

i Oi has any minimal elements; and otherwise:

L≤(O1, . . . , On) :=

n

• i=1

Oi.

Definition

Conditioning is said to be standard if the prior ≤ is well-founded.

Direct Solvability by Conditioning

Definition

Given a question Q on an epistemic space S = (S, O), any total order ⊆ Q × Q on Q induces in a canonical way a total preorder ≤ ⊆ S × S: s ≤ t iff As At. A problem P = (S, Q) is directly solvable by conditioning if it is solvable by conditioning with respect to a total order ⊆ Q × Q.

Ordering the States v. Ordering the Answers

t s u v V U P Q A1 A2

Linear Separation

Definition

A partition Q of a topological space (S, τ) is linearly separated if there exists some total order on Q and a map O : Q → τ, s.t. for all cells A, B ∈ Q:

• 1. A ⊆ OA;
• 2. if B ⊳ A then OA ∩ B = ∅ (where ⊳ is the corresponding strict order).

Theorem

P = (S, Q) is directly solvable by conditioning iff Q is linearly separated.

Universality: Building Stronger Refinements

Theorem

Conditioning is a universal problem solving method. not all solvable questions allow direct solvability directly solvable refinement locally closed sets linearly separated questions transform them? which ones do? implies

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

Logic for Learnability

Since learnability is about potentially successful changes of beliefs

• ne expects some doxastic logic to capture it and to reason about it.

Relational semantics for modal logic

Definition (Syntax)

Take countable set of propositional symbols P. ϕ := p | ¬ϕ | ϕ ∧ ϕ | ϕ, for all p ∈ P, the usual abbreviations are ∨, →, and ♦.

Relational semantics for modal logic

Definition (Syntax)

Take countable set of propositional symbols P. ϕ := p | ¬ϕ | ϕ ∧ ϕ | ϕ, for all p ∈ P, the usual abbreviations are ∨, →, and ♦.

Definition (Semantics)

Given a model M = (W , R, v), where v : P → ℘(W ), and a state x ∈ W : M, x | = p iff x ∈ v(p) for each p ∈ P M, x | = ¬ϕ iff not M, x | = ϕ M, x | = ϕ ∧ ψ iff M, x | = ϕ and M, x | = ψ M, x | = ϕ iff for all y ∈ W : if xRy then M, y | = ϕ and dually: M, x | = ♦ϕ iff there is y ∈ W : xRy and M, y | = ϕ

Some Axioms and Their Epistemic Meaning

Rules (MP) if ⊢ ϕ and ⊢ ϕ → ψ, then ⊢ ψ (N) if ⊢ ϕ, then ⊢ ϕ Axioms (K) (ϕ → ψ) → (ϕ → ψ) (omniscience) (T) ϕ → ϕ (truthfullness/reflexivity) (D) ϕ → ¬¬ϕ (consistency/seriality) (4) ϕ → ϕ (positive introspection/transitivity) (5) ¬ϕ → ¬ϕ (negative introspection/Euclidean-ness)

Some Axioms and Their Epistemic Meaning

Rules (MP) if ⊢ ϕ and ⊢ ϕ → ψ, then ⊢ ψ (N) if ⊢ ϕ, then ⊢ ϕ Axioms (K) (ϕ → ψ) → (ϕ → ψ) (omniscience) (T) ϕ → ϕ (truthfullness/reflexivity) (D) ϕ → ¬¬ϕ (consistency/seriality) (4) ϕ → ϕ (positive introspection/transitivity) (5) ¬ϕ → ¬ϕ (negative introspection/Euclidean-ness) Ax is a logic of a class of models M iff Ax is sound and complete wrt M.

Can we use modal logic on topologies?

Relational vs Topological := Int

ϕ ϕ

Can we use modal logic on topologies?

Relational vs Topological := Int

ϕ ϕ

Definition

Let P be a set of propositional symbols. A topological model (or a topo-model) M = (X, O, v) is a topological space τ = (X, O) together with a valuation function v : P → ℘(X).

Topological Topo-semantics for Modal Logic

Definition

Truth of modal formulas is defined inductively at points x in a topo-model M = (X, O, v) in the following way: M, x | = p iff x ∈ v(p) for each p ∈ P M, x | = ¬ϕ iff not M, x | = ϕ M, x | = ϕ ∧ ψ iff M, x | = ϕ and M, x | = ψ M, x | = ϕ iff there is U ∈ τ(x ∈ U and for all y ∈ U: M, y | = ϕ) and dually: M, x | = ♦ϕ iff for all U ∈ τ(x ∈ U → there is y ∈ U: M, y | = ϕ)

Sound and Complete Topo-Axiomatizations

Rules (MP) if ⊢ ϕ and ⊢ ϕ → ψ, then ⊢ ψ (N) if ⊢ ϕ, then ⊢ ϕ Axioms (K) (ϕ → ψ) → (ϕ → ψ) (T) ϕ → ϕ (D) ϕ → ¬¬ϕ (4) ϕ → ϕ (5) ¬ϕ → ¬ϕ

Sound and Complete Topo-Axiomatizations

Rules (MP) if ⊢ ϕ and ⊢ ϕ → ψ, then ⊢ ψ (N) if ⊢ ϕ, then ⊢ ϕ Axioms (K) (ϕ → ψ) → (ϕ → ψ) (T) ϕ → ϕ (4) ϕ → ϕ S4 is the topo-logic of all topological spaces (McKinsey & Tarski 1944).

Sound and Complete Topo-Axiomatizations

Rules (MP) if ⊢ ϕ and ⊢ ϕ → ψ, then ⊢ ψ (N) if ⊢ ϕ, then ⊢ ϕ Axioms (K) (ϕ → ψ) → (ϕ → ψ) (T) ϕ → ϕ (4) ϕ → ϕ

S 4 = T

• p
• S4 is the topo-logic of all topological spaces (McKinsey & Tarski 1944).

What about Td-spaces (the learning spaces)?

Td is not topo-definable. Learnable spaces are not topo-definable. Luckily, we can once again change the way we view . ϕ ϕ

Topological d-semantics

Definition

Truth of modal formulas is defined inductively at points x in a topo-model M = (X, τ, v) in the following way: M, x | =d p iff x ∈ v(p) for each p ∈ P M, x | =d ¬ϕ iff not M, x | =d ϕ M, x | =d ϕ ∧ ψ iff M, x | =d ϕ and M, x | =d ψ M, x | =d ϕ iff ∃U ∈ τ(x ∈ U & ∀y ∈ U − {x} M, y | =d ϕ) and dually: M, x | =d ♦ϕ iff ∀U ∈ τ(x ∈ U → ∃y ∈ U − {x} M, y | =d ϕ)

Sound and Complete d-Axiomatizations

Rules (MP) if ⊢ ϕ and ⊢ ϕ → ψ, then ⊢ ψ (N) if ⊢ ϕ, then ⊢ ϕ Axioms (K) (ϕ → ψ) → (ϕ → ψ) (T) ϕ → ϕ (D) ϕ → ¬¬ϕ (4) ϕ → ϕ (5) ¬ϕ → ¬ϕ (w) (ϕ ∧ ϕ) → ϕ (GL) (ϕ → ϕ) → ϕ

Sound and Complete d-Axiomatizations

Rules (MP) if ⊢ ϕ and ⊢ ϕ → ψ, then ⊢ ψ (N) if ⊢ ϕ, then ⊢ ϕ Axioms (K) (ϕ → ψ) → (ϕ → ψ) (D) ϕ → ¬¬ϕ (4) ϕ → ϕ (5) ¬ϕ → ¬ϕ

KD45=DSO

KD45 is the d-logic of DSO-spaces.

Sound and Complete d-Axiomatizations

Rules (MP) if ⊢ ϕ and ⊢ ϕ → ψ, then ⊢ ψ (N) if ⊢ ϕ, then ⊢ ϕ Axioms (K) (ϕ → ψ) → (ϕ → ψ) (GL) (ϕ → ϕ) → ϕ

GL=scattered

GL is the d-logic of scattered spaces.

Sound and Complete d-Axiomatizations

Rules (MP) if ⊢ ϕ and ⊢ ϕ → ψ, then ⊢ ψ (N) if ⊢ ϕ, then ⊢ ϕ Axioms (K) (ϕ → ψ) → (ϕ → ψ) (w) (ϕ ∧ ϕ) → ϕ

wK4=Topo

wK4 is the d-logic of all topological spaces.

Sound and Complete d-Axiomatizations

Rules (MP) if ⊢ ϕ and ⊢ ϕ → ψ, then ⊢ ψ (N) if ⊢ ϕ, then ⊢ ϕ Axioms (K) (ϕ → ψ) → (ϕ → ψ) (4) ϕ → ϕ

K4=Td

Finally, K4 is the d-logic of all Td-spaces.

KD45 Doxastic d-logic (Steinvold 2006)

Because independent reasons (e.g., Stalnaker) one may want B:= to be: (K) (ϕ → ψ) → (ϕ → ψ) (D) ϕ → ¬¬ϕ (4) ϕ → ϕ (5) ¬ϕ → ¬ϕ

Theorem (Steinsvold 2006)

KD45 is a sound and complete d-axiomatization of DSO spaces. DSO stands for ‘derived sets are open’. DSO are Td-spaces (by 4), in which all derived sets are open (5), except that there are no open singletons (D).

Questions

But DSO ⊂ Td. So what do we talk about when we talk about beliefs in learning? Should conjectures be interpreted as beliefs? What if one restricts conjectures to only those which are ‘proper’ beliefs?

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

Conclusions

◮ Topological characterization of learnability & solvability in the limit. ◮ Universality of conditioning as a problem solving method. ◮ Use of stratification-like topological techniques.

Moreover:

◮ Learnable spaces are Td. ◮ Td-spaces are not topo-definable. ◮ Learnability is not topo-definable. ◮ Learnability cannot be expressed by solely topo-definable belief operators. ◮ The existing topo- and d-logics of belief are to fluffy to capture learnability.

THANK YOU