learning in the limit, general topology and modal logicfitelson.org/bridges2/gierasimczuk.pdf ·...
TRANSCRIPT
![Page 1: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/1.jpg)
Learning in the Limit, General Topologyand Modal Logic
Nina Gierasimczukresults obtained jointly with A. Baltag and S. Smets
Institute for Logic, Language and ComputationUniversity of Amsterdam
Bridges 2Rutgers University, September 19th, 2015
![Page 2: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/2.jpg)
Outline
Introduction
Characterization of Learnability and Solvability
Constructive Order-driven Learning
Towards Epistemic Logic of Learnability
Intermediate (but Interesting ) Conclusions
![Page 3: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/3.jpg)
Outline
Introduction
Characterization of Learnability and Solvability
Constructive Order-driven Learning
Towards Epistemic Logic of Learnability
Intermediate (but Interesting ) Conclusions
![Page 4: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/4.jpg)
Background
I Learning and belief revision go their separate ways,
I conjecture dynamics is a common theme.
I What are the principles of this dynamics?
B B ′p
![Page 5: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/5.jpg)
Background
I Learning and belief revision go their separate ways,
I conjecture dynamics is a common theme.
I What are the principles of this dynamics?
B B ′ B ′′p
![Page 6: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/6.jpg)
Background
I Learning and belief revision go their separate ways,
I conjecture dynamics is a common theme.
I What are the principles of this dynamics?
B B ′ B ′′p
Truth-tracking!
![Page 7: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/7.jpg)
Epistemic Spaces and Observables
DefinitionAn epistemic space is a pair S = (S ,O) consisting of a state space (a set ofpossible worlds) S and a countable set of observable properties O ⊆ P(S).
s t u w
UV
![Page 8: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/8.jpg)
Learning: Streams of Observables
DefinitionLet S = (S ,O) be an epistemic space.
I A data stream is an infinite sequence ~O = (O0,O1, . . .) of data from O.
I A data sequence is a finite sequence σ = (σ0, . . . , σn).
DefinitionTake S = (S ,O) and s ∈ S . A data stream ~O is:
I sound with respect to s iff every element listed in ~O is true in s.
I complete with respect to s iff every observable true in s is listed in ~O.
We assume that data streams are sound and complete.
![Page 9: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/9.jpg)
Learning: Learners and Conjectures
DefinitionLet S = (S ,O) be an epistemic space and let σ0, . . . , σn ∈ O. A learner is afunction L that on the input of S and data sequence (σ0, . . . , σn) outputs someset of worlds L(S, (σ0, . . . , σn)) ⊆ S , called a conjecture.
DefinitionS = (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.
![Page 10: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/10.jpg)
Example of a Learnable Space
Let S = (S ,O) such that S = {sn | n ∈ N}, O = {pi | i ∈ N}, and for anyk ∈ N, pk = {si | 0 ≤ i ≤ k}. S is learnable.
p0 p1 p2 p3 p4. . .
s0 s2
s0 s1 s2 s3 s4
![Page 11: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/11.jpg)
Example of a Learnable Space
Let S = (S ,O) such that S = {sn | n ∈ N}, O = {pi | i ∈ N}, and for anyk ∈ N, pk = {si | 0 ≤ i ≤ k}. S is learnable.
p0 p1 p2 p3 p4. . .
s0 s2
s0 s1 s2 s3 s4
![Page 12: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/12.jpg)
Example of a Learnable Space
Let S = (S ,O) such that S = {sn | n ∈ N}, O = {pi | i ∈ N}, and for anyk ∈ N, pk = {si | 0 ≤ i ≤ k}. S is learnable.
p0 p1 p2 p3 p4. . .
s0
s2s0
s1 s2 s3 s4
![Page 13: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/13.jpg)
Example of a Learnable Space
Let S = (S ,O) such that S = {sn | n ∈ N}, O = {pi | i ∈ N}, and for anyk ∈ N, pk = {si | 0 ≤ i ≤ k}. S is learnable.
p0 p1 p2 p3 p4. . .
s0
s2s0 s1
s2
s3 s4
![Page 14: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/14.jpg)
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.
p0 p1 p2 p3. . .
s1 s2
s0 s1 s2 s3 s∞
![Page 15: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/15.jpg)
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.
p0 p1 p2 p3. . .
s1 s2
s0 s1 s2 s3 s∞
![Page 16: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/16.jpg)
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.
p0 p1 p2 p3. . .
s1
s2
s0
s1
s2 s3 s∞
![Page 17: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/17.jpg)
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.
p0 p1 p2 p3. . .
s1 s2
s0 s1 s2 s3 s∞
![Page 18: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/18.jpg)
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.
p0 p1 p2 p3. . .
s1
s2s0 s1
s2
s3 s∞
![Page 19: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/19.jpg)
Questions, Answers, and Problems
DefinitionA 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 .
DefinitionQ′ is a refinement of Q if all answers of Q is a disjoint union of answers of Q′.
DefinitionA 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.
![Page 20: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/20.jpg)
Illustration
t s
u v
VU
PQ
![Page 21: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/21.jpg)
Illustration
t s
u v
VU
PQ
![Page 22: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/22.jpg)
Solving in the Limit
DefinitionA 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.
![Page 23: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/23.jpg)
General Topology
DefinitionA 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.
DefinitionTake 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}.
![Page 24: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/24.jpg)
Separability by observations: Illustration
s t u w
UV
(a) t and u are not separable
t s
VU
(b) weakly separated space, T0
V
t
U
s
(c) strongly separated space, T1
![Page 25: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/25.jpg)
Locally Closed and Constructible Sets
DefinitionA 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.
DefinitionA 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.
An ω-constructible set is a countable union of locally closed sets.
![Page 26: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/26.jpg)
Outline
Introduction
Characterization of Learnability and Solvability
Constructive Order-driven Learning
Towards Epistemic Logic of Learnability
Intermediate (but Interesting ) Conclusions
![Page 27: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/27.jpg)
The Topology Associated with an Epistemic Space
DefinitionThe topology τS associated with an epistemic space S = (S ,O) is a collectionof subsets of S of the following properties:
1. for any O ∈ O it is the case that O ∈ τS2. ∅ ∈ 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
UV
![Page 28: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/28.jpg)
The Topology Associated with an Epistemic Space
DefinitionThe topology τS associated with an epistemic space S = (S ,O) is a collectionof subsets of S of the following properties:
1. for any O ∈ O it is the case that O ∈ τS2. ∅ ∈ 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
UV
![Page 29: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/29.jpg)
Characterization of Solvability in the Limit
TheoremA problem P = (S,Q) is solvable in the limit iff Q has a locally closedrefinement.
CorollaryAn epistemic space S = (S ,O) is learnable in the limit iff it satisfies the Td
separation axiom.
![Page 30: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/30.jpg)
Outline
Introduction
Characterization of Learnability and Solvability
Constructive Order-driven Learning
Towards Epistemic Logic of Learnability
Intermediate (but Interesting ) Conclusions
![Page 31: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/31.jpg)
Order-driven learning: Motivation
I Belief Revision: minimal states give beliefs.
I Computational Learning Theory: co-learning, learning by erasing.
I Philosophy of Science: Ockham’s razor.
![Page 32: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/32.jpg)
Conditioning
DefinitionConditioning 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 .
DefinitionConditioning is said to be standard if the prior ≤ is well-founded.
TheoremNon-standard conditioning is a universal problem solving method.
![Page 33: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/33.jpg)
Outline
Introduction
Characterization of Learnability and Solvability
Constructive Order-driven Learning
Towards Epistemic Logic of Learnability
Intermediate (but Interesting ) Conclusions
![Page 34: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/34.jpg)
Logic for Learnability
Since learnability is about potentially successful changes of beliefsone expects some doxastic logic to capture it and to reason about it.
![Page 35: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/35.jpg)
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 ∈ PM, x |= ¬ϕ iff not M, x |= ϕM, x |= ϕ ∧ ψ iff M, x |= ϕ and M, x |= ψM, x |= �ϕ iff for all y ∈W : if xRy then M, y |= ϕand dually:M, x |= ♦ϕ iff there is y ∈W : xRy and M, y |= ϕ
![Page 36: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/36.jpg)
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 ∈ PM, x |= ¬ϕ iff not M, x |= ϕM, x |= ϕ ∧ ψ iff M, x |= ϕ and M, x |= ψM, x |= �ϕ iff for all y ∈W : if xRy then M, y |= ϕand dually:M, x |= ♦ϕ iff there is y ∈W : xRy and M, y |= ϕ
![Page 37: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/37.jpg)
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.
![Page 38: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/38.jpg)
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.
![Page 39: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/39.jpg)
Can we use modal logic on topologies?Relational � vs Topological � := Int
�ϕ �ϕ
DefinitionLet P be a set of propositional symbols. A topological model (or atopo-model) M = (X ,O, v) is a topological space τ = (X ,O) together with avaluation function v : P → ℘(X ).
![Page 40: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/40.jpg)
Can we use modal logic on topologies?Relational � vs Topological � := Int
�ϕ �ϕ
DefinitionLet P be a set of propositional symbols. A topological model (or atopo-model) M = (X ,O, v) is a topological space τ = (X ,O) together with avaluation function v : P → ℘(X ).
![Page 41: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/41.jpg)
Topological Topo-semantics for Modal Logic
DefinitionTruth of modal formulas is defined inductively at points x in a topo-modelM = (X ,O, v) in the following way:
M, x |= p iff x ∈ v(p) for each p ∈ PM, x |= ¬ϕ iff not M, x |= ϕM, x |= ϕ ∧ ψ iff M, x |= ϕ and M, x |= ψM, 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)
![Page 42: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/42.jpg)
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
S4=Topo
S4 is the topo-logic of all topological spaces (McKinsey & Tarski 1944).
![Page 43: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/43.jpg)
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
S4=Topo
S4 is the topo-logic of all topological spaces (McKinsey & Tarski 1944).
![Page 44: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/44.jpg)
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
S4=Topo
S4 is the topo-logic of all topological spaces (McKinsey & Tarski 1944).
![Page 45: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/45.jpg)
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
![Page 46: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/46.jpg)
Topological d-semantics
DefinitionTruth of modal formulas is defined inductively at points x in a topo-modelM = (X , τ, v) in the following way:
M, x |=d p iff x ∈ v(p) for each p ∈ PM, x |=d ¬ϕ iff not M, x |=d ϕM, x |=d ϕ ∧ ψ iff M, x |=d ϕ and M, x |=d ψM, 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)
![Page 47: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/47.jpg)
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 is the d-logic of all Td -spaces.
K4=T
d
![Page 48: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/48.jpg)
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 allderived sets are open (5), except that there are no open singletons (D).
![Page 49: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/49.jpg)
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?
![Page 50: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/50.jpg)
Outline
Introduction
Characterization of Learnability and Solvability
Constructive Order-driven Learning
Towards Epistemic Logic of Learnability
Intermediate (but Interesting ) Conclusions
![Page 51: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/51.jpg)
Conclusions
I Topological characterization of learnability & solvability in the limit.
I Universality of conditioning as a problem solving method.
I Use of stratification-like topological techniques.
Moreover:
I Learnable spaces are Td .
I Td -spaces are not topo-definable.
I Learnability is not topo-definable.
I Learnability cannot be expressed by solely topo-definable belief operators.
I The existing topo- and d-logics of belief are to fluffy to capture learnability.
![Page 52: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/52.jpg)
THANK YOU!
A. Baltag, N. Gierasimczuk, and S. Smets. On the solvability of inductive problems: A
study in epistemic topology. Proceedings of TARK’15 (ILLC PP-2015-13), 2015.
A. Baltag, N. Gierasimczuk, and S. Smets. Truth tracking by belief revision. ILLC
PP-2014-20 (to appear in Studia Logica), 2015.
A. Baltag, N. Gierasimczuk, and S. Smets. Belief revision as a truth-tracking process. In
K. Apt, editor, Proceedings of TARK’11, pages 187-190. ACM, 2011.
A. Baltag, N. Bezhanishvili, A. Ozgun, and S. Smets. The Topology of Belief, Belief Revision
and Defeasible Knowledge. Proceedings of the LORI’13, 2013
M. de Brecht and A. Yamamoto. Topological properties of concept spaces. Information and
Computation, 208(4):327340, 2010.
N. Gierasimczuk. Knowing Ones Limits. Logical Analysis of Inductive Inference. PhD thesis,
Universiteit van Amsterdam, The Netherlands, 2010.
N. Gierasimczuk, D. de Jongh, and V. F. Hendricks. Logic and learning. In A. Baltag and S.
Smets, editors, Johan van Benthem on Logical and Informational Dynamics. Springer, 2014.
![Page 53: Learning in the Limit, General Topology and Modal Logicfitelson.org/bridges2/gierasimczuk.pdf · 2015-09-20 · Learning in the Limit, General Topology and Modal Logic Nina Gierasimczuk](https://reader034.vdocument.in/reader034/viewer/2022042313/5edd029cad6a402d6667efb0/html5/thumbnails/53.jpg)
THANK YOU!
A. Baltag, N. Gierasimczuk, and S. Smets. On the solvability of inductive problems: A
study in epistemic topology. Proceedings of TARK’15 (ILLC PP-2015-13), 2015.
A. Baltag, N. Gierasimczuk, and S. Smets. Truth tracking by belief revision. ILLC
PP-2014-20 (to appear in Studia Logica), 2015.
A. Baltag, N. Gierasimczuk, and S. Smets. Belief revision as a truth-tracking process. In
K. Apt, editor, Proceedings of TARK’11, pages 187-190. ACM, 2011.
A. Baltag, N. Bezhanishvili, A. Ozgun, and S. Smets. The Topology of Belief, Belief Revision
and Defeasible Knowledge. Proceedings of the LORI’13, 2013
M. de Brecht and A. Yamamoto. Topological properties of concept spaces. Information and
Computation, 208(4):327340, 2010.
N. Gierasimczuk. Knowing Ones Limits. Logical Analysis of Inductive Inference. PhD thesis,
Universiteit van Amsterdam, The Netherlands, 2010.
N. Gierasimczuk, D. de Jongh, and V. F. Hendricks. Logic and learning. In A. Baltag and S.
Smets, editors, Johan van Benthem on Logical and Informational Dynamics. Springer, 2014.