Learning Equivalence Structures Luca San Mauro (Vienna University of Technology) Logic Colloquium 2018 joint work with Ekaterina Fokina and Timo Koetzing
Computational Learning Theory Computational Learning Theory (CLT) is a vast research program that comprises different models of learning in the limit. It deals with the question of how a learner, provided with more and more data about some environment, is eventually able to achieve systematic knowledge about it. 1
Computational Learning Theory Computational Learning Theory (CLT) is a vast research program that comprises different models of learning in the limit. It deals with the question of how a learner, provided with more and more data about some environment, is eventually able to achieve systematic knowledge about it. • (Gold, 1967): language identification. 1
Computational Learning Theory Computational Learning Theory (CLT) is a vast research program that comprises different models of learning in the limit. It deals with the question of how a learner, provided with more and more data about some environment, is eventually able to achieve systematic knowledge about it. • (Gold, 1967): language identification. More recently researchers applied the machinery of CLT to algebraic structures: 1
Computational Learning Theory Computational Learning Theory (CLT) is a vast research program that comprises different models of learning in the limit. It deals with the question of how a learner, provided with more and more data about some environment, is eventually able to achieve systematic knowledge about it. • (Gold, 1967): language identification. More recently researchers applied the machinery of CLT to algebraic structures: • (Stephan, Ventsov, 2001): Learning ring ideals of commutative rings. • (Harizanov, Stephan, 2002): Learning subspaces of V ∞ . 1
Our framework • Let K be a class of structures with some uniform effective enumeration {C i } i ∈ ω of the computable structures from K , up to isomorphism. 2
Our framework • Let K be a class of structures with some uniform effective enumeration {C i } i ∈ ω of the computable structures from K , up to isomorphism. • A learner M is a total function which takes for its inputs finite substructures of a given structure S from K . 2
Our framework • Let K be a class of structures with some uniform effective enumeration {C i } i ∈ ω of the computable structures from K , up to isomorphism. • A learner M is a total function which takes for its inputs finite substructures of a given structure S from K . • If M ( S i ) ↓ = n , for finite S i ⊆ S , then n represents M ’s conjecture as to an index for S in the above enumeration. 2
Our framework • Let K be a class of structures with some uniform effective enumeration {C i } i ∈ ω of the computable structures from K , up to isomorphism. • A learner M is a total function which takes for its inputs finite substructures of a given structure S from K . • If M ( S i ) ↓ = n , for finite S i ⊆ S , then n represents M ’s conjecture as to an index for S in the above enumeration. = -learns S if, for all T ∼ • M InfEx ∼ = S , there exists n ∈ ω such that T ∼ = C n and M ( T i ) ↓ = n , for all but finitely many i . 2
Our framework • Let K be a class of structures with some uniform effective enumeration {C i } i ∈ ω of the computable structures from K , up to isomorphism. • A learner M is a total function which takes for its inputs finite substructures of a given structure S from K . • If M ( S i ) ↓ = n , for finite S i ⊆ S , then n represents M ’s conjecture as to an index for S in the above enumeration. = -learns S if, for all T ∼ • M InfEx ∼ = S , there exists n ∈ ω such that T ∼ = C n and M ( T i ) ↓ = n , for all but finitely many i . • A family of structures A is InfEx ∼ = -learnable if there is M that learns all A ∈ A . 2
Our framework • Let K be a class of structures with some uniform effective enumeration {C i } i ∈ ω of the computable structures from K , up to isomorphism. • A learner M is a total function which takes for its inputs finite substructures of a given structure S from K . • If M ( S i ) ↓ = n , for finite S i ⊆ S , then n represents M ’s conjecture as to an index for S in the above enumeration. = -learns S if, for all T ∼ • M InfEx ∼ = S , there exists n ∈ ω such that T ∼ = C n and M ( T i ) ↓ = n , for all but finitely many i . • A family of structures A is InfEx ∼ = -learnable if there is M that learns all A ∈ A . • InfEx ∼ = ( K ) denotes the class of families of K -structures that are InfEx ∼ = -learnable. 2
Our framework, continued Our notation comes from classical CLT: Inf (short for informant ) means that we receive both positive and negative information about S ; Ex (short for explanatory ) means that M shall converge on a single input for S . At the end, we will discuss learning classes obtained by choosing natural alternatives of Inf , Ex , and ∼ =. 3
Our framework, continued Our notation comes from classical CLT: Inf (short for informant ) means that we receive both positive and negative information about S ; Ex (short for explanatory ) means that M shall converge on a single input for S . At the end, we will discuss learning classes obtained by choosing natural alternatives of Inf , Ex , and ∼ =. Remark: our model shares many analogies with the First-order Framework introduced in (Martin, Osherson, 1998). 3
Equivalence structures • Denote by E the class of equivalence structures. Our main focus is on InfEx ∼ = ( E ). 4
Equivalence structures • Denote by E the class of equivalence structures. Our main focus is on InfEx ∼ = ( E ). • (Downey, Melnikov, Ng, 2016): there is a Friedberg enumeration of computable equivalence structures. 4
Equivalence structures • Denote by E the class of equivalence structures. Our main focus is on InfEx ∼ = ( E ). • (Downey, Melnikov, Ng, 2016): there is a Friedberg enumeration of computable equivalence structures. • A non-Friedberg one is of course much more easy to be defined, e.g., for all n , let the size of the E n -classes be the cardinality of the columns of W n . 4
Example of learnability, 1 A B 5
Example of learnability, 1 A B S M ( S ) = � A � 5
Example of learnability, 1 A B S M ( S ) = � A � 5
Example of learnability, 1 A B S M ( S ) = � A � 5
Example of learnability, 1 A B S M ( S ) = � A � 5
Example of learnability, 1 A B S M ( S ) = � A � 5
Example of learnability, 1 A B S M ( S ) = � A � 5
Example of learnability, 1 A B S M ( S ) = � A � 5
Example of learnability, 1 A B S M ( S ) = � A � 5
Example of learnability, 1 A B S M ( S ) = � B � 5
Example of learnability, 2 A B 6
Example of learnability, 2 A B S M ( S ) = � A � 6
Example of learnability, 2 A B S M ( S ) = � A � 6
Example of learnability, 2 A B S M ( S ) = � A � 6
Example of learnability, 2 A B S M ( S ) = � A � 6
Example of learnability, 2 A B S M ( S ) = � B � 6
Example of learnability, 2 A B S M ( S ) = � B � 6
Example of learnability, 2 A B S M ( S ) = � B � 6
Example of learnability, 2 A B S M ( S ) = � A � 6
Example of nonlearnability, 1 A B 7
Example of nonlearnability, 1 Strategy • Assume that M learns {A , B} , • Construct by stages a structure S ∈ {T : T ∼ = A ∨ T ∼ = B} such that M fails to learn S . 7
Example of nonlearnability, 1 A B S ∼ = B M ( S ) = � A � 7
Example of nonlearnability, 1 A B S ∼ = B M ( S ) = � A � 7
Example of nonlearnability, 1 A B S ∼ = B M ( S ) = � B � 7
Example of nonlearnability, 1 A B S ∼ = B M ( S ) = � B � 7
Example of nonlearnability, 1 A B S ∼ = A M ( S ) = � B � 7
Example of nonlearnability, 1 A B S ∼ = A M ( S ) = � B � 7
Example of nonlearnability, 1 A B S ∼ = A M ( S ) = � B � 7
Example of nonlearnability, 1 A B S ∼ = A M ( S ) = � A � 7
Example of nonlearnability, 1 A B S ∼ = A M ( S ) = � A � 7
Example of nonlearnability, 1 A B S ∼ = B M ( S ) = � A � 7
Example of nonlearnability, 2 Recall that the character c A of A is c A = {� k , i � : A has i equivalence classes of size k } . Define A = {A i } i ∈ ω + such that, for all A i ’s, c A i = {� k , 1 � : k � = i } . Proposition A / ∈ InfEx ∼ = ( E ). 8
Finite separability • S is a limit of a finite family A if there is A ∈ A such that → fin S ∧ A �∼ A ֒ = S , 9
Finite separability • S is a limit of a finite family A if there is A ∈ A such that → fin S ∧ A �∼ A ֒ = S , and c S ⊆ c A . 9
Finite separability • S is a limit of a finite family A if there is A ∈ A such that → fin S ∧ A �∼ A ֒ = S , and c S ⊆ c A . • S is a limit of an infinite family A if → fin S ∧ A �∼ ( ∀A ∈ A )( A ֒ = S ) 9
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