Set identification with errors

2 dezembro 2013, 10:30 • José Félix Costa

On lifting from functions to sets and from single-valued total sets to functions.

(Osherson and Weinstein, Case and Lynes) TxtEx^a- and TxtBc^a-identification, for a \in {*} \cup N. Classes TxtEx^a and TxtBc^a.

That TxtEx \subset TxtEx^1 \subset ... \subset TxtEx* (the non-collapsing character of TxtEx hierarchy).

That TxtBc \subset TxtBc^1 \subset ... \subset TxtBc* (the non-collapsing character of TxtBc hierarchy).

The interactions between the TxtEx and TxtBc hierarchies:

- That TxtEx* \notin TxtBc.

- That TxtBc \notin TxtEx*.

- That, for all n' \in N, for all n <= 2n', TxtEx^n \subseteq TxtBc^n'.

- That, for all n \in N, {L \in E: L =^{2n+1} N} \in TxtEx^{2n+1} - TxtBc^n.

Vacillatory identification

29 novembro 2013, 14:30 • José Félix Costa

(Barzdins and Podniecks) Concept of vacillatory identification (of a function, set of functions). Class Fex^a (with a \in {*} \cup N).

That Ex^a \subseteq Fex^a.

(Barzdins and Podniecks, Case and Smith) That Fex^a \subseteq Ex^a. (That vacillation does not improve the inferrence power of scientists in the case of function identification).

+++++++

Problem set VIII

Other strategies for learning

25 novembro 2013, 10:30 • José Félix Costa

Discussion of some strategies for learning such as conformal (consistent) and Popperian.

Some notes on operators and their fixed points.

General discussion on whether natural phenomena are describable by primitive recursive functions.

That Ex-identification is sufficient for primitive recursive functions (proof by enumeration of primitive recursive descriptions).

Conformal scientists

22 novembro 2013, 14:30 • José Félix Costa

The LPLA (logic problem of language acquisition). Empiricists and rationalists.

The attempt of solution via learning strategy: conformal learning.

(Bardins, Blum and Blum, Case) Conformal scientists. Class Conf.

The set FR of finite range self-referent recursive functions.

Thst Conf \subseteq Ex.

That FR \in Ex.

(Wiehagen) That FR \notin Conf.

That contradicting evidential facts improves inferring power.

Separations Bc^(n) =/= Bc^(n+1) and Bc^(n) =/= Bc*

18 novembro 2013, 10:30 • José Félix Costa

Bc^(n) and Bc*-identification.

The sets S^(n), for all n \in N, and S*.

(Case and Smith) That Bc^(n) =/= Bc^(n+1)

(Case and Smith) That Bc^(n) =/= Bc*, for all n \in N.

(Harrington) That R \in Bc* (that all recursive functions are identifiable if the scientist is tolerant to finitely many mistakes).

+++++++

Problem set VII

Ends course PART II.