Computing in the Limit (1303.4408v4)

Abstract: We define a class of functions termed "Computable in the Limit", based on the Machine Learning paradigm of "Identification in the Limit". A function is Computable in the Limit if it defines a property P_p of a recursively enumerable class A of recursively enumerable data sequences S in A, such that each data sequence S is generated by a total recursive function s that enumerates . Let the index s represent the data sequence S. The property P_p(s)=x is computed by a partial recursive function f_p(s,t) such that there exists a u where f_p(s,u)=x and for all t>=u, f_p(s,t)=x if it converges. Since the index s is known, this is not an identification problem - instead it is computing a common property of the sequences in A. We give a Normal Form Theorem for properties that are Computable in the Limit, similar to Kleene's Normal Form Theorem. We also give some examples of sets that are Computable in the Limit, and derive some properties of Canonical and Complexity Bound Enumerations of classes of total functions, and show that no full enumeration of all indices of Turing machines TM_i that compute a given total function can be Computable in the Limit.

• The paper introduces 'Computable in the Limit' functions, establishing a framework for computing stable properties from recursively enumerable data sequences.
• The paper’s Normal Form Theorem reformulates computation by replacing the μ operator with a λ function to capture final, stable outputs in iterative processes.
• Illustrative examples, including Kolmogorov sets and incompressible numbers, demonstrate adaptive computation’s potential in bridging theory and Machine Learning.

Computing in the Limit

The paper "Computing in the Limit" introduces a class of functions termed as "Computable in the Limit", creating a significant expansion of our understanding of computational properties within recursively enumerable data sequences. Inspired by Machine Learning paradigms like "Identification in the Limit", it explores functions that can be computed over time despite the need for ongoing inference and adjustment.

Definitions and Conceptual Framework

The paper begins by defining the core concept of functions Computable in the Limit, utilizing recursive sequences to determine properties of data sequences $S$. Each sequence is governed by a total recursive function $\phi_s$, and the computation of any property $P_p(s) = x$ is achieved through a partial recursive function $\phi_p(s, t)$. Crucially, there exists a time $u$ such that for all times $t \geq u$, $\phi_p(s, t)$ stabilizes at $x$. This contrasts sharply with identification problems, as the index $s$ is predefined, focusing solely on computing a property common to sequences within a given class.

The Normal Form Theorem

A critical contribution of the paper is the Normal Form Theorem for limit computability properties. This theorem parallels Kleene's Normal Form Theorem but uniquely employs the function $\lambda(x)$ for final guesses, rather than $\mu(x)$ for initial truths. This subtle shift provides a framework wherein functions Computable in the Limit possess descriptive power akin to effectively computable functions. By leveraging primitive recursive predicates $T'(e, x, y)$ and function $U(z)$, computational processes are formalized, demonstrating that even properties shifting over time can be reliably framed within established computational hierarchies.

Examples and Non-limit Computable Properties

The paper offers illustrative examples of properties Computable in the Limit and elaborates on their expressive capability beyond effectively computable functions. Notable examples include Kolmogorov sets, incompressible number sets, and enumeration of partial recursive functions by simplicity or domain finitude. These exemplify convergence on final values over increasing computation histories.

Conversely, the paper argues that certain properties cannot be Computable in the Limit—a claim substantiated by rigorous proofs, such as the impossibility of complete enumeration of all total recursive functions being l-total via a limit computable function.

Implications and Future Directions

The examination of Computable in the Limit functions opens avenues for integrating computational processes with theoretical Machine Learning constructs. It propels the paper beyond existing learning paradigms, inviting exploration into computational extensions and real-world applications, particularly in fields requiring adaptive and ongoing computation akin to human learning processes.

Conclusions

"Computing in the Limit" embarks on an innovative exploration of computational capabilities beyond intrinsically finite procedures, constructing a robust theoretical backdrop for learning models and systems that thrive on iterative convergence. The substitution of traditional forms for selffilled processes implicates expansive potential for further research and practical implications in computational theory and Machine Learning. The groundwork laid, paves future investigation of real-limit approximations within dynamic computational structures.