Some extensions are considered of Gold's influential model of language learning by machine from positive data. Studied are criteria of successful learning featuring convergence in the limit to vacillation between several alternative correct grammars. The main theorem of this paper is that there are classes of languages that can be learned if convergence in the limit to up to (n+1) exactly correct grammars is allowed but which cannot be learned if convergence in the limit is to no more than n grammars, where the no more than n grammars can each make finitely many mistakes. This contrasts sharply with results of Barzdin and Podnieks and, later, Case and Smith, for learnability from both positive and negative data. A subset principle from a 1980 paper of Angluin is extended to the vacillatory and other criteria of this paper. This principle, provides a necessary condition for circumventing overgeneralization in learning from positive data. It is applied to prove another theorem to the eff...

Kleene's Second Recursion Theorem provides a means for transforming any program p into a program e(p) which first creates a quiescent self copy and then runs p on that self copy together with any externally given input. e(p), in effect, has complete (low level) self knowledge, and p represents how e(p) uses its self knowledge (and its knowledge of the external world). Infinite regress is not required since e(p) creates its self copy outside of itself. One mechanism to achieve this creation is a self replication trick isomorphic to that employed by single-celled organisms. Another is for e(p) to look in a mirror to see which program it is. In 1974 the author published an infinitary generalization of Kleene's theorem which he called the Operator Recursion Theorem. It provides a means for obtaining an (algorithmically) growing collection of programs which, in effect, share a common (also growing) mirror from which they can obtain complete low level models of themselves and the other prog...

The set of minimal indices of a G#del numbering ' is deøned as MIN' = fe : (8i ! e)[' i 6= 'e ]g. It has been known since 1972 that MIN' jT ; 00 , but beyond this MIN' has remained mostly uninvestigated. This thesis collects the scarce results on MIN' from the literature and adds some new observations including that MIN' is autoreducible, but neither regressive nor (1; 2)- computable. We also study several variants of MIN' that have been deøned in the literature like size-minimal indices, shortest descriptions, and minimal indices of decision tables. Some challenging open problems are left for the adventurous reader. 1 Introduction How long is the shortest program that solves your problem? There are at least two ways to interpret this question depending on the type of problem involved. If the program's task is to output one speciøc object, we are looking for a shortest description of that object. This interpretation is closely related to Kolmogorov complexity. Although we have sev...

It is suggested that there are infinite computable sets of natural numbers with the property that no infinite subset can be computed more simply or more quickly than the whole set. Attempts to establish this without restricting in any way the computer involved in the calculations are not entirely successful. A hypothesis concerning the computer makes it possible to exhibit sets without simpler subsets. A second and analogous hypothesis then makes it possible to prove that these sets are also without subsets which can be computed more rapidly than the whole set. It is then demonstrated that there are computers which satisfy both hypotheses. The general theory is momentarily set aside and a particular Turing machine is studied. Lastly, it is shown that the second hypothesis is more restrictive then requiring the computer to be capable of calculating all infinite computable sets of natural numbers.

This paper provides a beginning study of the effects on inductive inference of paradigm shifts whose absence is approximately modeled by various formal approaches to forbidding large changes in the size of programs conjectured. One approach, called severely parsimonious, requires all the programs conjectured on the way to success to be nearly (i.e., within a recursive function of) minimal size. It is shown that this very conservative constraint allows learning infinite classes of functions, but not infinite r.e. classes of functions. Another approach, called non-revolutionary, requires all conjectures to be nearly the same size as one another. This quite conservative constraint is, nonetheless, shown to permit learning some infinite r.e. classes of functions. Allowing up to one extra bounded size mind change towards a final program learned certainly doesn’t appear revolutionary. However, somewhat surprisingly for scientific (inductive) inference, it is shown that there are classes learnable with the non-revolutionary constraint (respectively, with severe parsimony), up to (i + 1) mind changes, and no anomalies, which classes cannot be learned with no size constraint, an unbounded, finite number of anomalies in the final program, but with no more than i mind changes. Hence, in some cases, the possibility of one extra mind change is considerably more liberating than removal of very conservative size shift constraints. The proofs of these results are also combinatorially interesting. 1

We introduce a domain-specific language (DSL) for creating sets of tile types for simulations of the abstract Tile Assembly Model. The language defines objects known as tile templates, which represent related groups of tiles, and a small number of basic operations on tile templates that help to eliminate the error-prone drudgery of enumerating such tile types manually or with low-level constructs of general-purpose programming languages. The language is implemented as a class library in Python (a so-called internal DSL), but is presented independently of Python or object-oriented programming, with emphasis on support for a visual editing tool for creating large sets of complex tile types. 1

Suppose LC1 and LC2 are two machine learning classes each based on a criterion of success. Suppose, for every machine which learns a class of functions according to the LC1 criterion of success, there is a machine which learns this class according to the LC2 criterion. In the case where the converse does not hold LC1 is said to be separated from LC2. It is shown that for many such separated learning classes from the literature a much stronger separation holds: (∀C ∈ LC1)(∃C ′ ∈ (LC2 −LC1))[C ′ ⊃ C]. It is also shown that there is a pair of separated learning classes from the literature for which the stronger separation just above does not hold. A philosophical heuristic toward the design of artificially intelligent learning programs is presented with each strong separation result. 1

Learning Theory is the study of systems that implement functions from evidential states to theories. The theoretical framework developed in the theory makes possible the comparison of classes of algorithms which embody d i s t i n c t learning strategies along a variety of dimensions. Such comparisons yield valuable information to those concerned with inference problems in Cognitive Science and A r t i f i c i a l Intelligence. The present paper employs the framework of Learning Theory to study the design specifications of inductive systems

ing from concrete machine models the question translates into minimal indices with respect to a numbering of the computable, partial functions. The first part of the paper tells the history of this problem collecting the known results. The second part offers some new observations, and the last part concludes with a list of open problems. We will only consider Godel numberings. A Godel numbering is an effective numbering ' of all computable partial functions such that for every effective numbering / a '-index can be computed from a /-index. We will also use Kolmogorov numberings. A Godel numbering is a Kolmogorov numbering, if there is a linearly bounded computable function that transforms /-indices into '-indices. It is well known that Kolmogorov numberings exist. Definition 1.1 Let ' be a Godel numbering. Define MIN' := fe : (8i ! e)[' i 6= ' e ]g; the set of minimal indices of '. What would happen if instead of Godel numberings arbitrary numberings of the computable, partial fun...

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