Pavel Semukhin

Finite Automata Presentable Abelian Groups

We give new examples of FA presentable torsion-free abelian groups. Namely, for every n ? 2, we construct a rank nindecomposable torsion-free abelian group which has an FA presentation. We also construct an FA presentation of the group (?, + )2in which every nontrivial cyclic subgroup is not FA recognizable.

An Uncountably Categorical Theory Whose Only Computably Presentable Model Is Saturated

We build an א1-categorical but not א0-categorical theory whose only computably presentable model is the saturated one. As a tool, we introduce a notion related to limitwise monotonic functions. This research was partially supported by the Marsden Fund of New Zealand. The first author’s research was partially supported by the National Science Foundation of the USA under grant DMS-02-00465. The third author’s research was partially supported by RFFR grant No. 02-01-00593 and Council for Grants under RF President, project NSh-2112.2003.1. We thank the referee for providing a simpler proof of Lemma 3.4.

Uncountable automatic classes and learning

In this paper we consider uncountable classes recognizable by @w-automata and investigate suitable learning paradigms for them. In particular, the counterparts of explanatory, vacillatory and behaviourally correct learning are introduced for this setting. Here the learner reads in parallel the data of a text for a language L from the class plus an @w-index @a and outputs a sequence of @w-automata such that all but finitely many of these @w-automata accept the index @a if and only if @a is an index for L. It is shown that any class is behaviourally correct learnable if and only if it satisfies Angluins tell-tale condition. For explanatory learning, such a result needs that a suitable indexing of the class is chosen. On the one hand, every class satisfying Angluins tell-tale condition is vacillatorily learnable in every indexing; on the other hand, there is a fixed class such that the level of the class in the hierarchy of vacillatory learning depends on the indexing of the class chosen. We also consider a notion of blind learning. On the one hand, a class is blind explanatorily (vacillatorily) learnable if and only if it satisfies Angluins tell-tale condition and is countable; on the other hand, for behaviourally correct learning, there is no difference between the blind and non-blind version. This work establishes a bridge between the theory of @w-automata and inductive inference (learning theory).

Linear orders realized by C.e. Equivalence relations

Let E be a computably enumerable (c.e.) equivalence relation on the set ω of natural numbers. We say that the quotient set ω/E (or equivalently, the relation E) realizes a linearly ordered set L if there exists a c.e. relation E respecting E such that the induced structure (ω/E;E) is isomorphic to L. Thus, one can consider the class of all linearly ordered sets that are realized by ω/E; formally, K(E) = {L | the order-type L is realized by E}. In this paper we study the relationship between computability-theoretic properties of E and algebraic properties of linearly ordered sets realized by E. One can also define the following pre-order 6lo on the class of all c.e. equivalence relations: E1 6lo E2 if every linear order realized by E1 is also realized by E2. Following the tradition of computability theory, the lo-degrees are the classes of equivalence relations induced by the pre-order 6lo. We study the partially ordered set of lo-degrees. For instance, we construct various chains and antichains and show the existence of a maximal element among the lo-degrees.

Algebraic methods proving Sauer's bound for teaching complexity

Abstract This paper establishes an upper bound on the size of a concept class with given recursive teaching dimension (RTD, a teaching complexity parameter). The upper bound coincides with Sauers well-known bound on classes with a fixed VC-dimension. Our result thus supports the recently emerging conjecture that the combinatorics of VC-dimension and those of teaching complexity are intrinsically interlinked. We further introduce and study RTD-maximum classes (whose size meets the upper bound) and RTD-maximal classes (whose RTD increases if a concept is added to them), showing similarities but also differences to the corresponding notions for VC-dimension. Another contribution is a set of new results on maximal classes of a given VC-dimension. Methodologically, our contribution is the successful application of algebraic techniques, which we use to obtain a purely algebraic characterization of teaching sets (sample sets that uniquely identify a concept in a given concept class) and to prove our analog of Sauers bound for RTD. Such techniques have been used before to prove results relevant to computational learning theory, e.g., by Smolensky [13] , but are not standard in the field.

Sauer's bound for a notion of teaching complexity

This paper establishes an upper bound on the size of a concept class with given recursive teaching dimension (RTD, a teaching complexity parameter.) The upper bound coincides with Sauers well-known bound on classes with a fixed VC-dimension. Our result thus supports the recently emerging conjecture that the combinatorics of VC-dimension and those of teaching complexity are intrinsically interlinked. We further introduce and study RTD-maximum classes (whose size meets the upper bound) and RTD-maximal classes (whose RTD increases if a concept is added to them), showing similarities but also differences to the corresponding notions for VC-dimension. Another contribution is a set of new results on maximal classes of a given VC-dimension. Methodologically, our contribution is the successful application of algebraic techniques, which we use to obtain a purely algebraic characterization of teaching sets (sample sets that uniquely identify a concept in a given concept class) and to prove our analog of Sauers bound for RTD.

Automatic learning of subclasses of pattern languages

Automatic classes are classes of languages for which a finite automaton can decide membership question for the languages in the class, in a uniform way, given an index for the language. For alphabet size of at least 4, every automatic class of erasing pattern languages is contained, for some constant n, in the class of all languages generated by patterns which contain (1) every variable only once and (2) at most n symbols after the first occurrence of a variable. It is shown that such a class is automatically learnable using a learner with long-term memory bounded by the length of the first example seen. The study is extended to show the learnability of related classes such as the class of unions of two pattern languages of the above type.

Automatic learners with feedback queries

Automatic classes are classes of languages for which a finite automaton can decide whether a given element is in a set given by its index. The present work studies the learnability of automatic families by automatic learners which, in each round, output a hypothesis and update a long-term memory, depending on the input datum, via an automatic function. Many variants of automatic learners are investigated: where the long-term memory is restricted to be the current hypothesis whenever this exists, cannot be of length larger than the length of the longest datum seen, or has to consist of a constant number of examples seen so far. Learnability is also studied with respect to queries which reveal information about past data or past computation history; the number of queries per round is bounded by a constant. This paper first studies feedback queries for learning of automatic families.The types of feedback which allow to learn all automatic families are determined.Further variants like marked memory space are studied.

\(\Pi^0_1\)-Presentations of Algebras

In this paper we study the question as to which computable algebras are isomorphic to non-computable