Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces

Kalantari, Iraj; Retzlaff, Allen

doi:10.2307/2271869

Cited by 40 publications

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“…We construct W , a,nd W , with U = W , @ W , creative. (Using a. technique from KALANTARI-RETZLALFF [9], it is possible to make U maximal although we do not present details here; the construction is a straight forward combination of the ideas found below and those in [9].) We now show that the r.e.…”

Section: Vm E Ui + a 5 ( J U A ) * ( U I \ A ) + Amentioning

confidence: 75%

Direct Summands of Recursively Enumerable Vector Spaces

Retzlaff

1979

Mathematical Logic Qtrly

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in Purchase, New York (U.S.A.)l) g 1. Introduction and notation Let -!if'( V,) denote the modular lattice of recursively enumerable subspaces of a fixed, infinite dimensional vector space (called V,) under the operations of n and +. The structure of 9( V,) has been studied and compared with that of the somewhat similar lattice 8 of recursively enumerable sets of integers under n und U. Initial work in this area is due to RABIN [la], FROHLICH and SHEPHERDSON [5], DEKKER [3], HAMILTON [7], and R. GUHL [B]. The more recent work of METAKIDES and NERODE [12]. CROSSLEY and NERODE [23], REMMEL [15, 161 and KALANTARI [8] form the basis for current research in this area. Our notation is primarily that of ROGERS' book [19] as well as METAXIDES and NERODE [123. We have collected here examples, theorems and constructions concerning the elementary lattice operation + in 2' ( V,) which grew out of our investigations of this lattice. Of course the operation + for vector spaces corresponds in some sense to the operation of union for sets, however, the dissimilarity between the lattice of all vector subspaces of a given space and the lattice of all subsets of a given set is convincing evidence that the effective lattices of r.e. vector spaces and r.e. sets of integers (traditionally denoted by F ) could be quite different. This indeed proved to be the case, as we shall show.In section 1, the definitions and some of the basic results of the theory of recursively enumerable vector spaces are recalled. For further reference, the reader should consult KALANTARI [8] and REMMEL [15], [16]. Sections 2 and 3 contain two types of results.Theorem 3 and 4 delineate how the property of recursivity fails to be preserved under + .Infact, we show that any infinite dimensional recursively enumerable vector space over an infinite field can be written as a direct sum of two recursively enumerable. infinite dimensional subspaces whose domains are recursive sets of Godel numbers. This shows that the most obvious attempt to effectivize the theory of vector spaces by considering spaces whose domains are recursive sets (and whose basic operations are recursive functions) leads to a weak notion. The stronger, more reasonable notion of recursive space, which is due to METAXIDES and NERODE, namely a space which is complemented in 9 ( V _ ) gives much more familiar results in most cases (see [Sl, 1121, and [18]), however we show below that there are two recursive spaces whose direct sum is not a recursive space. Thus the lattice 9 ( V , ) is genuinely a new object whose theory is quite different from that of 8.The second type of result we have is that of splitting theorems (say A is split into B and C if A = B @ C where A , B, and C are vector spaces). The splitting theorems proven about & by SACKS and FRIEDBERG are recalled and the definitions of splitting are generalized to 9 ( V v , ) . In section 2, a special case of SACK'S splitting theorem is proven to illustrate the close recursion-theoretic relationship possible between r.e. spaces l ) This resea...

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Section: Vm E Ui + a 5 ( J U A ) * ( U I \ A ) + Amentioning

confidence: 75%

Direct Summands of Recursively Enumerable Vector Spaces

Retzlaff

1979

Mathematical Logic Qtrly

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“…Kalantari and Retzlaff [11] defined a space V ∈ L(V ∞ ) to be supermaximal if the dimension of V ∞ /V is infinite, and for every recursively enumerable space W ⊇ V , either W = V ∞ or W/V is finite-dimensional. Furthermore, for a natural number k 0, Kalantari and Retzlaff [11] introduced the concept of a k-thin space and showed its existence.…”

Section: Learnability and Types Of Quotient Spacesmentioning

confidence: 99%

On the learnability of vector spaces

Harizanov

Stephan

2007

Journal of Computer and System Sciences

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The central topic of the paper is the learnability of the recursively enumerable subspaces of V ∞ /V , where V ∞ is the standard recursive vector space over the rationals with (countably) infinite dimension and V is a given recursively enumerable subspace of V ∞ . It is shown that certain types of vector spaces can be characterized in terms of learnability properties: V ∞ /V is behaviourally correct learnable from text iff V is finite-dimensional, V ∞ /V is behaviourally correct learnable from switching the type of information iff V is finite-dimensional, 0-thin or 1-thin. On the other hand, learnability from an informant does not correspond to similar algebraic properties of a given space. There are 0-thin spaces W 1 and W 2 such that W 1 is not explanatorily learnable from an informant, and the infinite product (W 1 ) ∞ is not behaviourally correct learnable from an informant, while both W 2 and the infinite product (W 2 ) ∞ are explanatorily learnable from an informant.

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“…Let k be a natural number, possibly 0. In [14], Kalantari and Retzlaff introduced the following notion of a k-thin space. (i) The class L(V ) is BC-learnable from switching by an algorithmic learner.…”

Section: Characterizing Swbc-learnable Classes Of Vector Spacesmentioning

confidence: 99%

Inductive Inference Systems for Learning Classes of Algorithmically Generated Sets and Structures

Harizanov

2007

Induction, Algorithmic Learning Theory, and Philosophy

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Abstract. Computability theorists have extensively studied sets whose elements can be enumerated by Turing machines. These sets, also called computably enumerable sets, can be identified with their Gödel codes. Although each Turing machine has a unique Gödel code, different Turing machines can enumerate the same set. Thus, knowing a computably enumerable set means knowing one of its infinitely many Gödel codes. In the approach to learning theory stemming from E.M. Gold's seminal paper [9], an inductive inference learner for a computably enumerable set A is a system or a device, usually algorithmic, which when successively (one by one) fed data for A outputs a sequence of Gödel codes (one by one) that at certain point stabilize at codes correct for A. The convergence is called semantic or behaviorally correct, unless the same code for A is eventually output, in which case it is also called syntactic or explanatory. There are classes of sets that are semantically inferable, but not syntactically inferable.Here, we are also concerned with generalizing inductive inference from sets, which are collections of distinct elements that are mutually independent, to mathematical structures in which various elements may be interrelated. This study was recently initiated by F. Stephan and Yu. 30

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Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces

Cited by 40 publications

References 6 publications

Direct Summands of Recursively Enumerable Vector Spaces

Direct Summands of Recursively Enumerable Vector Spaces

On the learnability of vector spaces

Inductive Inference Systems for Learning Classes of Algorithmically Generated Sets and Structures

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