Recursive Equivalence Types Research Articles

Isols and maximal intersecting classes

AbstractIn transfinite arithmetic 2n is defined as the cardinality of the family of all subsets of some set v with cardinality n. However, in the arithmetic of recursive equivalence types (RETs) 2N is defined as the RET of the family of all finite subsets of some set v of nonnegative integers with RET N. Suppose v is a nonempty set. S is a class over v, if S consists of finite subsets of v and has v as its union. Such a class is an intersecting class (IC) over v, if every two members of S have a nonempty intersection. An IC over v is called a maximal IC (MIC), if it is not properly included in any IC over v. It is known and readily proved that every MIC over a finite set v of cardinality n ≥ 1 has cardinality 2n‐1. In order to generalize this result we introduce the notion of an ω‐MIC over v. This is an effective analogue ot the notion of an MIC over v such that a class over a finite set v is an ω‐MIC iff it is an MIC. We then prove that every ω‐MIC over an isolated set v of RET N ≥ 1 has RET 2N‐1. This is a generalization, for while there only are χ0 finite sets, there are ϰ isolated sets, where c denotes the cardinality of the continuum, namely all the finite sets and the c immune sets. MSC: 03D50.

Polynomial-time analogues of isolatedness

Recently, Nerode and Remmel have developed a polynomial-time version of the theory of recursive equivalence types and have defined two analogues of isolatedness for that setting. This paper examines various properties of those two analogues and investigates their relationship to additive cancellability.

Myhill's work in recursion theory

In this paper we discuss the following contributions to recursion theory made by John Myhill: (1) two sets are recursively isomorphic iff they are one-one equivalent; (2) two sets are recursively isomorphic iff they are recursively equivalent and their complements are also recursively equivalent; (3) every two creative sets are recursively isomorphic; (4) the recursive analogue of the Cantor–Bernstein theorem; (5) the notion of a combinatorial function and its use in the theory of recursive equivalence types.

The inclusion-exclusion principle for finitely many isolated sets

AbstractA nonnegative integer is called a number, a collection of numbers a set and a collection of sets a class. We write ε for the set of all numbers, o for the empty set, N(α) for the cardinality of α, ⊂ for inclusion and ⊂+ for proper inclusion. Let α, β1,…, βk be subsets of some set υ. Then α′ stands for υ−α and β1 … βk for β1 ∩ … ∩ βk. For subsets α1, …, αr of υ we write:Note that α0 = υ, hence s0 = N(υ). If the set υ is finite, the classical inclusion-exclusion principle (abbreviated IEP) statesIn this paper we generalize (a) and(b) to the case where α1, …, αr are subsets of some countable but isolated set υ. Then the role of the cardinality N(α) of the set α is played by the RET (recursive equivalence type) Req α of α. These generalizations of (a) and (b) are proved in §3. Since they involve recursive distinctness, this notion is discussed in §2. In §4 we consider a natural extension of “the sum of the elements of a finite set σ” to the case where σ is countable. §5 deals with valuations, i.e., certain mappings μ from classes of isolated sets into the collection Λ of all isols which permit us to further generalize IEP by substituting μ(α) for Req α.

The inclusion-exclusion principle for finitely many isolated sets

AbstractA nonnegative integer is called a number, a collection of numbers a set and a collection of sets a class. We write ε for the set of all numbers, o for the empty set, N(α) for the cardinality of α, ⊂ for inclusion and ⊂+ for proper inclusion. Let α, β1,…, βk be subsets of some set υ. Then α′ stands for υ−α and β1 … βk for β1 ∩ … ∩ βk. For subsets α1, …, αr of υ we write:Note that α0 = υ, hence s0 = N(υ). If the set υ is finite, the classical inclusion-exclusion principle (abbreviated IEP) statesIn this paper we generalize (a) and(b) to the case where α1, …, αr are subsets of some countable but isolated set υ. Then the role of the cardinality N(α) of the set α is played by the RET (recursive equivalence type) Req α of α. These generalizations of (a) and (b) are proved in §3. Since they involve recursive distinctness, this notion is discussed in §2. In §4 we consider a natural extension of “the sum of the elements of a finite set σ” to the case where σ is countable. §5 deals with valuations, i.e., certain mappings μ from classes of isolated sets into the collection Λ of all isols which permit us to further generalize IEP by substituting μ(α) for Req α.

Sound, totally sound, and unsound recursive equivalence types

An independent subset I of V ∞ is called sound is I is contained in a r.e. independent set. If not, we shall call I unsound. We call an RET A totally sound if every independent set in A is sound. Clearly every RET containing an r.e. set is totally sound, and it had been suggested that the converse held: viz, every totally sound RET contained a recursive set. However, we show that there are 2 χ 0 sets { A π } such that if B ⩽ m A π then RET( B) is totally sound. In the co-r.e. case it is shown that if A is co-r.e. nonrecursive and nonisolic, then RET( A) is not totally sound. Indeed, RET( A) contains an independent set which is a basis of a subspace (of codimension 1 in V ∞), no basis of which is sound. This result is deduced from a construction of a new type of r.e. subspace of V ∞. That is, we show that if R is a fully co-r.e. nonrecursive subspace of V ∞ there existse an r.e. subspace V sich that V⊕ R = V ∞ and V has the property that if W is an r.e. subspace with W⊃ V, then dim( V ∞/ W) < ∞ implies W = V ∞. On the other hand , the co-simple isols provide a nice dichotomy. First, it is shown that (in each high r.e. degree) there is a co-simple totally sound RET. Second it is shown that if a is any nonzero r.e. degree, there exists a nonzero r.e. degree ⩽ T a such that b bounds no nontrivial totally sound co-r.e. RET. That is, if B is r.e. with Turing degree b and C is co-r.e. with C⩽ T B, then either (i) C is recursive or (ii) RET( C) is not totally sound. Finally, we extend our results to general algebraic settings; first to a class of Steinitz closure systems, and later to a general model-theoretic setting (without dependence).

Recursive equivalence types and octahedra

AbstractLet the word “graph” be used in the sense of a countable, connected, simple graph with at least one vertex. We write Qn and Ocn for the graphs associated with the n-cube Qn and the n-octahedron Ocn respectively. In a previous paper (Dekker, 1981) we generalized Qn and Qn to a graph QN and a cube QN, for any nonzero recursive equivalence type N. In the present paper we do the same for Ocn and Ocn. We also examine the nature of the duality between QN and OcN, in case N is an infinite isol. There are c RETs, c denoting the cardinality of the continuum.

An introduction to ω-extensions of ω-groups

Let ε stand for the set of nonnegative integers (numbers), V for the class of all subcollections of ε(sets), Λ for the set of isols, ΛR for the set of regressive isols, and for the set of mappings from a subset of ε into ε (functions). If ƒ is a function we write δƒ and ρƒ for its domain and range, respectively. We denote the inclusion relation by ⊃ and proper inclusion by ⊊. The sets α and β are recursively equivalent [written: α ≃ β], if δƒ = α and ρƒ = β for some function ƒ with a one-to-one partial recursive extension. We denote the recursive equivalence type of a set α, {σ ∈ V ∣ σ ≃ α} by Req(α). The reader is assumed to be familiar with the contents of [1], [2], [3], and [6].The concept of an ω-group was introduced in [6], and that of an ω-homomorphism in [1]. However, except for a few examples, very little is known about the structure of ω-groups. If G is an ω-group and Π is an ω-homomorphism, then it follows that K = Ker Π and H = Π(G) are ω-groups. The question arises that if we know the structure of K and H, then what can we say about the structure of G? In this paper we will begin the study of ω-extensions, which will give us a partial answer to this question.

Twilight graphs

AbstractThis paper deals primarily with countable, simple, connected graphs and the following two conditions which are trivially satisfied if the graphs are finite:(a) there is an edge-recognition algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to decide whether they are adjacent,(b) there is a shortest path algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to find a minimal path joining them.A graph G = ‹ν, η› with ν as set of vertices and η as set of edges is an α-graph if it satisfies (a); it is an ω-graph if it satisfies (b). G is called r.e. (isolic) if the sets ν and η are r.e. (isolated). While every ω-graph is an α-graph, the converse is false, even if G is r.e. or isolic. Several basic properties of finite graphs do not generalize to ω-graphs. For example, an ω-tree with more than one vertex need not have two pendant vertices, but may have only one or none, since it may be a 1-way or 2-way infinite path. Nevertheless, some elementary propositions for finite graphs G = ‹ν, η› with n = card(ν), e = card(η), do generalize to isolic ω-graphs, e.g., n − 1 ≤ e ≤ n(n − 1)/2. Let N and E be the recursive equivalence types of ν and η respectively. Then we have for an isolic α-tree G = ‹ν, η›: N = E + 1 iff G is an ω-tree. The theorem that every finite graph has a spanning tree has a natural, effective analogue for ω-graphs.The structural difference between isolic α-graphs and isolic ω-graphs will be illustrated by: (i) every infinite graph is isomorphic to some isolic α-graph; (ii) there is an infinite graph which is not isomorphic to any isolic ω-graph.An isolic α-graph is also called a twilight graph. There are c such graphs, c denoting the cardinality of the continuum.

Admissible sets and recursive equivalence types.

Admissible sets and recursive equivalence types.

Recursive equivalence types on recursive manifolds.

Recursive equivalence types on recursive manifolds.

Controlling the dependence degree of a recursively enumerable vector space

Early work combining recursion theory and algebra had (at least) two different sets of motivations. First the precise setting of recursion theory offered a chance to make formal classical concerns as to the effective or algorithmic nature of algebraic constructions. As an added benefit the formalization gives one the opportunity of proving that certain constructions cannot be done effectively even when the original data is presented in a recursive way. One important example of this sort of approach is the work of Frohlich and Shepardson [1955] in field theory. Another motivation for the introduction of recursion theory to algebra is given by Rabin [1960]. One hopes to mathematically enrich algebra by the additional structure provided by the notion of computability much as topological structure enriches group theory. Another example of this sort is provided in Dekker [1969] and [1971] where the added structure is that of recursive equivalence types. (This particular structural view culminates in the monograph of Crossley and Nerode [1974].)More recently there is the work of Metakides and Nerode [1975], [1977] which combines both approaches. Thus, for example, working with vector spaces they show in a very strong way that one cannot always effectively extend a given (even recursive) independent set to a basis for a (recursive) vector space.

Infinite sums and products of isol integers

A definition is provided of k-fold infinite sums and products of isol integers, and certain properties of such sums and products are proven. The definition is based on the concept of generalized recursive equivalence types, introduced at the beginning of this paper. The validity of the properties, connected with the canonical extensions of functions on isols, is established for arbitrary functions, and not only for general recursive functions, since the generalized recursive equivalence types permit their incorporation without the Nerode metatheorem in the proof of such properties.

Hereditarily universal sets

AbstractAn immune set is found such that the recursive equivalence type of its infinite subsets are universal in a very strong sense.

Recursive equivalence types of vector spaces

We consider subspaces of a vector spaceUF, which is countably infinite dimensional over a recursively enumerable fieldFwith recursive operations, where the operations inUFare also recursive, and where, of course,FandUFare sets of natural numbers. It is the object of this paper to investigate recursive equivalence types of such vector spaces and the ways in which their properties are analogous to and depend on properties of recursive equivalence types of sets.

A characterization of the orders of regressive ω-groups

Let IE, ∧ and ∧R denote the collections of all non-negative integers, isols and regressive isols respectively. An ω-group is a pair (α, p) where (1) α ⊆ E, (2) p(x, y) is a partial recursive group multiplication for α and (3) the function which maps each element of α to its inverse under p has a partial recursive extension. If G = (α, p) is an ω-group, we call the recursive equivalence type of a the RET or order of G (written o(G)). Let GR = {T∈∧R′T = o(G) for some ω-group G}. It follows from the version of the Lagrange Theorem given in [4] that ∧R − GR is non-empty and has cardinality c. In this paper we characterise the isols in GR as follows: A regressive isol T belongs to GR if and only if T∈E or T is infinite and there exist a regressive isol U ≦ T and a function an from E into E − {0} such that U ≦*an and T = ΠUan. (The “≦*” is denned in [2]). In presenting the proof of this result, we shall assume that the reader is familiar with either [3] or [4]. The proof that, given an and U ≦*an, a group of order ΠUan exists is based on the natural trick—one constructs a direct product of disjoint cyclic groups of order a0, a1,… indexed by elements of a set of RET U. The proof that any regressive group G has order of the form ΠUan is trivial for finite groups; the proof for infinite regressive groups is based upon the construction of an ascending chain of finite subgroups Gi of G such that and .

ω-homomorphisms and ω-groups

Let ε stand for the set of nonnegative integers (numbers), V for the class of all subcollections of ε (sets), Λ for the set of isols, and for the set of mappings from a subset of ε into ε (functions). I f is a function we write δf and ρf for its domain and range respectively. The relation of inclusion is denoted by ⊂ and that of proper inclusion by ⊊. The sets α and β are recursively equivalent [written: α ≃ β], if δf = α and ρf = β for some function f with a one-to-one partial recursive extension f. We denote the recursive equivalence type of α, {σ ∈ V ∣ ≃ α}, by Req(α). Also R stands for Req(ε), while ΛR denotes the collection of all regressive isols. The reader is assumed to be familiar with the contents of [1] and [6].

A universal embedding property of the RETs

Recursive equivalence types are an effective or recursive analogue of cardinal numbers. They were introduced by Dekker in the early 1950's. The richness of various theories related to the recursive equivalence types is demonstrated in this paper by showing that the theory of any countable relational structure can be embedded in or interpreted in these theories. A more complete summary is presented in the last paragraph of this section.Let E = {0,1, 2, …} be the natural numbers. If α ⊆ E, β ⊆ E, and there is a 1-1 partial recursive function f such that the image under f of α is β, α and β are called recursively equivalent (see [3]). The recursive equivalence type or RET of α, denoted 〈α〉, is the class of all β recursively equivalent to α. Addition of RETs is defined by 〈α〉 + 〈β〉 = 〈{2x ∣ x ∈ α} ∪ 〈{2x + 1 ∣ x ∈ β}〉. The partial ordering ≤ is defined on the RETs by A ≤ B iff (EC)(A + C = B). An RET, X, is called an isol if X ≠ X + 1 or, equivalently, if no representative of X is recursively equivalent to a proper subset of itself. The isols are thus the recursive analogue of the Dedekind-finite cardinals.

A universal embedding property of the RETs

Recursive equivalence types are an effective or recursive analogue of cardinal numbers. They were introduced by Dekker in the early 1950's. The richness of various theories related to the recursive equivalence types is demonstrated in this paper by showing that the theory of any countable relational structure can be embedded in or interpreted in these theories. A more complete summary is presented in the last paragraph of this section.Let E = {0,1, 2, …} be the natural numbers. If α ⊆ E, β ⊆ E, and there is a 1-1 partial recursive function f such that the image under f of α is β, α and β are called recursively equivalent (see [3]). The recursive equivalence type or RET of α, denoted 〈α〉, is the class of all β recursively equivalent to α. Addition of RETs is defined by 〈α〉 + 〈β〉 = 〈{2x ∣ x ∈ α} ∪ 〈{2x + 1 ∣ x ∈ β}〉. The partial ordering ≤ is defined on the RETs by A ≤ B iff (EC)(A + C = B). An RET, X, is called an isol if X ≠ X + 1 or, equivalently, if no representative of X is recursively equivalent to a proper subset of itself. The isols are thus the recursive analogue of the Dedekind-finite cardinals.

Recursive equivalence types and groups

Let ε and Λ denote the sets of all nonnegative integers and isols respectively. Let σ be any subset of ε. Following [6], we shall denote the recursive equivalence type of a by Req(σ) and write σ≃τ when Req(σ) = Req(τ).