A splitting theorem for simple Π11 Sets

1971 ◽  
Vol 36 (3) ◽  
pp. 433-438 ◽  
Author(s):  
James C. Owings
Keyword(s):  
Analogous Result ◽  
Recursion Theory ◽  
The Other ◽  
Splitting Theorem ◽  
One To One ◽  
Priority Argument ◽  
Recursive Set

As was first mentioned in [3, §5], if A is any – set, A is the union of two disjoint – sets B(0), B(1). In metarecursion theory this is proven as follows. Let ƒ be a one-to-one metarecursive function whose range is A, let R be an unbounded metarecursive set whose complement is also unbounded, and set B(0) = f(R), B(1) = f(). The corresponding fact of ordinary recursion theory, namely that any r.e. but not recursive set can be split into two other such sets, was proved by Friedberg [2, Theorem 1], using a clever priority argument. Sacks [7, Corollary 2] then showed that any r.e. but not recursive set is the union of two disjoint r.e. sets neither of which was recursive in the other, a much stronger result. In this paper we attempt to prove the analogous result for – sets A, but succeed only in the case A is simple; i.e., the complement of A contains no infinite subset. As a corollary we show the metadegrees are dense, a fact already announced by Sacks [8, Corollary 1], but only proven by him for nonzero metadegrees.

The thickness lemma from P− + IΣ1 + ¬BΣ2

1995 ◽  
Vol 60 (2) ◽  
pp. 505-511
Author(s):  
Yue Yang
Keyword(s):  
General Theorem ◽  
Recursion Theory ◽  
The Other ◽  
Splitting Theorem ◽  
Natural Question ◽  
Other Hand ◽  
Turing Reducibility ◽  
Induction Scheme

Let P− denote the Peano axioms minus the induction scheme. Let IΣn, (I∏n), BΣn (B∏n), LΣn (L∏n denote the induction scheme, the collection scheme, and the least number principle for Σn-(∏n-) formulas respectively. Paris and Kirby [3] studied the relative proof-theoretic strengths of those schemes. The general theorem states that IΣn, I∏n, LΣn, and L∏n are equivalent; IΣn implies BΣn implies IΣn–1; but not conversely.In recent years, people have been interested in doing recursion theory on fragments of arithmetic. One of the purposes of this study is to understand the priority methods. Much work has been done in this area. For example, M. Mytilinaios [5] showed that the Sacks splitting theorem can be proven in P− + IΣ1. Later, J. Mourad showed that the Sacks splitting theorem is indeed equivalent to IΣ1 [4]. M. Groszek and M. Mytilinaios [1] showed that P− + IΣ2 is sufficient to prove the existence of a high incomplete r.e. set. On the other hand, M. Mytilinaios and T. Slaman [6] showed that P− + IΣ1 is too weak to prove the existence of such a set. A natural question to ask is if the existence of such a set implies IΣ2. In this paper, we will show the answer is negative by constructing a model of P− + IΣ1 + ¬BΣ2 which has a high incomplete r.e. set. Notice that, as shown by M. Groszek and T. Slaman in [2], P− + IΣ1 is too weak to show the transitivity of weak Turing reducibility on Σ2-sets.

Σn sets which are Δn-incomparable (uniformly)

1974 ◽  
Vol 39 (2) ◽  
pp. 295-304 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Set Theory ◽  
Recursion Theory ◽  
The Other ◽  
Direct Generalization ◽  
Natural Notion ◽  
A Priority ◽  
Definition Of ◽  
Admissible Ordinals ◽  
Priority Argument

In this paper we will present an application of generalized recursion theory to (noncombinatorial) set theory. More precisely we will combine a priority argument in α-recursion theory with a forcing construction to prove a theorem about the interdefinability of certain subsets of admissible ordinals.Our investigation was prompted by G. Sacks and S. Simpson asking [6] if it is obvious that there are, for each Σn-admissible α, Σn (over Lα) subsets of α which are Δn-incomparable. If one understands “B is Δn in C” to mean that there are Σn/Lα reduction procedures which put out B and when one feeds in C, then the answer is an unqualified “yes.” In this sense “Δn in” is a direct generalization of “α-recursive in” (replace Σ1 by Σn in the definition) and so amenable to the methods of [7, §§3, 5]. Indeed one simply chooses a complete Σn−1 set A and mimics the construction of [6] as modified in [7, §5] to produce two α-A-r.e. sets B and C neither of which is α-A-recursive in the other. By the remarks on translation [7, §3] this will immediately give the desired result for this definition of “Δn in.”There is, however, the more obvious and natural notion of “Δn in” to be considered: B is Δn in C iff there are Σn and Πn formulas of ⟨Lα, C⟩ which define B.

α-degrees of α-theories

1972 ◽  
Vol 37 (4) ◽  
pp. 677-682 ◽  
Author(s):  
George Metakides
Keyword(s):  
Initial Segment ◽  
Recursion Theory ◽  
The Other ◽  
Infinite Length ◽  
Infinitary Logic ◽  
Admissible Set ◽  
Recursively Enumerable ◽  
Limit Ordinal ◽  
The Subject ◽  
Further Development

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

Development of a Single-Code/Default Coding Strategy in Pigeons

2000 ◽  
Vol 11 (3) ◽  
pp. 261-264 ◽  
Author(s):  
Tricia S. Clement ◽  
Thomas R. Zentall
Keyword(s):  
Conditional Discrimination ◽  
The Other ◽  
Matching Accuracy ◽  
Efficient Coding ◽  
Coding Strategy ◽  
One To One ◽  
Retention Intervals ◽  
The Many ◽  
The One ◽  
Better Than

We tested the hypothesis that pigeons could use a cognitively efficient coding strategy by training them on a conditional discrimination (delayed symbolic matching) in which one alternative was correct following the presentation of one sample (one-to-one), whereas the other alternative was correct following the presentation of any one of four other samples (many-to-one). When retention intervals of different durations were inserted between the offset of the sample and the onset of the choice stimuli, divergent retention functions were found. With increasing retention interval, matching accuracy on trials involving any of the many-to-one samples was increasingly better than matching accuracy on trials involving the one-to-one sample. Furthermore, following this test, pigeons treated a novel sample as if it had been one of the many-to-one samples. The data suggest that rather than learning each of the five sample-comparison associations independently, the pigeons developed a cognitively efficient single-code/default coding strategy.

On the constitution of the resins. Part I

1843 ◽  
Vol 4 ◽  
pp. 135-136
Keyword(s):  
The Other ◽  
Chemical Investigation ◽  
Prolonged Action ◽  
Acid Properties ◽  
Relative Composition ◽  
General Investigation ◽  
Theoretical View ◽  
Common Principle ◽  
One To One ◽  
The One

The object of the general investigation, of which the commencement is given in this paper, is to determine the relative composition of the various resins which occur in nature, and to trace the analogies they exhibit in their constitution; and also to ascertain how far they may be regarded as being derived from one common principle, and whether they admit of being all represented by one or more general formulæ. The chemical investigation of the resin of mastic shows that this substance consists of two resins; the one soluble, and acid; the other insoluble, and having no acid properties. The formulæ expressing the analysis of each of these are given by the author. He also shows that a series of analyses may be obtained which do not indicate the true constitution of a resin. The soluble resin, when exposed to the prolonged action of a heat exceeding 300° Fahr. is partly converted into a resin containing three, and partly into one containing five equivalent parts of oxygen, the proportion of carbon remaining constant. The same resin combines with bases, so as to form four series of salts; which, in the case of oxide of lead, consist of equivalents of resin and of oxide in the proportions, respectively, of two to one; three to two; one to one; and one to two. This soluble resin in combining with bases does not part with any of its oxygen; but if any change takes place in its constitution, it consists in the hydrogen being replaced by an equivalent proportion of a metal; and formulæ are given representing the salts of lead on this theoretical view. By boiling the resin in contact with ammonia and nitrate of silver, or perhaps with nitrate of ammonia, it is converted into a resin which forms a bisalt with oxide of silver, in winch there is also an apparent replacement of hydrogen by silver .

Albertus (1573) and Ölinger (1574)

2001 ◽  
Vol 28 (1-2) ◽  
pp. 7-38
Author(s):  
Nicola McLelland
Keyword(s):  
The Other ◽  
German Language ◽  
Form Function ◽  
Other Hand ◽  
Structural Principles ◽  
One To One ◽  
Function Relationship ◽  
Creative Construction

Summary This article adapts Linn’s ‘stylistics of standardization’ concept, which Linn (1998) has used to compare Norwegian and Faroese grammarians, to look at grammaticization processes in the first two grammars of German (Albertus 1573, Ölinger 1574). While both are clearly indebted to traditional Latin grammar and humanist ideals, these two grammars differ interestingly in the picture of the language that emerges from their metalanguage and structural principles. In his reflection on the language, his structuring and naming of linguistic phenomena and his attitudes to variation, Ölinger is the practical pedagogue, who imposes systematicity and aims for a one-to-one form-function relationship. Albertus on the other hand, though he too envisages his grammar being used for learning German, has a more cultural patriotic motivation, celebrating the richness and variety of German, worthy to be ranked alongside Latin, Greek and Hebrew. Albertus and Ölinger thus come up with quite different versions of the (as yet arguably non-existent) High German language. Each grammar yields a different subset of possible forms, reminding us that grammar-writing is always a task of creative construction.

scholarly journals On the endomorphism near-ring of a free group

1980 ◽  
Vol 23 (1) ◽  
pp. 103-121 ◽  
Author(s):  
R. Warwick Zeamer
Keyword(s):  
Wreath Product ◽  
Free Group ◽  
The Other ◽  
Finite Support ◽  
Finite Product ◽  
Prime Factorization ◽  
Infinite Rank ◽  
One To One ◽  
Almost All ◽  
Countably Infinite

Suppose F is an additively written free group of countably infinite rank with basis T and let E = End(F). If we add endomorphisms pointwise on T and multiply them by map composition, E becomes a near-ring. In her paper “On Varieties of Groups and their Associated Near Rings” Hanna Neumann studied the sub-near-ring of E consisting of the endomorphisms of F of finite support, that is, those endomorphisms taking almost all of the elements of T to zero. She called this near-ring Φω. Now it happens that the ideals of Φω are in one to one correspondence with varieties of groups. Moreover this correspondence is a monoid isomorphism where the ideals of φω are multiplied pointwise. The aim of Neumann's paper was to use this isomorphism to show that any variety can be written uniquely as a finite product of primes, and it was in this near-ring theoretic context that this problem was first raised. She succeeded in showing that the left cancellation law holds for varieties (namely, U(V) = U′(V) implies U = U′) and that any variety can be written as a finite product of primes. The other cancellation law proved intractable. Later, unique prime factorization of varieties was proved by Neumann, Neumann and Neumann, in (7). A concise proof using these same wreath product techniques was also given in H. Neumann's book (6). These proofs, however, bear no relation to the original near-ring theoretic statement of the problem.

Identifying logical dummy activities in an activity-on-arrow network

2008 ◽  
Vol 35 (7) ◽  
pp. 739-743
Author(s):  
Kyunghwan Kim
Keyword(s):  
Construction Projects ◽  
Heuristic Method ◽  
The Other ◽  
Multiple Relationships ◽  
Critical Problem ◽  
Stepwise Manner ◽  
The Creation ◽  
One To One

The activity-on-arrow (AOA) method can be used in construction projects depending on the project condition, although it is relatively inefficient as compared to the activity-on-node (AON) method. The most critical problem of the AOA method that is yet to be resolved is the application of a logical dummy activity (dummy). This paper presents a heuristic method that can be used for systematically identifying and applying logical dummies during the creation of an AOA network based on predefined activity dependencies. The heuristic method first identifies logical dummies of one-to-one relationships, and then detects the other multiple relationships in a stepwise manner to remove the redundancies therein.

scholarly journals Polynomial Solution to the Position Analysis of Two Assur Kinematic Chains With Four Loops and the Same Topology

2009 ◽  
Vol 1 (2) ◽  
Author(s):  
Júlia Borràs ◽  
Raffaele Di Gregorio
Keyword(s):  
Polynomial Solution ◽  
Large Family ◽  
The Other ◽  
End Effector ◽  
Closed Loops ◽  
Kinematic Chains ◽  
Position Analysis ◽  
One To One ◽  
Actuated Joints

The direct position analysis (DPA) of a manipulator is the computation of the end-effector poses (positions and orientations) compatible with assigned values of the actuated-joint variables. Assigning the actuated-joint variables corresponds to considering the actuated joints locked, which makes the manipulator a structure. The solutions of the DPA of a manipulator one to one correspond to the assembly modes of the structure that is generated by locking the actuated-joint variables of that manipulator. Determining the assembly modes of a structure means solving the DPA of a large family of manipulators since the same structure can be generated from different manipulators. This paper provides an algorithm that determines all the assembly modes of two structures with the same topology that are generated from two families of mechanisms: one planar and the other spherical. The topology of these structures is constituted of nine links (one quaternary link, four ternary links, and four binary links) connected through 12 revolute pairs to form four closed loops.

Minimal α-recursion theoretic degrees

1973 ◽  
Vol 38 (1) ◽  
pp. 18-28 ◽  
Author(s):  
John M. MacIntyre
Keyword(s):  
Recursive Function ◽  
Regular Cardinal ◽  
Recursion Theory ◽  
Minimal Degree ◽  
The Other ◽  
Ordinal Numbers ◽  
Definition Of

This paper investigates the problem of extending the recursion theoretic construction of a minimal degree to the Kripke [2]-Platek [5] recursion theory on the ordinals less than an admissible ordinal α, a theory derived from the Takeuti [11] notion of a recursive function on the ordinal numbers. As noted in Sacks [7] when one generalizes the recursion theoretic definition of relative recursiveness to α-recursion theory for α > ω the two usual definitions give rise to two different notions of reducibility. We will show that whenever α is either a countable admissible or a regular cardinal of the constructible universe there is a subset of α whose degree is minimal for both notions of reducibility. The result is an excellent example of a theorem of ordinary recursion theory obtainable via two different constructions, one of which generalizes, the other of which does not. The construction which cannot be lifted to α-recursion theory is that of Spector [10]. We sketch the reasons for this in §3.

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