Enumeration reducibility, nondeterministic computations and relative computability of partial functions

1990 ◽  
pp. 57-110 ◽  
Author(s):  
S. Barry Cooper
Keyword(s):  
Partial Functions ◽  
Enumeration Reducibility

A survey of partial degrees

1975 ◽  
Vol 40 (2) ◽  
pp. 130-140 ◽  
Author(s):  
Leonard P. Sasso
Keyword(s):  
Equivalence Classes ◽  
Turing Machines ◽  
Systems Of Equations ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Upper Semilattice ◽  
Enumeration Reducibility ◽  
Turing Reducibility ◽  
Recursive Operators ◽  
Relative Enumerability

Partial degrees are equivalence classes of partial natural number functions under some suitable extension of relative recursiveness to partial functions. The usual definitions of relative recursiveness, equivalent in the context of total functions, are distinct when extended to partial functions. The purpose of this paper is to compare the upper semilattice structures of the resulting degrees.Relative partial recursiveness of partial functions was first introduced in Kleene [2] as an extension of the definition by means of systems of equations of relative recursiveness of total functions. Kleene's relative partial recursiveness is equivalent to the relation between the graphs of partial functions induced by Rogers' [10] relation of relative enumerability (called enumeration reducibility) between sets. The resulting degrees are hence called enumeration degrees. In [2] Davis introduces completely computable or compact functionals of partial functions and uses these to define relative partial recursiveness of partial functions. Davis' functionals are equivalent to the recursive operators introduced in Rogers [10] where a theorem of Myhill and Shepherdson is used to show that the resulting reducibility, here called weak Turing reducibility, is stronger than (i.e., implies, but is not implied by) enumeration reducibility. As in Davis [2], relative recursiveness of total functions with range ⊆{0, 1} may be defined by means of Turing machines with oracles or equivalently as the closure of initial functions under composition, primitive re-cursion, and minimalization (i.e., relative μ-recursiveness). Extending either of these definitions yields a relation between partial functions, here called Turing reducibility, which is stronger still.

Jumps of quasi-minimal enumeration degrees

1985 ◽  
Vol 50 (3) ◽  
pp. 839-848 ◽  
Author(s):  
Kevin McEvoy
Keyword(s):  
Turing Degree ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Enumeration Degree ◽  
Enumeration Reducibility ◽  
Degree Structure ◽  
Usual Topology ◽  
Definition Of ◽  
Second Category ◽  
Degrees Of Unsolvability

Enumeration reducibility is a reducibility between sets of natural numbers defined as follows: A is enumeration reducible to B if there is some effective operation on enumerations which when given any enumeration of B will produce an enumeration of A. One reason for interest in this reducibility is that it presents us with a natural reducibility between partial functions whose degree structure can be seen to extend the structure of the Turing degrees of unsolvability. In [7] Friedberg and Rogers gave a precise definition of enumeration reducibility, and in [12] Rogers presented a theorem of Medvedev [10] on the existence of what Case [1] was to call quasi-minimal degrees. Myhill [11] also defined this reducibility and proved that the class of quasi-minimal degrees is of second category in the usual topology. As Gutteridge [8] has shown that there are no minimal enumeration degrees (see Cooper [3]), the quasi-minimal degrees are very much of interest in the study of the structure of the enumeration degrees. In this paper we define a jump operator on the enumeration degrees which was introduced by Cooper [4], and show that every complete enumeration degree is the jump of a quasi-minimal degree. We also extend the notion of a high Turing degree to the enumeration degrees and construct a “high” quasi-minimal enumeration degree—a result which contrasts with Cooper's result in [2] that a high Turing degree cannot be minimal. Finally, we use the Sacks' Jump Theorem to characterise the jumps of the co-r.e. enumeration degrees.

Methods and clinical applications in nuclear cardiology: a position statement

2002 ◽  
Vol 41 (01) ◽  
pp. 3-13 ◽  
Author(s):  
M. Schäfers
Keyword(s):  
Nuclear Cardiology ◽  
Clinical Medicine ◽  
Imaging Techniques ◽  
Cost Savings ◽  
Cost Benefit ◽  
Position Statement ◽  
Electron Beam Tomography ◽  
Partial Functions ◽  
Cost Benefit Ratio ◽  
Medicine Today

SummaryNuclear cardiological procedures have paved the way for non-invasive diagnostics of various partial functions of the heart. Many of these functions cannot be visualised for diagnosis by any other method (e. g. innervation). These techniques supplement morphological diagnosis with regard to treatment planning and monitoring. Furthermore, they possess considerable prognostic relevance, an increasingly important issue in clinical medicine today, not least in view of the cost-benefit ratio.Our current understanding shows that effective, targeted nuclear cardiology diagnosis – in particular for high-risk patients – can contribute toward cost savings while improving the quality of diagnostic and therapeutic measures.In the future, nuclear cardiology will have to withstand mounting competition from other imaging techniques (magnetic resonance imaging, electron beam tomography, multislice computed tomography). The continuing development of these methods increasingly enables measurement of functional aspects of the heart. Nuclear radiology methods will probably develop in the direction of molecular imaging.

On the action of the implicative closure operator on the set of partial functions of the multivalued logic

2021 ◽  
Vol 31 (3) ◽  
pp. 155-164
Author(s):  
Sergey S. Marchenkov
Keyword(s):  
Closure Operator ◽  
Multivalued Logic ◽  
Partial Functions ◽  
Logical Implication

Abstract On the set P k ∗ $\begin{array}{} \displaystyle P_k^* \end{array}$ of partial functions of the k-valued logic, we consider the implicative closure operator, which is the extension of the parametric closure operator via the logical implication. It is proved that, for any k ⩾ 2, the number of implicative closed classes in P k ∗ $\begin{array}{} \displaystyle P_k^* \end{array}$ is finite. For any k ⩾ 2, in P k ∗ $\begin{array}{} \displaystyle P_k^* \end{array}$ two series of implicative closed classes are defined. We show that these two series exhaust all implicative precomplete classes. We also identify all 8 atoms of the lattice of implicative closed classes in P 3 ∗ $\begin{array}{} \displaystyle P_3^* \end{array}$ .

scholarly journals Quotienting the delay monad by weak bisimilarity

2017 ◽  
Vol 29 (1) ◽  
pp. 67-92 ◽  
Author(s):  
JAMES CHAPMAN ◽  
TARMO UUSTALU ◽  
NICCOLÒ VELTRI
Keyword(s):  
Computer Science ◽  
Equivalence Relation ◽  
Type Theory ◽  
Partial Functions ◽  
Homotopy Type Theory ◽  
Inductive Type ◽  
Alternative Approach ◽  
The One ◽  
Axiom Of Countable Choice

The delay datatype was introduced by Capretta (Logical Methods in Computer Science, 1(2), article 1, 2005) as a means to deal with partial functions (as in computability theory) in Martin-Löf type theory. The delay datatype is a monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay datatype quotiented by weak bisimilarity is still a monad–a constructive alternative to the maybe monad. In this paper, we consider the alternative approach of Hofmann (Extensional Constructs in Intensional Type Theory, Springer, London, 1997) of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the (semi-classical) axiom of countable choice. With the aid of these principles, we also prove that the quotiented delay datatype delivers free ω-complete pointed partial orders (ωcppos).Altenkirch et al. (Lecture Notes in Computer Science, vol. 10203, Springer, Heidelberg, 534–549, 2017) demonstrated that, in homotopy type theory, a certain higher inductive–inductive type is the free ωcppo on a type X essentially by definition; this allowed them to obtain a monad of free ωcppos without recourse to a choice principle. We notice that, by a similar construction, a simpler ordinary higher inductive type gives the free countably complete join semilattice on the unit type 1. This type suffices for constructing a monad, which is isomorphic to the one of Altenkirch et al. We have fully formalized our results in the Agda dependently typed programming language.

scholarly journals Simultaneously vanishing higher derived limits

2021 ◽  
Vol 9 ◽  
Author(s):  
Jeffrey Bergfalk ◽  
Chris Lambie-Hanson
Keyword(s):  
Euclidean Space ◽  
Metric Spaces ◽  
Inverse System ◽  
Finite Support ◽  
Partial Functions ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Strong Homology ◽  
Finite Support Iteration ◽  
Weakly Compact Cardinal

Abstract In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system $\textbf {A}$ with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that $\lim ^n\textbf {A}$ (the nth derived limit of $\textbf {A}$ ) vanishes for every $n>0$ . Since that time, the question of whether it is consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for every $n>0$ has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces. We show that assuming the existence of a weakly compact cardinal, it is indeed consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for all $n>0$ . We show this via a finite-support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration, a condition equivalent to $\lim ^n\textbf {A}=0$ will hold for each $n>0$ . This condition is of interest in its own right; namely, it is the triviality of every coherent n-dimensional family of certain specified sorts of partial functions $\mathbb {N}^2\to \mathbb {Z}$ which are indexed in turn by n-tuples of functions $f:\mathbb {N}\to \mathbb {N}$ . The triviality and coherence in question here generalise the classical and well-studied case of $n=1$ .

Arterial hypertension in patients with active crystal formation

1999 ◽  
Vol 80 (6) ◽  
pp. 415-416
Author(s):  
A. N. Maksudova ◽  
I. G. Salihov ◽  
O. N. Sigitova
Keyword(s):  
Arterial Hypertension ◽  
Interstitial Nephritis ◽  
Purine Metabolism ◽  
Crystal Formation ◽  
Partial Functions ◽  
Ultrasonic Examination ◽  
Active Crystal ◽  
Fast Development

The results of examination of 70 patients with urinary syndrome as oxaliccalcic and uratic crystalluria and in the projection of pelvic system by ultrasonic examination data were analyzed. The study of partial functions of kidneys, purine and oxalic metabolism was performed to estimate hypertension syndrome in patients with dysmetabolic disorders. The comparison of patients with arterial hypertension (15) and normotonia (55) showed the changes in the first group as fast development of dysmetabolic disorders and significant disorder of purine metabolism. The data obtained show the relation between hypertension and interstitial nephritis activity.

scholarly journals Realization of CyberRail Partial Functions in Actual Environment

2005 ◽  
Vol 46 (1) ◽  
pp. 18-22 ◽  
Author(s):  
Takahiko OGINO ◽  
Ryuji TSUCHIYA
Keyword(s):  
Partial Functions ◽  
Actual Environment
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