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hierarchy of functions


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hierarchy of functions

hierarchy of functions

Hierarchical multi-label prediction of gene function

The redshift evolution of the mass function of cold gas in hierarchical galaxy formation models

The SWELLS survey – VI. Hierarchical inference of the initial mass functions of bulges and discs

Hierarchy and size distribution function of star formation regions in the spiral galaxy NGC 628

Faint galaxy counts as a function of morphological type in a hierarchical merger model

Exploring the Hierarchy of Mobility Performance in High-Functioning Older Women

Constant Hierarchic Patterns of Physical Functioning Across Seven Populations in Five Countries1

An Integrative Theory of Anterior Cingulate Cortex Function: Option Selection in Hierarchical Reinforcement Learning

P-receiver functions of anisotropic continental crust: a hierarchic catalogue of crustal models and azimuthal waveform patterns

The angular correlation function and hierarchical moments of ∼70 000 faint galaxies to R=23.5

Development of a Local Size Hierarchy Causes Regular Spacing of Trees in an Even-aged Abies Forest: Analyses Using Spatial Autocorrelation and the Mark Correlation Function

Developmental changes in shoot N dynamics of lucerne (Medicago sativa L.) in relation to leaf growth dynamics as a function of plant density and hierarchical position within the canopy

Developmental changes in shoot N dynamics of lucerne (Medicago sativa L.) in relation to leaf growth dynamics as a function of plant density and hierarchical position within the canopy

 

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A sequence of sets of functions F0, F1, F2,… with the property thatF0F1F2 ⊆ … (see subset). Typically the functions in F0 will include certain initial functions; the sets of functions F1, F2,… are normally defined by combining initial functions in some way.

F0F1F2 ⊆ …

Hierarchies of primitive recursive functions can be defined by letting Fi represent those functions that can be computed by programs containing at most i loops nested one within the other. thenFiFi+1for all integers i > 0. The union of all these sets includes all the primitive recursive functions and only those functions. Consequently the hierarchy is often called a subrecursive hierarchy. This same hierarchy can be expressed in a slightly different form, so resulting in the Grzegorczyk hierarchy.

FiFi+1

In an attempt to circumvent problems caused by recursion, Bertrand Russell invented a theory of types, which essentially imposed a hierarchy on the set of functions; functions at one level could be defined only in terms of functions at lower levels.

The study of hierarchies of functions dates from work of David Hilbert around 1926 on the foundations of mathematics. More recent interest stems from their applicability to computational complexity.

Subjects: Computing.


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