A continuous functional with noncollapsing hierarchy

In [5] S. S. Wainer introduces a hierarchy for arbitrary type-2-functionals. Given F, he defines a set of ordinal notations OF, and for each a ∈ OF, a function fa recursive in F and an ordinal ∣a∣F < For any f recursive in F there is an a ∈ OF such that f is primitive recursive in fa.Let ρF be the least ordinal α such that for any f recursive in F there is an α ∈ OF with ∣a∣F ≤ α such that f is primitive recursive in fa. If ρF < the hierarchy breaks down. In Bergstra and Wainer [2] ρF is described as “the real ordinal of the 1-section of F”.Using standard methods (originally due to Kleene) one may prove that if F is normal, then ρF = Feferman has proved that if F is recursive, then ρF = ω2.Let 1-section (F) = l-sc(F) = {f; f is recursive in F} where f is a total object of type 1.Grilliot [4] proved that F ↾ 1-sc(F) is continuous if and only if F is not normal.Let h be an associate for a given functional F, and assume that h is recursive in the jump of an element of 1-sc(F).

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Of ants and academics: The computational power of external representation

Behavioral and Brain Sciences ◽

10.1017/s0140525x97370022 ◽

1997 ◽

Vol 20 (1) ◽

pp. 78-79

Author(s):

Jon Oberlander

Keyword(s):

Real World ◽

External Representation ◽

External Representations ◽

Computational Power ◽

The Real

Clark & Thornton speculate that intervening in the real world might be a way of transforming type-2 problems into type-1, but they state that they are not aware of any definite cases. It is argued that the active construction of external representations often performs exactly this function, and that recoding via the real world is therefore common, if not ubiquitous.

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Application of Type-1 and Type-2 Fuzzy Logic Controller for The Real Swarm Robot

International Journal of Online and Biomedical Engineering (iJOE) ◽

10.3991/ijoe.v15i06.10075 ◽

2019 ◽

Vol 15 (06) ◽

pp. 83 ◽

Cited By ~ 1

Author(s):

Ade Silvia Handayani ◽

Nyayu Latifah Husni ◽

Siti Nurmaini ◽

Irsyadi Yani

Keyword(s):

Fuzzy Logic ◽

Fuzzy Logic Controller ◽

Complex Problem ◽

The Real ◽

Advantages And Disadvantages ◽

Path Navigation ◽

Swarm Robot ◽

Interval Type

Navigation is one of the typical problem domains occurred in studying swarm robot. This task needs a special ability in avoiding obstacles. This research presents the navigation techniques using type 1 fuzzy logic and interval type 2 fuzzy logic. A comparison of those two fuzzy logic performances in controlling swarm robot as tools for complex problem modeling, especially for path navigation is presented in this paper. Each hierarchical of fuzzy logic shows its advantages and disadvantages. For testing the robustness of type-1 fuzzy logic and interval type-2 fuzzy logic algorithms, 3 robots for the real swarm robot experiment are used. Each is equipped with one compass sensor, three distance sensors, and one X-Bee communication module. The experimental results show that type-2 fuzzy logic has better performance than type-1 fuzzy logic.

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Term extraction and Ramsey's theorem for pairs

Journal of Symbolic Logic ◽

10.2178/jsl/1344862165 ◽

2012 ◽

Vol 77 (3) ◽

pp. 853-895 ◽

Cited By ~ 2

Author(s):

Alexander P. Kreuzer ◽

Ulrich Kohlenbach

Keyword(s):

Uniform Method ◽

Second Step ◽

Functional Interpretation ◽

Ramsey's Theorem ◽

Term Extraction ◽

Ramsey’S Theorem ◽

Skolem Functions ◽

Primitive Recursive

AbstractIn this paper we study with proof-theoretic methods the function(al)s provably recursive relative to Ramsey's theorem for pairs and the cohesive principle (COH).Our main result on COH is that the type 2 functional provably recursive fromare primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact thatis-conservative over PRA.Recent work of the first author showed thatis equivalent to a weak variant of the Bolzano-Weierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs.For Ramsey's theorem for pairs and two colorswe obtain the upper bounded that the type 2 functional provable recursive relative toare inT1. This is the fragment of Gödel's systemTcontaining only type 1 recursion—roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction ofT1-bounds from proofs that use. Moreover, this yields a new proof of the fact thatis-conservative over.The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel's functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using-comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard's ordinal analysis of bar recursion (for).

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Entropy Optimization, Maxwell–Boltzmann, and Rayleigh Distributions

Entropy ◽

10.3390/e23060754 ◽

2021 ◽

Vol 23 (6) ◽

pp. 754

Author(s):

Nicy Sebastian ◽

Arak M. Mathai ◽

Hans J. Haubold

Keyword(s):

Communication Theory ◽

Complex Domain ◽

Cauchy Matrix ◽

The Real ◽

Entropy Optimization ◽

Engineering Statistics

In physics, communication theory, engineering, statistics, and other areas, one of the methods of deriving distributions is the optimization of an appropriate measure of entropy under relevant constraints. In this paper, it is shown that by optimizing a measure of entropy introduced by the second author, one can derive densities of univariate, multivariate, and matrix-variate distributions in the real, as well as complex, domain. Several such scalar, multivariate, and matrix-variate distributions are derived. These include multivariate and matrix-variate Maxwell–Boltzmann and Rayleigh densities in the real and complex domains, multivariate Student-t, Cauchy, matrix-variate type-1 beta, type-2 beta, and gamma densities and their generalizations.

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