Speculative Computing of Recursive Functions Taking Values from Finite Sets

Recursion, or the capacity of ‘self-reference’, has played a central role within mathematical approaches to understanding the nature of computation, from the general recursive functions of Alonzo Church to the partial recursive functions of Stephen C. Kleene and the production systems of Emil Post. Recursion has also played a significant role in the analysis and running of certain computational processes within computer science (viz., those with self-calls and deferred operations). Yet the relationship between the mathematical and computer versions of recursion is subtle and intricate. A recursively specified algorithm, for example, may well proceed iteratively if time and space constraints permit; but the nature of specific data structures—viz., recursive data structures—will also return a recursive solution as the most optimal process. In other words, the correspondence between recursive structures and recursive processes is not automatic; it needs to be demonstrated on a case-by-case basis.

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Algorithmic Logic with Recursive Functions

Fundamenta Informaticae ◽

10.3233/fi-1981-4411 ◽

1981 ◽

Vol 4 (4) ◽

pp. 975-995

Author(s):

Andrzej Szałas

Keyword(s):

Completeness Theorem ◽

Inference Rules ◽

Partial Correctness ◽

Recursive Functions ◽

Algorithmic Logic ◽

Block Structured

A language is considered in which the reader can express such properties of block-structured programs with recursive functions as correctness and partial correctness. The semantics of this language is fully described by a set of schemes of axioms and inference rules. The completeness theorem and the soundness theorem for this axiomatization are proved.

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Algebraic singularities of scattering amplitudes from tropical geometry

Journal of High Energy Physics ◽

10.1007/jhep04(2021)002 ◽

2021 ◽

Vol 2021 (4) ◽

Author(s):

James Drummond ◽

Jack Foster ◽

Ömer Gürdoğan ◽

Chrysostomos Kalousios

Keyword(s):

Scattering Amplitudes ◽

Natural Language ◽

Tropical Geometry ◽

Cluster Algebras ◽

Finite Alphabet ◽

Square Root ◽

Finite Sets ◽

Yang Mills ◽

Square Roots ◽

Algebraic Singularities

Abstract We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.

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Blind Separation for Multiple Moving Sources with Labeled Random Finite Sets

IEEE/ACM Transactions on Audio Speech and Language Processing ◽

10.1109/taslp.2021.3087003 ◽

2021 ◽

pp. 1-1

Author(s):

Jonah Ong ◽

Ba Tuong Vo ◽

Sven Erik Nordholm

Keyword(s):

Blind Separation ◽

Finite Sets ◽

Moving Sources ◽

Random Finite Sets

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Representation and manipulation of finite sets

ACM SIGAPL APL Quote Quad ◽

10.1145/586169.586171 ◽

1980 ◽

Vol 10 (4) ◽

pp. 8-12 ◽

Cited By ~ 1

Author(s):

B. L. McAllister

Keyword(s):

Finite Sets

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New refinements of the discrete Jensen’s inequality generated by finite or infinite permutations

Aequationes Mathematicae ◽

10.1007/s00010-019-00696-z ◽

2019 ◽

Vol 94 (6) ◽

pp. 1109-1121

Author(s):

László Horváth

Keyword(s):

Banach Spaces ◽

Jensen’S Inequality ◽

Vector Spaces ◽

Real Vector ◽

Finite Sets ◽

Jensen's Inequality

AbstractIn this paper some new refinements of the discrete Jensen’s inequality are obtained in real vector spaces. The idea comes from some former refinements determined by cyclic permutations. We essentially generalize and extend these results by using permutations of finite sets and bijections of the set of positive numbers. We get refinements of the discrete Jensen’s inequality for infinite convex combinations in Banach spaces. Similar results are rare. Finally, some applications are given on different topics.

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