scholarly journals On Polynomial Recursive Sequences

2021 ◽  
Author(s):  
Michaël Cadilhac ◽  
Filip Mazowiecki ◽  
Charles Paperman ◽  
Michał Pilipczuk ◽  
Géraud Sénizergues
Keyword(s):  
Expressive Power ◽  
Weighted Automata ◽  
Recursive Sequences ◽  
Recursive Sequence ◽  
Nonlinear Extensions

AbstractWe study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences. These sequences arise naturally in the study of nonlinear extensions of weighted automata, where (non)expressiveness results translate to class separations. A typical example of a polynomial recursive sequence is bn = n!. Our main result is that the sequence un = nn is not polynomial recursive.

scholarly journals Inequalities approach in determination of convergence of recurrence sequences

2021 ◽  
Vol 5 (1) ◽  
pp. 65-72
Author(s):  
Albert Adu-Sackey ◽  
◽  
Francis T. Oduro ◽  
Gabriel Obed Fosu ◽  
◽  
...  
Keyword(s):  
Mathematical Expression ◽  
Mathematical Induction ◽  
Recurrence Sequence ◽  
Recursive Sequences ◽  
Recursive Sequence ◽  
Leading Term ◽  
Geometric Sequence ◽  
Newton Raphson ◽  
Recurrence Sequences

The paper proves convergence for three uniquely defined recursive sequences, namely, arithmetico-geometric sequence, the Newton-Raphson recursive sequence, and the nested/composite recursive sequence. The three main hurdles for this prove processes are boundedness, monotonicity, and convergence. Oftentimes, these processes lie in the predominant use of prove by mathematical induction and also require some bit of creativity and inspiration drawn from the convergence monotone theorem. However, these techniques are not adopted here, rather, as a novelty, extensive use of basic manipulation of inequalities and useful equations are applied in illustrating convergence for these sequences. Moreover, we established a mathematical expression for the limit of the nested recurrence sequence in terms of its leading term which yields favorable results.

scholarly journals Non-Linear Recursive Sequences

1959 ◽  
Vol 11 ◽  
pp. 370-378 ◽  
Author(s):  
Elbert A. Walker
Keyword(s):  
Maximum Length ◽  
Unpublished Work ◽  
Recursive Sequences ◽  
Non Linear ◽  
Recursive Sequence

The purpose of this paper is to investigate non-linear recursive sequences of maximum length with elements from GF(2). In particular, the question of whether or not a recursive sequence of maximum length can be equal to its dual is settled. This question, as far as the author knows, was originally asked by Rosser. Part I contains the necessary background for Part II, and in the main is a condensation of some unpublished work (1955) of W. A. Blankinship and R. P. Dilworth.

RECURSIVE SEQUENCES IDENTIFYING THE NUMBER OF EMBEDDED COALITIONS

2008 ◽  
Vol 10 (01) ◽  
pp. 129-136
Author(s):  
DAVID W. K. YEUNG
Keyword(s):  
Coalition Structures ◽  
Recursive Sequences ◽  
Strategic Behaviors ◽  
Recursive Sequence ◽  
Individual Player ◽  
Player Game ◽  
The Individual

Strategic behaviors of players depend crucially on the coalition structures of a game. A recursive sequence identifying the number of embedded coalitions in a n-player game is derived. The paper also derives a recursive sequence identifying the number of embedded coalitions in a n-player game where the position of the individual player in a partition counts.

Recursive generation in language

2017 ◽  
Author(s):  
David J. Lobina
Keyword(s):  
Mathematical Logic ◽  
Production System ◽  
General Property ◽  
Recent Literature ◽  
Expressive Power ◽  
The Self ◽  
Recursive Structure ◽  
Embedded Structures ◽  
Mechanical Procedure ◽  
The 1950S

The introduction of recursion into linguistics was the result of applying some of the results of mathematical logic to the study of language. In particular, recursion was introduced in the 1950s as a general property of the mechanical procedure underlying the grammar, in order to account for language’s discrete infinity and expressive power—in the 1950s, this mechanical procedure was a production system, whereas more recently, of course, it is the set-operator merge. Unfortunately, the recent literature has confused the general recursive property of a grammar with specific instances of (recursive) rules/operations within a grammar; more worryingly still, there has been a general conflation of these recursive rules with some of the self-embedded structures these rules can generate, adding to the confusion. The conflation is manifold but always fallacious. Moreover, language manifests a much more generally recursive structure than is usually recognized: bundles of the universal (Specifier)-Head-Complement(s) geometry.

scholarly journals Probability logic: A model-theoretic perspective

2020 ◽  
Author(s):  
M Pourmahdian ◽  
R Zoghifard
Keyword(s):  
Characterization Theorem ◽  
Expressive Power ◽  
Probability Logic ◽  
Theoretic Analysis ◽  
Compactness Property ◽  
Probability Models ◽  
Finitely Additive Probability ◽  
Additive Probability ◽  
Basic Probability ◽  
Abstract Logics

Abstract This paper provides some model-theoretic analysis for probability (modal) logic ($PL$). It is known that this logic does not enjoy the compactness property. However, by passing into the sublogic of $PL$, namely basic probability logic ($BPL$), it is shown that this logic satisfies the compactness property. Furthermore, by drawing some special attention to some essential model-theoretic properties of $PL$, a version of Lindström characterization theorem is investigated. In fact, it is verified that probability logic has the maximal expressive power among those abstract logics extending $PL$ and satisfying both the filtration and disjoint unions properties. Finally, by alternating the semantics to the finitely additive probability models ($\mathcal{F}\mathcal{P}\mathcal{M}$) and introducing positive sublogic of $PL$ including $BPL$, it is proved that this sublogic possesses the compactness property with respect to $\mathcal{F}\mathcal{P}\mathcal{M}$.

The Complexity of Model-Checking Tail-Recursive Higher-Order Fixpoint Logic

2021 ◽  
Vol 178 (1-2) ◽  
pp. 1-30
Author(s):  
Florian Bruse ◽  
Martin Lange ◽  
Etienne Lozes
Keyword(s):  
Model Checking ◽  
Lower Bounds ◽  
Expressive Power ◽  
Higher Order ◽  
Order Logic ◽  
High Complexity ◽  
Transition Systems ◽  
Recursive Formulas ◽  
Second Order Logic ◽  
Monadic Second Order Logic

Higher-Order Fixpoint Logic (HFL) is a modal specification language whose expressive power reaches far beyond that of Monadic Second-Order Logic, achieved through an incorporation of a typed λ-calculus into the modal μ-calculus. Its model checking problem on finite transition systems is decidable, albeit of high complexity, namely k-EXPTIME-complete for formulas that use functions of type order at most k < 0. In this paper we present a fragment with a presumably easier model checking problem. We show that so-called tail-recursive formulas of type order k can be model checked in (k − 1)-EXPSPACE, and also give matching lower bounds. This yields generic results for the complexity of bisimulation-invariant non-regular properties, as these can typically be defined in HFL.

scholarly journals Log-concavity of P-recursive sequences

2021 ◽  
Vol 107 ◽  
pp. 251-268
Author(s):  
Qing-hu Hou ◽  
Guojie Li
Keyword(s):  
Recursive Sequences

scholarly journals Audio-to-score singing transcription based on a CRNN-HSMM hybrid model

2021 ◽  
Vol 10 ◽  
Author(s):  
Ryo Nishikimi ◽  
Eita Nakamura ◽  
Masataka Goto ◽  
Kazuyoshi Yoshii
Keyword(s):  
Hybrid Model ◽  
Viterbi Algorithm ◽  
Language Model ◽  
Expressive Power ◽  
Acoustic Model ◽  
Musical Language ◽  
Musical Score ◽  
Singing Voice ◽  
Generative Process ◽  
Music Signal

This paper describes an automatic singing transcription (AST) method that estimates a human-readable musical score of a sung melody from an input music signal. Because of the considerable pitch and temporal variation of a singing voice, a naive cascading approach that estimates an F0 contour and quantizes it with estimated tatum times cannot avoid many pitch and rhythm errors. To solve this problem, we formulate a unified generative model of a music signal that consists of a semi-Markov language model representing the generative process of latent musical notes conditioned on musical keys and an acoustic model based on a convolutional recurrent neural network (CRNN) representing the generative process of an observed music signal from the notes. The resulting CRNN-HSMM hybrid model enables us to estimate the most-likely musical notes from a music signal with the Viterbi algorithm, while leveraging both the grammatical knowledge about musical notes and the expressive power of the CRNN. The experimental results showed that the proposed method outperformed the conventional state-of-the-art method and the integration of the musical language model with the acoustic model has a positive effect on the AST performance.

scholarly journals Modeling of safe time Petri nets by interval weighted automata

2020 ◽  
Vol 53 (4) ◽  
pp. 187-192
Author(s):  
Jan Komenda ◽  
Aiwen Lai ◽  
José Godoy Soto ◽  
Sébastien Lahaye ◽  
Jean-louis Boimond
Keyword(s):  
Petri Nets ◽  
Time Petri Nets ◽  
Weighted Automata
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