Transducer degrees: atoms, infima and suprema

Endrullis, Jörg; Klop, Jan Willem; Bakhshi, Rena

doi:10.1007/s00236-019-00353-7

Cited by 6 publications

(6 citation statements)

References 31 publications

(55 reference statements)

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“…that can be found by this lemma as q ǫ (n), as in [3]. The following lemma is due to Robert Israel in [5].…”

Section: Weight Products and Useful Resultsmentioning

confidence: 82%

Classifying All Degrees Below $N^3$

Noah¹

2021

Preprint

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We answer an open question in the theory of transducer degrees initially posed in [3], on the structure of polynomial transducer degrees, in particular the question of what degrees, if any, lie below the degree of n 3 . Transducer degrees are the equivalence classes formed by word transformations which can be realized by a finite-state transducer. While there are no general techniques to tell if a word w1 can be transformed into w2 via an FST, the work of Endrullis et al. in [2] provides a test for the class of spiralling functions, which includes all polynomials. We classify fully the degrees of all cubic polynomials which are below n 3 , and many of the methods can also be used to classify the degrees of polynomials of higher orders.

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“…that can be found by this lemma as q ǫ (n), as in [3]. The following lemma is due to Robert Israel in [5].…”

Section: Weight Products and Useful Resultsmentioning

confidence: 82%

Classifying All Degrees Below $N^3$

Noah¹

2021

Preprint

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“…A less obvious property is that f zip is not symmetric in general, so f zip(f, g) need not equal f zip(g, f ) . This was proven inadvertently in [5], where a careful reading of the main proof in terms of f zip gives us a specific example of two functions f, g which satisfy f zip(f, g) = f zip(g, f ) . However, there are two classes of functions which do allow f zip to be symmetric, as detailed in the following lemma.…”

Section: Basic Resultsmentioning

confidence: 95%

“…We will give some preliminary definitions with the goal of understanding the definition of a weight product, the key operation for all the results in this paper. For more definitions and background in this area, see [1,2,3,4,5]. We begin by setting 2 = {0,1}, which we will use as our input and output alphabet for all of our transducers.…”

Section: Definitionsmentioning

confidence: 99%

A Diamond Structure in the Transducer Hierarchy

Noah¹

2021

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We answer an open question in the theory of transducer degrees initially posed in [1] on the existence of a diamond structure in the transducer hierarchy. Transducer degrees are the equivalence classes formed by word transformations which can be realized by a finite state transducer, which form an order based on which words can be transformed into other words. We provide a construction which proves the existence of a diamond structure, while also introducing a new function on streams which may be useful for proving more results about the transducer hierarchy.

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“…In the context of neuromorphic computing and recurrent neural networks, a dynamical systems interpretation of "computing" seems more adequate than a function-evaluation interpretation. Some models of "computing" have been proposed in theoretical computer science which account for continual online processing of unbounded-length, symbolic input streams, in particular interactive Turing machines (van Leeuwen and Wiedermann, 2001) and more recently stream automata (Endrullis et al, 2019). In followship of the traditional questions that are considered in classical complexity theory, this research aims at classifying continual input-output stream processing tasks into complexity classes.…”

Section: Time Complexitymentioning

confidence: 99%

Dimensions of Timescales in Neuromorphic Computing Systems

Jaeger,

Doorakkers,

Lawrence

et al. 2021

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This article is a public deliverable of the EU project Memory technologies with multi-scale time constants for neuromorphicarchitectures (MeMScales, memscales.eu/, Call ICT-06-2019 Unconventional Nanoelectronics, project number 871371). This arXiv version is a verbatim copy of the deliverable report, with administrative information stripped. It collects a wide and varied assortment of phenomena, models, research themes and algorithmic techniques that are connected with timescale phenomena in the fields of computational neuroscience, mathematics, machine learning and computer science, with a bias toward aspects that are relevant for neuromorphic engineering. It turns out that this theme is very rich indeed and spreads out in many directions which defy a unified treatment. We collected several dozens of sub-themes, each of which has been investigated in specialized settings (in the neurosciences, mathematics, computer science and machine learning) and has been documented in its own body of literature. The more we dived into this diversity, the more it became clear that our first effort to compose a survey must remain sketchy and partial. We conclude with a list of insights distilled from this survey which give general guidelines for the design of future neuromorphic systems.

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Transducer degrees: atoms, infima and suprema

Cited by 6 publications

References 31 publications

Classifying All Degrees Below $N^3$

Classifying All Degrees Below $N^3$

A Diamond Structure in the Transducer Hierarchy

Dimensions of Timescales in Neuromorphic Computing Systems

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