Degrees of Infinite Words, Polynomials and Atoms

Endrullis, Jörg; Karhumäki, Juhani; Klop, Jan Willem; Saarela, Aleksi

doi:10.1142/s0129054118420066

Cited by 6 publications

(41 citation statements)

References 12 publications

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“…For the Transducer degrees, the existence of atoms has been investigated in the papers [12,17,19,20]. We give a short account of our findings there.…”

Section: Atomsmentioning

confidence: 84%

“…In Sect. 3, we introduce the partial order of Transducer degrees arising from the stream transformation realised by finite state transducers, and we describe the results of an investigation [12,17,19,20] of the appearance of atoms in the hierarchy of degrees. In Sects.…”

Section: Outline and Contributionmentioning

confidence: 99%

“…The partial order of Mealy degrees is studied by G. Rayna [31] and A. Belov [2]. The Transducer degrees have been studied in [12,17,19,20,23]. The papers [12,17,19,20] are concerned with atoms in this hierarchy.…”

Section: Related Workmentioning

confidence: 99%

“…The Transducer degrees have been studied in [12,17,19,20,23]. The papers [12,17,19,20] are concerned with atoms in this hierarchy. A comparison of Transducer degrees with the Turing degrees can be found in [23].…”

Section: Related Workmentioning

confidence: 99%

“…The Transducer degrees are the partial order Figure 2 displays some initial results about the hierarchy of Transducer degrees that have been obtained in [12,17,19,20,23], and points to some open problems. We use the following notation: for f : N → N we define the stream…”

Section: Definitionmentioning

confidence: 99%

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Transducer degrees: atoms, infima and suprema

2019

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Although finite state transducers are very natural and simple devices, surprisingly little is known about the transducibility relation they induce on streams (infinite words). We collect some intriguing problems that have been unsolved since several years. The transducibility relation arising from finite state transduction induces a partial order of stream degrees, which we call Transducer degrees, analogous to the well-known Turing degrees or degrees of unsolvability. We show that there are pairs of degrees without supremum and without infimum. The former result is somewhat surprising since every finite set of degrees has a supremum if we strengthen the machine model to Turing machines, but also if we weaken it to Mealy machines.

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“…For the Transducer degrees, the existence of atoms has been investigated in the papers [12,17,19,20]. We give a short account of our findings there.…”

Section: Atomsmentioning

confidence: 84%

Section: Outline and Contributionmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

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Transducer degrees: atoms, infima and suprema

2019

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Braids via term rewriting

Endrullis

Klop

2019

Theoretical Computer Science

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We present a brief introduction to braids, in particular simple positive braids, with a double emphasis: first, we focus on term rewriting techniques, in particular, reduction diagrams and decreasing diagrams. The second focus is our employment of the colored braid notation next to the more familiar Artin notation. Whereas the latter is a relative, position dependent, notation, the former is an absolute notation that seems more suitable for term rewriting techniques such as symbol tracing. Artin's equations translate in this notation to simple word inversions. With these points of departure we treat several basic properties of positive braids, in particular related to the word problem, confluence property, projection equivalence, and the congruence property. In our introduction the beautiful diamond known as the permutohedron plays a decisive role.

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Classifying all transducer degrees below N3

Noah

2022

Journal of Logic and Computation

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We answer an open question in the theory of transducer degrees initially posed in Endrullis et al. (2016, Proc. Conf. on Developments in Language Theory (DLT 2016), 164–176), on the structure of polynomial transducer degrees. Transducer degrees are the equivalence classes formed by word transformations which can be realized by a finite-state transducer (FST). While there are no general techniques to tell if a word $\sigma $ can be transformed into $\tau $ via an FST, the work of Endrullis et al. (2010, J. Int., 11B.A6, 164–176) provides a test for the class of streams determined by spiralling functions, which includes all streams determined by polynomials. We fully classify the degrees of all cubic polynomial streams which are below the stream corresponding to $n^3$, and many of the methods can also be used to classify the degrees of polynomial streams of higher orders.

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Degrees of Infinite Words, Polynomials and Atoms

Cited by 6 publications

References 12 publications

Transducer degrees: atoms, infima and suprema

Transducer degrees: atoms, infima and suprema

Braids via term rewriting

Classifying all transducer degrees below N3

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