Finite Computational Structures and Implementations: Semigroups and Morphic Relations

Egri-Nagy, Attila

doi:10.15803/ijnc.7.2_318

Cited by 3 publications

(9 citation statements)

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“…In this sense, we say that semigroups measure computation. Turing completeness is just a binary measure, hence the need for a more fine-grained scale [10]. Thus, we can ask questions like "What is the smallest semigroup that can represent a particular computation?…”

Section: Flip-flopmentioning

confidence: 99%

“…In other words, isomorphisms are strictly structure preserving, while homomorphisms can be structure forgetting down to the extreme of mapping everything to a single state and to the identity operation. The technical details can be complicated due to clustering states (surjective homomorphism) and by the need of turning around homomorphism we also consider homomorphic relations [9,10].…”

Section: ϕ(Xy) = ϕ(X)ϕ(y)mentioning

confidence: 99%

“…For instance, reversible system can carry out irreversible computation by a carefully chosen output encoding [10,31]. A morphism can only produces irreversible systems out irreversible systems (algebraically: the image of a group homomorphism is a group).…”

Section: Interpretationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…As such, we will mainly focus on classical digital computation and many aspects of computation (e.g., engineering practices and their social impacts) will not be discussed here. We will use only minimal mathematical formalism here; for technical details, see [9,10].First we generalize traditional models of computation to composition tables that describe semigroups. We show how these abstract semigroups can cover the wide spectrum of computational phenomena.…”

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The Algebraic View of Computation: Implementation, Interpretation and Time

Egri-Nagy

2018

Philosophies

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Abstract:Computational implementations are special relations between what is computed and what computes it. Though the word "isomorphism" appears in philosophical discussions about the nature of implementations, it is used only metaphorically. Here we discuss computation in the precise language of abstract algebra. The capability of emulating computers is the defining property of computers. Such a chain of emulation is ultimately grounded in an algebraic object, a full transformation semigroup. Mathematically, emulation is defined by structure preserving maps (morphisms) between semigroups. These are systematic, very special relationships, crucial for defining implementation. In contrast, interpretations are general functions with no morphic properties. They can be used to derive semantic content from computations. Hierarchical structure imposed on a computational structure plays a similar semantic role. Beyond bringing precision into the investigation, the algebraic approach also sheds light on the interplay between time and computation.Keywords: computation; abstract algebra; semigroup theory; finite automata; homomorphism; hierarchical structure; finiteness and universality The steam engine replaced muscle power. It did not just make life easier, but a whole bunch of impossible things became possible. Curiously, it was invented before we understood how it worked. Afterward, trying to make it more efficient led to thermodynamics and indirectly to a deeper understanding of the physical world. Similarly, computers replace brain power, but we still do not have a full comprehension of computation. Trying to make computation more efficient and to find its limits is taking us to a deeper understanding of not just computer science, but of other branches of science (e.g., biology, physics, and mathematics). Just as physics advanced by focusing on the very small (particles) and on the very large (universe), studying computers should also focus on basic building blocks (finite state computations) and on large-scale abstract structures (hierarchical (de)compositions).Another parallel with physics is that the underlying theory of computation is mathematical. Algebraic automata theory is a well-established part of theoretical computer science [1][2][3][4][5]. From a strictly technical point of view, it is not necessary to discuss computation at the algebraic level. Several excellent textbooks on the theory of computation do not even mention semigroups [6][7][8]. However, we argue here that for doing a more philosophical investigation of computation, semigroup theory provides the right framework.The following theses summarize the key points of the algebraic view of computation:1. Computation has an abstract algebraic structure. Semigroups (sets with associative binary operation) are natural generalizations of models of computation (Turing machines, λ-calculus, finite state automata, etc.).Philosophies 2018, 3, 00; doi:10.3390/philosophies3020015 www.mdpi.com/journal/philosophies Philosophies 2018, 3, 00 2 of 14 2. Algebraic...

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Section: Flip-flopmentioning

confidence: 99%

Section: ϕ(Xy) = ϕ(X)ϕ(y)mentioning

confidence: 99%

Section: Interpretationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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The Algebraic View of Computation: Implementation, Interpretation and Time

Egri-Nagy

2018

Philosophies

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The Algebraic View of Computation

Egri-Nagy

2017

Preprint

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|{Math, Philosophy, Programming, Writing}| = 1

Egri-Nagy¹

2018

Preprint

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Philosophical thinking has a side effect: by aiming to find the essence of a diverse set of phenomena, it o en makes it difficult to see the differences between them.is can be the case with Mathematics, Programming, Writing and Philosophy itself. eir unified essence is having a shared understanding of the world helped by off-loading our cognitive efforts to suitable languages."Whenever I try make a philosophical argument, it transforms into a mathematical question, and to answer that I end up writing code. My fate. "the author's tweet-sized self-introduction.Mathematicians, philosophers, programmers and writers -is there anything common in what they do? On the surface, it seems, they all do same: si ing in front of computers and typing on keyboards. Joking aside, here we claim that these are very similar activities in their inner workings. Rather, they are the essentially the same, if we investigate them at a suitable level of abstraction. By suitable we mean general enough to capture all of them, but not too much, so it still carries some useful information.What can we see from such a unifying perspective? e differences are obvious. e demarcation lines are institutionalized. In formal education one has to choose between these professions early on. erefore, we want to draw a ention to the parallels instead.is may serve as an antidote for any fixed ideas about the strict borderlines between these fields. e provocative arguments below can be helpful for people who think in terms of "I'm good at X but not at Y . " For instance, learning an abstract and thus seemingly useless piece of mathematics could improve programming skills. Alternatively, one might try writing essays as well. In reverse, writing computer programs may help a writer, since working with a very limited language could give new appreciation of a natural language. In any case, the statements below are more tentative than definitive, and meant to be more like challenges than confrontations.Cognitive metaphors [11] made their way into mathematics [12] and into the computer science discourse [17]. ese are knowledge transferring devices. One can view the ideas below as a network of metaphors, with the purpose of sharing knowledge between the fields.

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Finite Computational Structures and Implementations: Semigroups and Morphic Relations

Cited by 3 publications

References 32 publications

The Algebraic View of Computation: Implementation, Interpretation and Time

The Algebraic View of Computation: Implementation, Interpretation and Time

The Algebraic View of Computation

|{Math, Philosophy, Programming, Writing}| = 1

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