Enumerating transformation semigroups

East, James; Egri-Nagy, Attila; Mitchell, James D.

doi:10.1007/s00233-017-9869-2

Cited by 9 publications

(6 citation statements)

References 27 publications

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“…The concept of transformation semigroups has played an important role in the theory of finite automata [17,18]. East et al, [21] describe general methods for enumerating sub semigroups of finite semigroups and techniques to improve the algorithmic efficiency of the calculations. In addition, Julatha and Rimcholakarn [22] investigate the characterizing semigroups by their generalized fuzzy ideals.…”

Section: Literature Reviewmentioning

confidence: 99%

General Fuzzy Switchboard Transformation Semigroup

Nur Ain Ebas,

Kavikumar Jacob,

Mohd Saifullah Rusiman

et al. 2024

ARASET

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Commutative and switching processing are central to computation. However, some problems arise when classical versions do not reflect the true needs of current computer science, whereby they cannot predict the flow of the next input information in the specified output. The Switchboard State Machine is a controller that controls the direct flow of information from one state to another and also plays the main part in communication between the subsystems. If the simple system satisfies the two properties of the switching and commutative state machines, the system is therefore a finite switchboard state machine. The general fuzzy switchboard automation (GFSA) was introduced by incorporating the switchboard into the general fuzzy automation. This paper is intended to introduce the concept of a general fuzzy switchboard transformation semigroup (GFSTS) by combining the GFSA and the transformation semigroup. Some related definitions and properties are established. The example of certain products, such as GFSTS cascade products, has also been studied. General Fuzzy Switchboard Poly-Transformation Semigroup (GFSPS) is also introduced since it fulfils the switchboard state machine properties. Some of the definitions and properties associated with the GFSPS are defined. Applications for general fuzzy switchboard automata, such as washing machines, are also provided.

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Section: Literature Reviewmentioning

confidence: 99%

General Fuzzy Switchboard Transformation Semigroup

Nur Ain Ebas,

Kavikumar Jacob,

Mohd Saifullah Rusiman

et al. 2024

ARASET

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“…Algorithmically, this is a graph search problem: the nodes are subsemigroups and the directed edges are labeled by adding an additional element to a subsemigroup (source) generating another bigger subsemigroup (target). This strategy led to the successful enumeration of all 132 069 776 transformation semigroups with 4 states [12]. As a first step towards a classification of these we can draw the size distribution (Fig.…”

Section: Enumeration and Classification Of Computational Structuresmentioning

confidence: 99%

Finite Computational Structures and Implementations: Semigroups and Morphic Relations

Egri-Nagy

2017

IJNC

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What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only partial answers to these questions. To make these problems more precise and easier to tackle, we describe an abstract algebraic definition of classical computation by generalizing traditional models to semigroups. This way implementations are morphic relations between semigroups. The mathematical abstraction also allows the investigation of different computing paradigms (e.g. cellular automata, reversible computing) in the same framework. While semigroup theory helps in clarifying foundational issues about computation, at the same time it has several open problems that require extensive computational efforts. This mutually beneficial relationship is the central tenet of the described research.

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“…For computational projects, where a bespoke search algorithm is needed (e.g. [9]), SAT-solvers still can serve as validation methods.…”

Section: Search Problems Manualmentioning

confidence: 99%

Declarativeness: the work done by something else

Egri-Nagy¹

2017

Preprint

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Being declarative means that we do computer programming on higher levels of abstraction. This vague definition identifies declarativeness with the act of ignoring details, but it is a special case of abstraction. The unspecified part is some computational work.Automating computations and offloading mental processing are essentially the same concept, which is fundamental for both computational and mathematical thinking. This shows that declarativeness is not just a particular style, but it is the core idea of programming.Here we demonstrate this argument and examine its consequences for teaching by a systematic study of coding examples from an introductory programming course. The chosen language is Clojure, as it is proven to be accessible for novices.

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Enumerating transformation semigroups

Cited by 9 publications

References 27 publications

General Fuzzy Switchboard Transformation Semigroup

General Fuzzy Switchboard Transformation Semigroup

Finite Computational Structures and Implementations: Semigroups and Morphic Relations

Declarativeness: the work done by something else

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