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Decidability and Topological Complexity: Lecture Notes in Computer Science 9802

Anthony Wells

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Published in Descriptive Set Theoretic Methods In Automata Theory: Decidability And Topological Complexity (Lecture Notes In Computer Science 9802)

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The book 'Decidability and Topological Complexity: Lecture Notes in Computer Science 9802', edited by Antti Kuusisto and Juha Kontinen, is a collection of articles that explore the relationship between decidability and topology. Decidability is the study of whether a given problem can be solved by a computer, while topology is the study of the properties of geometric objects. The book covers a wide range of topics, including the decidability of topological properties, the topological complexity of decision problems, and the relationship between decidability and topology.

Descriptive Set Theoretic Methods in Automata Theory: Decidability and Topological Complexity (Lecture Notes in Computer Science 9802)

Descriptive Set Theoretic Methods in Automata Theory: Decidability and Topological Complexity (Lecture Notes in Computer Science Book 9802)

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Language	:	English
File size	:	12302 KB
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Print length	:	230 pages

Decidability of Topological Properties

One of the main topics covered in the book is the decidability of topological properties. A topological property is a property of a geometric object that is invariant under continuous deformations. For example, the property of being connected is a topological property, as it is not affected by stretching or bending the object.

The decidability of a topological property is the question of whether there is an algorithm that can determine whether a given object has the property. For some topological properties, such as the property of being connected, there are simple algorithms that can decide the property. However, for other topological properties, such as the property of being homeomorphic to a sphere, there are no known algorithms that can decide the property.

Topological Complexity of Decision Problems

Another topic covered in the book is the topological complexity of decision problems. The topological complexity of a decision problem is the minimum number of topological operations that are required to solve the problem. For example, the topological complexity of the problem of determining whether two objects are homeomorphic is at least two, as it requires at least two topological operations to determine whether the objects are homeomorphic.

The topological complexity of decision problems is important because it can provide insights into the difficulty of solving the problems. For example, if a decision problem has a high topological complexity, then it is likely to be difficult to solve.

Relationship Between Decidability and Topology

The book also explores the relationship between decidability and topology. Decidability is a topological property, as it is a property of the set of all objects that have a given property. However, decidability is not a topological invariant, as it can change under continuous deformations.

The relationship between decidability and topology is complex and not fully understood. However, the book provides a number of insights into the relationship, and it is a valuable resource for researchers working in the area of decidability and topological complexity.

The book 'Decidability and Topological Complexity: Lecture Notes in Computer Science 9802' is a comprehensive and up-to-date overview of the field of decidability and topological complexity. The book covers a wide range of topics, and it is a valuable resource for researchers working in the area.

A Complex Geometric Object Descriptive Set Theoretic Methods In Automata Theory: Decidability And Topological Complexity (Lecture Notes In Computer Science 9802)