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Sinopsis
Excerpt from Groups With Solvable Word Problems
Equating truth in G with derivability from a set of axioms suggests the possibility of formulating the word problem for G as the derivability problem for some type of formal system. The first part of the paper is concerned with this. Once this is done, we can consider the questions usually asked about formal systems, such as consistency, decid ability, and completeness, and use results about particular formal systems to obtain results about groups and semigroups corresponding to these systems. In this analogy, nontrivial groups correspond to consistent systems, groups with solvable word problem to decidable systems, and simple groups to complete systems. The last analogy is particularly striking, for, strangely enough, although it has been known for a long time that complete recursively axiomatized theories are decidable, it was only recently noted that recursively presented simple groups have solvable word problems.
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Reseña del editor
Excerpt from Groups With Solvable Word Problems
Equating truth in G with derivability from a set of axioms suggests the possibility of formulating the word problem for G as the derivability problem for some type of formal system. The first part of the paper is concerned with this. Once this is done, we can consider the questions usually asked about formal systems, such as consistency, decid ability, and completeness, and use results about particular formal systems to obtain results about groups and semigroups corresponding to these systems. In this analogy, nontrivial groups correspond to consistent systems, groups with solvable word problem to decidable systems, and simple groups to complete systems. The last analogy is particularly striking, for, strangely enough, although it has been known for a long time that complete recursively axiomatized theories are decidable, it was only recently noted that recursively presented simple groups have solvable word problems.
About the Publisher
Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com
This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
Reseña del editor
Excerpt from Groups With Solvable Word Problems
This paper considers two aspects of the word problem for groups: first, some similarities between derivations from the relations of a group and proofs from a set of axioms in logic, and second, the computational difficulty of word problems and the problem of constructing groups with solvable word problems of some preassigned degree of difficulty.
Given a presentation of a group in terms of generators and relations, we define two words on the generators to be equal if and only if one can be transformed into the other in a finite number of steps using the relations of the group. Defining equality in a group as derivability from a set of axioms suggests formulating the word problem for a group as the derivability problem for a formal system. The system we choose is equational logic. We show that there exist some striking analogies between results from logic and results about groups: nontrivial groups correspond to consistent systems, groups with solvable word problem to decidable systems, and simple groups to complete systems. This suggests that results about formal systems could be used to obtain results about decidability in groups.
Hence there exist groups with arbitrarily difficult, but solvable word problem.
About the Publisher
Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com
This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
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Groups With Solvable Word Problems (Classic Reprint)
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Paperback. Condición: New. Print on Demand. This book explores the fascinating connections between formal systems and the computational complexity of word problems in groups. The author establishes striking analogies between theorems from logic and results about groups, revealing that groups with solvable word problems correspond to decidable systems and simple groups to complete systems. The author investigates the concept of decidability in groups, defining a group as having a word problem in level n of the Grzegorczyk hierarchy if the running time of the algorithm solving the word problem is in 8n. A significant contribution of this book is the construction of finitely presented groups with word problems in 8n (n ⥠2). This is achieved by utilizing a technique for constructing semigroups following J. Robinson's method of functional equations, which provides a concrete representation of the groups. The author demonstrates that if a semigroup has a word problem in 8n (and not lower), then the resulting group also has a word problem in 8n (and not lower). The insights presented in this book not only contribute to the understanding of word problems in groups but also shed light on the relationship between formal systems and decidability, offering valuable perspectives for further research and applications in computer science and mathematics. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book. print-on-demand item. Nº de ref. del artículo: 9781330197349_0
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Groups With Solvable Word Problems Classic Reprint
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PAP. Condición: New. New Book. Shipped from UK. Established seller since 2000. Nº de ref. del artículo: LW-9781330197349
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Groups With Solvable Word Problems Classic Reprint
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PAP. Condición: New. New Book. Shipped from UK. Established seller since 2000. Nº de ref. del artículo: LW-9781330197349
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