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Turing machines to word problems

Published online by Cambridge University Press:  05 June 2014

By
Charles F. Miller
Edited by
Rod Downey
Charles F. Miller
Affiliation:
University of Melbourne
Rod Downey
Affiliation:
Victoria University of Wellington
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Summary

Abstract. We trace the emergence of unsolvable problems in algebra and topology from the unsolvable halting problem for Turing machines.

§1. Introduction. Mathematicians have always been interested in being able to calculate with or about the things they study. For instance early developers of number theory and the calculus apparently did extensive calculations. By the early 1900s a number of problems were introduced asking for general algorithms to do certain calculations. In particular the tenth problem on Hilbert's influential list asked for an algorithm to determine whether an integer polynomial in several variables has an integer solution.

The introduction by Poincaré of the fundamental group as an invariant of a topological space which can often be finitely described by generators and relations led to Dehn's formulation of the word and isomorphism problem for groups. To make use of such group invariants we naturally want to calculate them and determine their properties. It turns out many of these problems do not have algorithmic solutions and we will trace the history and some of the ideas involved in showing these natural mathematical problems are unsolvable.

In the 1930s several definitions of computable functions emerged together with the formulation of the Church-Turing Thesis that these definitions captured intuitive notions of computability. Church and independently Turing showed that there is no algorithm to determine which formulas of first-order logic are valid, that is, the Entscheidungsproblem is unsolvable.

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Turing's Legacy
Developments from Turing's Ideas in Logic
 , pp. 329 - 385
Publisher: Cambridge University Press
Print publication year: 2014

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