Combinatorics on Words. Progress and Perspectives by Larry J. Cummings

By Larry J. Cummings

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Magnus showed that the word problem for groups with a single defining relation is decidable (see [28]). Adjan [l] showed thai the word problem for special Thue systems with a single rule is reducible to the word problem for groups with a single defining relation and so is solvable. It would be of interest to have an explicit algorithm for this problem, but none is known at this time. Further, it would be of interest to know the inherent computational complexity of this problem but that appears to be beyond our present ability (but one should see [3] for partial results).

Set w=w0wl · · · w^ Clearly M, starting in any state, will enter one of the states ^ during its run on w. Let u be a random ω-word. The finite word w occurs as a block infinitely often in u. Therefore, M will enter one of the states ^ when it first encounters a block w. Each time thereafter when M encounters a block w it cannot have left the essential class 5 so it must once again enter state

A substructure of a structure A is a structure whose universe is a subset of the universe of A, and whose relations, operations and constants agree with those of A on this subset. Suppose that u is an α-word, α = < Α , < > . Define the structure A u with A as its universe, a binary relation < (the total order on A) and for each letter aÇE a unary relation Ra = {i£A:u(i) = a}. By considering structures one may study various properties of words expressable in the different logics studied by model theorists.

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