TWiki> CoWiki Web>ImportantOpenProblems>LargestIndependentSystem (2011-08-08, StepanHolub) EditAttach

What is the size of largest independent system of word equations?

A system of word equations is independent if the set of solutions changes when any one
equation is removed. Given *n* > 2, what is the size of the largest independent system of equations over *n* variables? Or can such a system be arbitrarily large? (It is known it cannot be infinite.)
The question is open already for *n* = 3.

A related question is the analogous question for the size of test sets. For some language *L*, a test set is a set of words *T* ⊂ *L* such that if two morphisms are equivalent on *T*, then they are equivalent on *L*. Again, every language over a finite alphabet has a finite test set. A minimal test set is analogous to an independent system of equations. The question is: What is the largest size of a minimal test set for languages over *n* letters? Is the size limited?

Bounded languages seem to be a good special case to start with. A language is bounded if it is a subset of
*w*_{1}^{*}*w*_{2}^{*}...*w*_{k}^{*} for some words *w*_{i}. See a particular open problem of this kind

-- JeffreyShallit - 13 Jul 2011

-- StepanHolub - 13 Jul 2011

Topic revision: r4 - 2011-08-08 - StepanHolub

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