Test Sets for Large Families of Languages

European Journal of Combinatorics

2015

a b s t r a c tWe develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions of these results. Finally, we obtain the first nontrivial upper bounds for the fundamental problem of the maximal size of independent systems. These bounds depend quadratically on the size of the shortest equation. No methods of having such bounds have been known before.

Section: Theorem 42 If a Pair Of Nontrivial Equations On N Unknowns H...mentioning

confidence: 74%

Systems of word equations, polynomials and linear algebra: A new approach

European Journal of Combinatorics

2015

“…In [18] it was also stated that the arithmetization and matrix techniques in [24] would give a simpler proof of a weaker result. Similar questions have been studied in [14] and there are relations to independent systems [22].…”

Section: Theorem 42 If a Pair Of Nontrivial Equations On N Unknownsmentioning

confidence: 74%

Systems of Word Equations and Polynomials: a New Approach

Electron. Proc. Theor. Comput. Sci.

2011

“…The idea behind this construction (both the original and the modified) is that (ababa) k = (ab) k a k (ba) k holds for k < 3, but not for k = 3. It was noted in [18] that if we could find words u i such that (u 1 . .…”

Section: Lower Boundsmentioning

confidence: 99%

On maximal chains of systems of word equations

Karhumäki

Proc. Steklov Inst. Math.

2011

We consider systems of word equations and their solution sets. We discuss some fascinating properties of those, namely the size of a maximal independent set of word equations, and proper chains of solution sets of those. We recall the basic results, extend some known results and formulate several fundamental problems of the topic.Comment: 10 pages. This is a journal article published in 2011, together with a note about a missing reference added in 201