Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let T = A ∪ B ∪ C be an alphabet that is partitioned into three subalphabets. The mixing product of a word g over A ∪ B and of a word d over A ∪ C is the set of words w over T such that its projection onto A ∪ B gives g and its projection onto A ∪ C gives d.Let R be a regular language over T such that xbcy is in R if and only if xcby is in R for any two letters b in B and c in C. In other words, R is commutative over B and C. Is this property "structural" in the sense that R can then be obtained as a mixing product of a regular language over A ∪ B and of a regular language over A ∪ C?This question has a rather easy ansr, but there are many cases where the ansr is negative. A more interesting question is whether R can be represented as a finite union of mixed products of regular languages. For the moment,  do not have an ansr to this question. Hover,  prove that it is decidable whether, for a given k, the language R is a union of at most k mixed products of regular languages.
['Jean Berstel', 'Luc Boasson', 'Michel Latteux']
I am writing a research paper on a idea Mixed languages
[{"from": "human", "value": "I am writing a research paper on a idea Mixed languages"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let T = A \u222a B \u222a C be an alphabet that is partitioned into three subalphabets. The mixing product of a word g over A \u222a B and of a word d over A \u222a C is the set of words w over T such that its projection onto A \u222a B gives g and its projection onto A \u222a C gives d.Let R be a regular language over T such that xbcy is in R if and only if xcby is in R for any two letters b in B and c in C. In other words, R is commutative over B and C. Is this property \"structural\" in the sense that R can then be obtained as a mixing product of a regular language over A \u222a B and of a regular language over A \u222a C?This question has a rather easy ansr, but there are many cases where the ansr is negative. A more interesting question is whether R can be represented as a finite union of mixed products of regular languages. For the moment,  do not have an ansr to this question. Hover,  prove that it is decidable whether, for a given k, the language R is a union of at most k mixed products of regular languages."}]