Normal forms for linear displacement context-free grammars

In this paper we prove several results on normal forms for linear displacement context-free grammars. The results themselves are rather simple and use well-known techniques, but they are extensively used in more complex constructions. Therefore this article mostly serves educational and referential purposes. 1 Displacement context-free grammars Displacement context-free grammars (DCFGs) are a reformulation of well-nested multiple context-free grammars. In this draft we use tuple notation. Let Σ be a finite alphabet, then Σ ̊ denotes the set of all words with letters in Σ, ε being the empty string. When Σ is fixed, Θk denotes the set of all tuples of the form pu0, . . . , ukq, ui P Σ ̊ and Θ “ Ť kPN Θk. We call k the rank of the tuple u “ pu0, . . . , ukq and denote it by rkpuq. The length |u| of a tuple |u| is the sum of lengths of all its components, we denote by Θplq the set of all tuples of length l. The notation Θpďlq and Θpělq are also understood in a natural way. We use the displacement context-free languages notation for well-nested MCFLs. We consider tuples of strings instead of gapped strings. Let Σ be a finite alphabet, then Σ ̊ denotes the set of all words with letters in Σ, ε being the empty string. When Σ is fixed, Θk denotes the set of all tuples of the form pu0, . . . , ukq, ui P Σ ̊ and Θ “ Ť kPN Θk. We call k the rank of the tuple u “ pu0, . . . , ukq and denote it by rkpuq. The length |u| of a tuple |u| is the sum of lengths of all its components, we denote by Θplq the set of all tuples of length l and also write Θpďlq for Ť jďl Θpjq. On the set of tuples we define the concatenation operation ̈ : Θi ˆ Θj Ñ Θi`j and the countable set of intercalation operations dl : Θi ˆΘj Ñ Θi`j ́1: px0, . . . , xiq ̈ py0, . . . , yjq “ px0, . . . , xiy0, . . . , yjq px0, . . . , xiq dl py0, . . . , yjq “ px0, . . . xl ́1y0, y1, . . . , yjxl, . . . , xiq Let N be a finite ranked set of nonterminals and rk : N Ñ N be the rank function. Let Opk “ t ̈,d1, . . . ,dku, the set TmkpN,Σq of k-correct terms is defined as follows: 1. @j ď k pΘj Ă TmkpN,Σq. 2. If α, β P Tmk and rkpαq ` rkpβq ď k, then pα ̈ βq P Tmk, rkpα ̈ βq “ rkpαq ` rkpβq. 3. If j ď k, α, β P Tmk, rkpαq ` rkpβq ď k ` 1, rkpαq ě j, then pαdj βq P Tmk, rkpα ̈ βq “ rkpαq ` rkpβq ́ 1. We assume that all the operation symbols are leftassociative and concatenation has greater priority then intercalation. We may also omit the ̈ symbol, so the notation Ad2 BC d1 D means pAd2 ppB ̈ Cqq d1 Dq. Let Var “ tx1, x2, . . .u be a countable ranked set of variables, such that for every k there is an infinite number of variables having rank k. A context Crxs is a term where a variable x occurs in a leaf position, the rank of x must respect the constraints of term construction. Provided β P Tmk and rkpxq “ rkpβq, Crβs denotes the