1 Introduction and definitions
Union | \(g_{\cup }^{{{\,\mathrm{nsc}\,}}}(n,m) =\{ 1,2,\ldots , m+n+1\}\) for \(n\ge 2\), \(m\ge 2\) |
[12] |
Intersection | \(g_{\cap }^{{{\,\mathrm{nsc}\,}}}(n,m) =\{ 1,2,\ldots , m\cdot n\}\) for \(n\ge 1\), \(m\ge 1\) |
[12] |
Concatenation | \(g_{\cdot }^{{{\,\mathrm{nsc}\,}}}(n,m) =\{ 1,2,\ldots , m+n\}\) for \(n\ge 2\), \(m\ge 2\) |
[13] |
Reversal | \(g_{R}^{{{\,\mathrm{nsc}\,}}}(n) =\{ n-1,n,n+1\}\) for \(n\ge 2\) |
[14] |
Kleene-closure | \(g_{*}^{{{\,\mathrm{nsc}\,}}}(n) =\{ 1,2,\ldots , n+1\}\) for \(n\ge 2\) |
[14] |
Union | \(g_{\cup }^{{{\,\mathrm{Varcf}\,}}}(n,m) =\{ 1,2,\ldots , m+n+1\}\) for \(n\ge 1\), \(m\ge 1\) |
concatenation | \(\{ 1\} \cup \{ \max \{n,m\},\ldots , m+n+1\}\subseteq g_{\cdot }^{{{\,\mathrm{Varcf}\,}}}(n,m)\) |
for \(n\ge 1\), \(m\ge 1\) | |
Reversal | \(g_{R}^{{{\,\mathrm{Varcf}\,}}}(n) =\{n\}\) for \(n\ge 1\) |
Kleene-closure* | \(g_{*}^{{{\,\mathrm{Varcf}\,}}}(n) =\{ 1,2,\ldots , n+1\}\) for \(n\ge 1\) |
2 The complexity of some special languages
3 Behaviour under operations
3.1 Reversal
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if \(A\rightarrow w\in P\), \(w\in T^*\) and \(A\in N\), then \(S^R\rightarrow w^RA\in P^R\),
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if \(A\rightarrow wB\in P\), \(w\in T^*\), \(A\in N\), and \(B\in N\), then \(B\rightarrow w^RA\in P^R\),
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if \(S\rightarrow w\in P\) and \(w\in T^*\), then \(S^R\rightarrow w^R\in P^R\), and
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\(S\rightarrow \lambda \in P^R\).
3.2 Kleene-closure
3.3 Concatenation
3.4 Intersection
\(n\ge m\ge 1\)
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\(L_n=P(1,2,\ldots ,n)\)
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\(k=0\)
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\(K_m=P(n+1,n+2,\ldots ,n+m)\)
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\(L_n\cap K_m=\emptyset \)
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\(n\ge m\ge 1\)
| \(L_n=P'(1,2,\ldots , n)\), |
\(1\le k\le m\)
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\(K_m=P'(1,2,\ldots k,n+1,\ldots ,n+m-k)\)
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\(L_n\cap K_m=P'(1,2,\ldots , k)\)
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\(n\ge m\ge 3\)
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\(L_n=U(1,2,\ldots ,m-2) \cup P(m-1,m,\ldots ,m+k'-1)\)
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\(\cup P(m+k',m+k'+1,\ldots ,n+m-4)\) | |
\(0\le k'\le n-4\)
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\(K_m=P(1,2,\ldots ,m-2) \cup U(m-1,\ldots ,m+k'-1)\)
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\(m\le k=m+k'\le m+n-4\)
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\(L_n\cap K_m=P(1,\ldots ,m-2) \cup P(m-1,\ldots ,m+k'-1)\)
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\(n\ge m\ge 3\)
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\(L_n=U(1,2,\ldots ,m-2) \cup P(m-1,m,\ldots ,m+n-4)\)
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\(k=n+m-3\)
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\(K_m=P(1,2,\ldots ,m-2) \cup U(m-1,\ldots ,m+n-4)\)
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\(L_n\cap K_m=P(1,\ldots ,m-2) \cup P(m-1,\ldots ,m+n-4)\)
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\(n\ge 3\), \(m=2\) |
\(L_n=P(1,2,\ldots ,k) \cup h(P(k+1,k+2,\ldots ,n-1))\)
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\(1\le k\le n-2\)
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\(K_m=\{a\}\{a,b\}^*\)
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\(L_n\cap K_m=P(1,2,\ldots ,k)\)
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\(n\ge 2\), \(m=2\) |
\(L_n=\{c,\lambda \}P(1,2,\ldots ,n-1)\)
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\(k=n-1\)
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\(K_m=\{a\}\{a,b\}^*\)
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\(L_n\cap K_m=P(1,2,\ldots ,n-1)\)
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\(n\ge 1\), \(m=2\) |
\(L_n=P(1,2,\ldots ,n)\)
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\(k=n\)
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\(K_m=\{a\}\{a,b\}^*\)
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\(L_n\cap K_m=P(1,2,\ldots ,n)\)
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\(n\ge 3\), \(m=1\) |
\(L_n=P(1,2,\ldots ,k) \cup h(P(k+1,\ldots ,n-1))\)
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\(1\le k\le n-2\)
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\(K_m=\{a,b\}^*\)
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\(L_n\cap K_m=P(1,2,\ldots ,k)\)
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\(n\ge 2\), \(m=1\) |
\(L_n=\{c,\lambda \}P(1,2,\ldots ,n-1)\)
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\(k=n-1\)
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\(K_m=\{a,b\}^*\)
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\(L_n\cap K_m=P(1,2,\ldots ,n-1)\)
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\(n\ge 1\), \(m=1\) |
\(L_n=P(1,2,\ldots ,n)\)
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\(k=n\)
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\(K_m=\{a,b\}^*\)
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\(L_n\cap K_m=P(1,2,\ldots ,n)\)
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3.5 Union
\(n\ge 3\), \(m\ge 1\) |
\(L_n=P(1)\cup P(2,3,\ldots , n-1)\)
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\(k=n+m\)
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\(K_m=P(n+1,n+2,\ldots , n+m)\)
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\(P(1)\cup P(2,3,\ldots , n-1)\cup P(n+1,n+2,\ldots , n+m)\)
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\(n\ge m\ge 3\)
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\(L_n=P(1)\cup P(2)\cup \cdots \cup P(n-1)\)
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\(k=n+m-1\)
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\(K_m=P(n)\cup P(n+1) \cup \cdots \cup P(n+m-2)\)
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\(P(1)\cup P(2)\cup \cdots \cup P(n+m-2)\)
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\(n\ge 3\), \(m=2\) |
\(L_n=P(1)\cup P(2)\cup \cdots \cup P(n-1)\)
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\(k=n+m-1=n+1\)
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\(K_m=\{c\}P(n)\)
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\(P(1)\cup P(2)\cup \cdots \cup P(n-1)\cup \{c\}P(n)\)
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\(n\ge 3\), \(m=1\) |
\(L_n=P(1)\cup P(2)\cup \cdots \cup P(n-1)\)
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\(k=n+m-1=n\)
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\(K_m=P(1)\)
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\(P(1)\cup P(2)\cup \cdots \cup P(n-1)\)
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\(n\ge m\ge 3\)
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\(L_n=P(1)\cup P(2)\cup \cdots \cup P(n-1)\)
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\(n+1\le k\le m+n-2\)
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\(K_m=P(n+1)\cup \cdots \cup P(n+k')\cup P(1)\cup \cdots \cup P(m-k'-1)\)
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\(k'=k-n\)
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\(P(1)\cup \cdots \cup P(n-1)\cup P(n+1)\cup \cdots \cup P(n+k')\)
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\(n\ge m\ge 3\)
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\(L_n=P(1)\cup \cdots \cup P(n-2)\cup U(n+1,\cdots ,n+m-2)\)
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\(3\le k\le n\)
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\(K_m=U(k-2,\cdots , n-2)\cup P(n+1)\cup \cdots \cup P(n+m-2)\)
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\(P(1)\cup \cdots \cup P(k-3)\cup U(k-2,\ldots , n-2)\)
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\(\cup \, U(n+1,\ldots ,n+m-2)\) | |
\(n\ge 3\), \(m\ge 3\) |
\(L_n=\{c\}P(1,2,\ldots ,n-2) \cup \{cb\}\{a,b\}^*\)
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\(k=2\)
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\(K_m=h(\{c\}P(1,2,\ldots ,m-2) \cup \{cb\}\{a,b\}^*)\)
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\(\{c\}\{a,b\}^+\)
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\(n\ge 3\), \(m\ge 3\) |
\(L_n=P(1,2,\ldots ,n-2) \cup \{b\}\{a,b\}^*\)
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\(k=1\)
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\(K_m=h(P(1,2,\ldots ,m-2) \cup \{b\}\{a,b\}^*)\)
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\(\{a,b\}^+\)
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\(n\ge 3\), \(m=2\) |
\(L_n=P(1)\cup \cdots \cup P(k-2)\cup \{c\}P(k-1)\cup \cdots \cup \{c\}P(n-1)\)
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\(3\le k\le n\)
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\(K_m=\{c\}U(k-1,k,\ldots , n-1)\)
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\(P(1)\cup P(2)\cup \cdots \cup P(k-2)\cup \{c\}U(k-1,k,\ldots , n-1)\)
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\(n\ge 3\), \(m=2\) |
\(L_n=\{c\}P(1,2,\ldots ,n-2) \cup \{cb\}\{a,b\}^*\)
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\(k=2\)
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\(K_m=h(\{cb\}\{a,b\}^*)\)
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\(\{c\}\{a,b\}^+\)
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\(n\ge 3\), \(m=2\) |
\(L_n=P(1,2,\ldots ,n-2) \cup \{b\}\{a,b\}^*\)
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\(k=1\)
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\(K_m=h(\{b\}\{a,b\}^*)\)
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\(\{a,b\}^+\)
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\(n\ge 3\), \(m=1\) |
\(L_n=P(1)\cup \cdots \cup P(k-2)\cup P(k-1)\cup \cdots \cup P(n-1)\)
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\(3\le k\le n\)
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\(K_m=U(k-1,k,\ldots , n-1)\)
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\(P(1)\cup P(2)\cup \cdots \cup P(k-2)\cup U(k-1,k,\ldots , n-1)\)
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\(n\ge 3\), \(m=1\) |
\(L_n=P(1)\cup \cdots \cup P(n-2)\cup \{c\}\{a,b\}^+\)
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\(k=2\)
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\(K_m=\{a,b\}^+\)
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\((\{c\}\cup \{\lambda \})\{a,b\}^+\)
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\(n\ge 3\), \(m=1\) |
\(L_n=P(1)\cup \cdots \cup P(k-2)\cup P(k-1)\cup \cdots \cup P(n-1)\)
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\(k=1\)
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\(K_m=\{a,b\}^+\)
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\(\{a,b\}^+\)
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\(n=2\), \(m\ge 1\) | \(L_n=\{c\}P(1)\), \(K_m=P(2,3\ldots ,m+1)\) |
\(k=n+m\)
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\(\{c\}P(1)\cup P(2,3\ldots ,m+1)\)
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\(n=m=1\)
| \(L_n=\{a\}\), \(K_m=\{ a^{3i} \mid i\ge 1\}\) |
\(k=n+m=2\)
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\(\{a\}\cup \{ a^{3i} \mid i\ge 1\}\)
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\(n=m=2\)
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\(L_n=K_m=\{c\}P(1)\)
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\(k=2\)
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\(\{c\}P(1)\)
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\(n=m=2\)
| \(L_n=\{b\}\{a,b\}^*\), \(K_m=\{a\}\{a,b\}^*\) |
\(k=1\)
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\(\{a,b\}^+\)
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\(n=2\), \(m=1\) | \(L_n=\{b\}\{a\}^+\), \(K_m=\{a\}^+\) |
\(k=2\)
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\(\{b,\lambda \}\{a\}^+\)
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\(n=2\), \(m=1\) | \(L_n=\{b\}\{a\}^+\), \(K_m=\{a,b\}^+\) |
\(k=1\)
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\(\{a,b\}^+\)
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\(n=m=1\)
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\(L_n=K_m=\{a\}^+\)
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\(k=1\)
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\(\{a\}^+\)
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4 Conclusions
Union | \(g_{\cup }^{{{\,\mathrm{Var}\,}}}(n,m)=\{1,2,\ldots ,n+m+1\}\) for \(n\ge 1\), \(m\ge 1\) |
Intersection | \(\{0,1,2,\ldots , n+m-3\}\subseteq g_{\cap }^{{{\,\mathrm{Var}\,}}}(n,m)\) for \(n\ge 3\), \(m\ge 3\) |
Concatenation | \(\{1\}\cup \{\min \{m,n\}, \min \{m,n\}+1,\ldots ,m+n\}\subseteq g_{\cdot }^{{{\,\mathrm{Var}\,}}}(m,n)\) |
for \(n\ge 1\), \(m\ge 1\) | |
Reversal | \(g_{R}^{{{\,\mathrm{Var}\,}}}(n)=\{n-1,n,n+1\}\) for \(n\ge 2\) |
Kleene-closure+ | \(g_{+}^{{{\,\mathrm{Var}\,}}}(n)=\{1,2,\ldots ,n\}\) for \(n\ge 1\) |
Kleene-closure* | \(g_{*}^{{{\,\mathrm{Var}\,}}}(n)=\{1,2,\ldots ,n+1\}\) for \(n\ge 1\) |