• Hyp cos

    The (n+1)-separator in DAN cannot be embedded into the n-th system of Taranovsky's ordinal notation. In Taranovsky's ordinal notation, consider this kind of expressions: let \(f(x)=C(C(C(C(x,\beta_t),\beta_{t-1})\cdots,\beta_2),\beta_1)\), and \(\alpha>f(\alpha)\), then \(f(f(\cdots f(f(\alpha))))\) (with n+1 f's) is not standard in the n-th system.

    Suppose that the DAN rules (especially the case of the (n+1)-ple comma) apply on C-expressions, with \(\alpha\) to be an (n+1)-separator term and \(f(\alpha)\) to be an n-separator term. Consider the least and simplest situation: if we need adding rule from n-separator and below, we will get \(f^m(\alpha)\) as an (n-m+1)-separator term, and finally \(f^{n+1}(\alpha)\) as a "0-separator" term. Fo…

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  • Hyp cos

    The term "extended Kruskal theorem" showed in Bird's introduction page of his array notation (page 27). It generates functions with growth rate \(\vartheta(\Omega_\omega)\), much stronger than the TREE function.

    (The definitions of embeddings and the order types are derived here.)

    In a tree, let inf(x,y) be the nearest common ancestor of node x and y. So inf(x,y)=x iff x is ancestor of y. In labeled tree, let l(x) be the label of node x. Here we discuss trees with labels in positive integers.

    Labeled tree A is strongly gap-embeddable into B if there exists an injection f from nodes of A to nodes of B such that:

    1. For any x and y of A, f(inf(x,y)) = inf(f(x),f(y))
    2. For any x of A, l(x) = l(f(x))
    3. For any x of A, let x' be the parent of x; for any node t of B, if f…

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  • Hyp cos

    Define SCGD(n) as the maximal length of sequence of graphs (G1, G2, ..., Gm) such that

    1. The degree of every vertex of every Gi is at most 3.
    2. For all i, Gi has at most n+i vertices.
    3. For all i < j, Gi is not graph minor of Gj.
    4. For all i, the diamond graph (K4 with one edge removal) is not graph minor of Gi.

    SCGD function grows much slower than SCG function, but not slower than TREE function.

    Here I'll show encoding of n-colored trees (the objects of TREE function) into subcubic graphs without diamond graph as minor (the objects of SCGD function).

    The root (colored m) with k children is encoded into(if it has less than 2 children, still let k=2)

    A non-root node (colored m) with k children is encoded into(if it has less than 2 children, still let k=2)

    To deco…

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  • Hyp cos

    Catching hierarchy fails

    January 16, 2018 by Hyp cos

    Catching hierarchy fails beyond \(C_1(\Omega)\) (or \(C_{\Omega_2}(\Omega)\)).

    Recall the definition:

    1. \(C_\pi(0)=\psi_\pi(\Omega_\omega)\).
    2. If \(C_\pi(\alpha)=\psi_\pi(\beta)\), then \(C_\pi(\alpha+1)=\psi_\pi(\gamma)\) where \(\psi(\gamma)\) is the least ordinal that \(g_{\psi(\gamma)}\) is comparable to \(f_{\psi(\gamma)}\) and \(\gamma>\beta\), and both \(\psi_\pi(\beta)\) and \(\psi_\pi(\gamma)\) are full-simplified.
    3. \(\pi\) is the diagonalizer of \(C_\pi()\) function.
    4. For limit \(\alpha\), \(C_\pi(\alpha)[n]=C_\pi(\alpha[n])\).

    But what's \(C_1(\Omega)\)? The \(\Omega\) works as the diagonalizer of C(), which is outside, so \(C_1(\Omega)\) refers to the outside C().

    Now consider \(C_1(\Omega+1)\). \(C_1(\Omega+1)=\psi_1(\gamma)\) where \(\ps…

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  • Hyp cos

    Monotonic FS systems

    January 11, 2018 by Hyp cos

    I don't know how you understand "a monotonic FS system", but in this blog post, I'll use "monotonic" for some special FS systems.

    Definition 1: In a fundamental sequence system \(S:\mu\cap\text{Lim}\times\mathbb N\rightarrow\mu\), let \(\lambda\alpha\ge0\).

    1. If \(\beta\) is a successor ordinal, then let \(\beta=\gamma+1=\lambda[n_1]_S[n_2]_S\cdots[n_k]_S[]_S\) where \(n_k>0\) (\(k\ge0\)). \(\alpha
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