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311708c8637d52c11480b51ffc7f7795f7b3edb7 | subsection | 18 | 57 | Four point form factor | Now we only need to consider the first two terms.In this part we need to compute \mathcal {R}_4 in GOE, which is{\mathcal {R}_{4}}=\sum \limits _{a,b,c,d=1}^{L}{\int {D\lambda }{{e}^{i({{\lambda }_{a}}+{{\lambda }_{b}}-{{\lambda }_{c}}-{{\lambda }_{d}})t}}}Take a look at the classifications of combinations in {\mathcal... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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993920c71209395ad61d1a57e0caf5c14e962bb1 | subsection | 19 | 57 | Four point form factor | Add them together we get& {\mathcal {R}_{4}}=L(L-1)(L-2)(L-3)\int {D\lambda }{{e}^{i({{\lambda }_{1}}+{{\lambda }_{2}}-{{\lambda }_{3}}-{{\lambda }_{4}})t}} \\
& +2L(L-1)(L-2)\operatorname{Re}\int {D\lambda }{{e}^{i(2{{\lambda }_{1}}-{{\lambda }_{2}}-{{\lambda }_{3}})t}} \\
& +L(L-1)\int {D\lambda }{{e}^{i(2{{\lambda }... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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05438c0e6845d659d0945c99592e6b553f898329 | subsection | 20 | 57 | Four point form factor | Now we only need to consider the first two terms.The four point form factor has the universal form&{{\cal R}_4} = {L^4}|{r_1}(t){|^4}\\
&- 2{L^3}{\mathop {\rm Re}\nolimits } (r_1^2(t)){r_2}(t){r_3}(2t) - 4{L^3}|{r_1}(t){|^2}{r_2}(t) + 2{L^3}{\rm {Re}}({r_1}(2t)r_1^{*2}(t)) + 4{L^3}|{r_1}(t){|^2}\\
&+ 2{L^2}r_2^2(t) + {... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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22dcee36e331240198ddfecaf886a0be01bdac55 | subsection | 21 | 57 | Four point form factor | For all three ensembles we still have&{{r}_{1}}(t)={{e}^{2it}}({{J}_{0}}(2t)-i{{J}_{1}}(2t))\\
&{r_3}(t) =\frac{{\sin (t/2\rho (u))}}{{t/2\rho (u)}}For LUE we have&{r_{3,1}}(t) = {r_{3,2}}(t) =r_{3,3}(t/3)= {r_4}(t/2) = {r_2}(t) =\left\lbrace \begin{array}{*{35}{l}}
1-\frac{t}{2\pi L\rho (u)} & \text{for}~~0<t<2\pi L\r... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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daa4015ce307821d702894af7492b30c3282b051 | subsection | 22 | 57 | The first term | The first term is an actual four point function.L(L-1)(L-2)(L-3)\int {D\lambda }{{e}^{i({{\lambda }_{1}}+{{\lambda }_{2}}-{{\lambda }_{3}}-{{\lambda }_{4}})t}}When expanding the determinant, the terms could be summarized as the following,4-type: In this case we have
& -2\int {d\lambda _1d\lambda _2d\lambda _3d\lambda ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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4c28984a0ad9d0841884fc0c0e915c7ff4863940 | subsection | 23 | 57 | The first term | We can separately discuss these terms as the following,4-type: In this case we only have the T_4, In this case, all six terms in the expansion give the same answer, which is
-\int {d}{{\lambda }_{1}}d{{\lambda }_{2}}d{{\lambda }_{3}}d{{\lambda }_{4}}{{T}_{4}}({{\lambda }_{1}},{{\lambda }_{2}},{{\lambda }_{3}},{{\lambd... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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e4a551a5cbb63cb52ee4c9467103427ff8864071 | subsection | 24 | 57 | The first term | Thus we have
& -\int {d}{{\lambda }_{1}}d{{\lambda }_{2}}d{{\lambda }_{3}}d{{\lambda }_{4}}{{T}_{1}}({{\lambda }_{1}}){{T}_{1}}({{\lambda }_{2}}){{T}_{2}}({{\lambda }_{3}},{{\lambda }_{4}}){{e}^{i({{\lambda }_{1}}+{{\lambda }_{2}}-{{\lambda }_{3}}-{{\lambda }_{4}})t}} \\
& -\int {d}{{\lambda }_{1}}d{{\lambda }_{2}}d{{... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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b5cb217cb9917369735d95f50ca07c2ff02b22c8 | subsection | 25 | 57 | The first term | Thus we have
& +\int {d}{{\lambda }_{1}}d{{\lambda }_{2}}d{{\lambda }_{3}}d{{\lambda }_{4}}{{T}_{2}}({{\lambda }_{1}},{{\lambda }_{2}}){{T}_{2}}({{\lambda }_{3}},{{\lambda }_{4}}){{e}^{i({{\lambda }_{1}}+{{\lambda }_{2}}-{{\lambda }_{3}}-{{\lambda }_{4}})t}} \\
& +\int {d}{{\lambda }_{1}}d{{\lambda }_{2}}d{{\lambda }_... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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bfe86805dadbf4ac389d669f3b1b64a20190ad6d | subsection | 26 | 57 | The first term | Thus we have
& +\int {d}{{\lambda }_{1}}d{{\lambda }_{2}}d{{\lambda }_{3}}d{{\lambda }_{4}}{{T}_{3}}({{\lambda }_{2}},{{\lambda }_{3}},{{\lambda }_{4}}){{T}_{1}}({{\lambda }_{1}}){{e}^{i({{\lambda }_{1}}+{{\lambda }_{2}}-{{\lambda }_{3}}-{{\lambda }_{4}})t}} \\
& +\int {d}{{\lambda }_{1}}d{{\lambda }_{2}}d{{\lambda }_... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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fd5d665b771cf06f36feceafbf418211e2ad2333 | subsection | 27 | 57 | The second term | In this part we will evaluate the second term2L(L-1)(L-2)\operatorname{Re}\int {D\lambda }{{e}^{i(2{{\lambda }_{1}}-{{\lambda }_{2}}-{{\lambda }_{3}})t}}Let us firstly consider it without a factor of 2L(L-1)(L-2)\operatorname{Re}\int {D\lambda }{{e}^{i(2{{\lambda }_{1}}-{{\lambda }_{2}}-{{\lambda }_{3}})t}}Then we obta... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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d1163df9975fd4b1bfa103992d55601e28bc3115 | subsection | 28 | 57 | The second term | 1-1-1-type: In this case we have
\operatorname{Re}\int {d}{{\lambda }_{1}}d{{\lambda }_{2}}d{{\lambda }_{3}}{{T}_{1}}({{\lambda }_{1}}){{T}_{1}}({{\lambda }_{2}}){{T}_{1}}({{\lambda }_{3}}){{e}^{i(2{{\lambda }_{1}}-{{\lambda }_{2}}-{{\lambda }_{3}})t}}={{L}^{3}}{{r}_{1}}(2t)r_{1}^{2}(t)
2-1-type: In this case we hav... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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56ddebb3e05696da7e09e77ea72646b5aefcb8e6 | subsection | 29 | 57 | The final result | Here we make a summary. The final result for \mathcal {R}_4 is&{{\mathcal {R}}_{4}}={{L}^{4}}|{{r}_{1}}(t){{|}^{4}} \\
&-2{{L}^{3}}\text{Re}(r_{1}^{2}(t)){{r}_{2}}(t){{r}_{3}}(2t)-4{{L}^{3}}|{{r}_{1}}(t){{|}^{2}}{{r}_{2}}(t)+2{{L}^{3}}\text{Re}({{r}_{1}}(2t)r_{1}^{*2}(t))+4{{L}^{3}}|{{r}_{1}}(t){{|}^{2}} \\
&+2{{L}^{2}... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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327b7b3152774c96bf4d03d3686ab9eda558bfab | subsection | 30 | 57 | Finite temperature result | Finally, we will take a look at the finite temperature result, where this result will also rely on the refined kernel and the interval splitting technology as mentioned before, so here we only precisely compute the two point case. | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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5b89092a0c99af0b447155e2c50a70bac131bb35 | subsection | 31 | 57 | Finite temperature result | The definition of the finite temperature two point form factor is{\mathcal {R}_2} = \sum \limits _{i,j} {\int {D\lambda } {e^{i({\lambda _i} - {\lambda _j})t}}{e^{ - \beta ({\lambda _i} - {\lambda _j})}}}Following from a simple analysis we have&{\mathcal {R}_2} = \sum \limits _{i,j} {\int {D\lambda } {e^{i({\lambda _i}... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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c5fbd3ed2baf9657b21ed5b770613089113b290e | subsection | 32 | 57 | Finite temperature result | However, one can expand it over small \beta . We have&{L^2}\int {d{u_1}d{u_2}\frac{{{{\sin }^2}(\pi L{u_1}\rho ({u_2}))}}{{{{(\pi L{u_1})}^2}}}{e^{i{u_1}t}}{e^{ - 2\beta {u_2}}}} \\
&= \left\lbrace {\begin{array}{*{20}{c}}
\begin{array}{l}
\frac{2}{\pi }L{\rm {arccsc}}\left( {\frac{{2L}}{{\sqrt{4{L^2} - {t^2}} }}} \rig... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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89f42a63caaa7c98405df56084d4deacba72d7c4 | subsection | 33 | 57 | Random matrix theory review | GOE and GSE describe physical systems with discrete antiunitary symmetries. Here we will briefly review the mathematical construction. We define the joint distribution of eigenvalues for GOE and GSE asP(\lambda _1,\ldots ,\lambda _L) \sim e^{-\tilde{\beta }\frac{L}{4} \sum _i \lambda _i^2} \prod _{i<j} (\lambda _i-\lam... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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d9f406d7bba0b4ecf1fcf9e75f1e58517e52cc06 | subsection | 34 | 57 | Random matrix theory review | For instance, for a cycle t like:t: ~~~a\rightarrow b \rightarrow c \rightarrow \cdots \rightarrow d \rightarrow athe corresponding contribution in the product is{{\text{cy}{{\text{c}}_{t}}(\sigma ,Q)}}=(q_{ab}q_{bc}\cdots q_{cd}q_{da})^{(0)}where the upper index {(0)} means the scalar part, or equivalently(q_{ab}q_{bc... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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2b9b662548c002b877e95aa56cb8b32ec6104b3e | subsection | 35 | 57 | Random matrix theory review | In fact, define the following function:&\hat{s}(r)=\frac{\sin (r)}{r}~~~~~~\epsilon (r)=\frac{1}{2}\text{sign}(r)\\
&\mathbf {D}\hat{s}(r)=\partial _r \hat{s}(r)~~~~~~\mathbf {I}\hat{s}(r)=\int _{0}^{r}\hat{s}(t)dtThus the quaternion kernel for GOE isK({\lambda _i},{\lambda _j}) \equiv \left\lbrace {\begin{array}{*{20}... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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2452d6d6d99617b259cacba1058d55f2de53f0a8 | subsection | 36 | 57 | Random matrix theory review | It is not called the determinantal point process in random matrix theory literature, but it is called the Pfaffian point process. For our practical motivation, we may define the joint eigenvalue distribution as some linear combination of cluster function T,{{\rho }^{(n)}}({{\lambda }_{1}},\ldots ,{{\lambda }_{n}})=\fra... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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0dca240d9df7456cf9a0c6726e72e055cf3cb1e5 | subsection | 37 | 57 | Random matrix theory review | There are some simplest examples for those formulas, for instance,& {{\rho }^{(1)}}({{\lambda }_{1}})=\frac{1}{L}\times \frac{1}{2}\text{Tr}\left( K({{\lambda }_{1}},{{\lambda }_{1}}) \right) \\
& {{\rho }^{(2)}}({{\lambda }_{1}},{{\lambda }_{2}})=\frac{1}{L(L-1)}\times \left( \frac{1}{4}\text{Tr}\left( K({{\lambda }_{... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
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5e75d3b0b26f30a8b3aaca8d11fee4c31029641f | subsection | 38 | 57 | Random matrix theory review | Similarly, for LUE it is a determinant point process, so we could determine the correlation functions as{\rho ^{(n)}}({\lambda _1}, \ldots ,{\lambda _n}) = \frac{{(L - n)!}}{{L!}}\det (K({\lambda _i},{\lambda _j}))_{i,j = 1}^nwhereK({\lambda _i},{\lambda _j}) \equiv \left\lbrace {\begin{array}{*{20}{l}}
{\frac{{\sin (L... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
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59c64bd1efbec47da930715b0e5ba2f8af92bf9d | subsection | 39 | 57 | Random matrix theory review | However, here in the range with a positive definite eigenvalue we cannot use such a prescription.Similarly, for the LOE case we haveK({\lambda _i},{\lambda _j}) \equiv \left\lbrace {\begin{array}{*{20}{l}}
{L\rho (u)\left( {\begin{array}{*{20}{c}}
{\hat{s}(L\rho (u)\pi ({\lambda _i} - {\lambda _j}))}&{{\bf {D}}\hat{s}(... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
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94f90d3bff2df3e210b5dfb98335535a6749e153 | subsection | 40 | 57 | Random matrix theory review | Based on these knowledge, we could start to summarize the results for form factors in the case of Wishart-Laguerre matrices. | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
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569141db6061633e93ba5c2781de973b236db43a | subsection | 41 | 57 | Theorems | It is straightforward to generalize our previous formula of convolution kernels to the quaternion matrix theory. We haveTheorem 3.1 (Convolution formula for GOE)
We have&\int {\prod \limits _{i=1}^{m}{d{{\lambda }_{i}}}}{K}({{\lambda }_{1}},{{\lambda }_{2}}){K}({{\lambda }_{2}},{{\lambda }_{3}})\ldots {K}({{\lambda }_... | {
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77b96474f0d159b121ce0ba187e1bd1f445b6ada | subsection | 42 | 57 | Two point form factor | Based on our GUE knowledge, we will briefly describe how to compute form factors.We start by computing \mathcal {R}_2 at infinite temperature for GOE.{{\mathcal {R}}_{2}}(t)=L+\int {d}{{\lambda }_{1}}d{{\lambda }_{2}}\left( \frac{1}{4}\text{Tr}\left( K({{\lambda }_{1}},{{\lambda }_{1}}) \right)\text{Tr}\left( K({{\lamb... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
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33022965edcdf6ad7d922768a24ea3f9008511a8 | subsection | 43 | 57 | Two point form factor | It is because of a pole 1/k in the expression of H(k). However, that is an artifact of the Fourier transformation of the integral of the sine kernel \text{sinc}(x). Besides the methods of explicitly computing the Fourier transform, we could also understand the time before 2L as a continuation. As a result, there is a p... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
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b0e10510fbc829ddcbabc3107d067181adeac582 | subsection | 44 | 57 | Final expression and summary | From those calculations we could obtain the final expression for \mathcal {R}_4, which is& {{\mathcal {R}}_{4}}=+{{L}^{4}}r_{1}^{4}(t) \\
& -2{{L}^{3}}r_{1}^{2}(t){{r }_{2}}(t){{r}_{3}}(2t)-4{{L}^{3}}r_{1}^{2}(t){{r }_{2}}(t)+2{{L}^{3}}{{r}_{1}}(2t)r_{1}^{2}(t)+4{{L}^{3}}r_{1}^{2}(t) \\
& +2{{L}^{2}}r _{2}^{2}(t)+{{L}^... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
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02e1ba674764a0734415652bdaf68f60aad205c6 | subsection | 45 | 57 | GSE | In GSE the computations are very similar, and we have to replace these block functions by{{r }_{4}}(t)=\left\lbrace \begin{matrix}
1-\frac{1}{2}\frac{t}{L}+\frac{3}{16}\frac{t}{L}\log \left| \frac{t}{L}-1 \right| & t<2L \\
0 & t>2L \\
\end{matrix} \right.r_{3,1}=r_4r_{3,2}=r_2{{r }_{3,3}}=\left\lbrace \begin{matrix}
1-... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
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160034b64ee1bc1858ec203f50578ec5e931376a | subsection | 46 | 57 | Refined two point form factor | Now we discuss the trick that is similar with our previous improvement. Let us start from GOE. We will use the short distance refined kernel,&\tilde{ K}({\lambda _i},{\lambda _j}) \equiv L\rho ((\lambda _i+\lambda _j)/2)\times \\
&\left( {\begin{array}{*{20}{c}}
{\hat{s}(\pi L\rho (({\lambda _i} + {\lambda _j})/2)({\la... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
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377113dc44712ff32b3720f237c8a6645f058603 | subsection | 47 | 57 | Refined two point form factor | We have&{{\cal R}_2}(t,\beta )\\
&=L{r_1}(2i\beta ) + {L^2}{r_1}(t + i\beta ){r_1}(t - i\beta )\\
&- \frac{1}{2}\int d {\lambda _1}d{\lambda _2}\left( {{\rm {Tr}}\left( {K({\lambda _1},{\lambda _2})K({\lambda _2},{\lambda _1})} \right)} \right){e^{i({\lambda _1} - {\lambda _2})t}}{e^{ - \beta ({\lambda _1} + {\lambda _... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
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70f99f225b056a25d05ba43545ebdff9cf5d8c1d | subsection | 48 | 57 | Refined two point form factor | For LUE, we have&{{\cal R}_2}(t,\beta ) = L{r_1}(2i\beta ) + {L^2}{r_1}(t + i\beta ){r_1}(t - i\beta ) - \int {d{u_2}{e^{ - 2\beta {u_2}}}\max \left( {L\rho ({u_2}) - \frac{t}{{2\pi }},0} \right)}When \beta =0 the integral is\int {d{u_2}{e^{ - 2\beta {u_2}}}\max \left( {L\rho ({u_2}) - \frac{t}{{2\pi }},0} \right)} = \... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
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81d687b478469ac05837c82df5751053b0ed2a1d | subsection | 49 | 57 | Figures | We obtain numerous analytic results in the previous sections. In this section, we will try to plot some of those results.Figures in REF are describing the two point spectral form factors in Gaussian ensembles. One could observe that the main difference among three ensembles is the behavior around the plateau time. We h... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
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1f70a49448d3285e9d926e364fa3ba62cebdee22 | subsection | 50 | 57 | Figures | We choose L=100.][Figure: LOE(up), LUE(middle) and LSE(down) connected form factor \mathcal {R}_2^\text{conn}(t) with different approximations in the infinite temperature. We choose L=100. For LUE case by choosing u in the box approximation the Taylor expansion curve and the box approximation curve are the same, so two... | {
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} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
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63e97b52eefb321aed96e4660c7ba0ea9c0ebc0c | subsection | 51 | 57 | Applications | The spectral form factors of the random matrix theory in the standard ensembles have wide applications in many areas of late time quantum chaos. In this section, we will review and summarize some of the applications with recent interests. | {
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} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
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f7ddbceadfe2489318fea91128dafe7613e1e2e3 | subsection | 52 | 57 | SYK-like models and classifications | One direct application of the random matrix theory form factor results will be matching the qualitative and quantitative behaviors of the spectral form factor of the SYK model. In the majonara SYK model, there exists an eight-fold classification of random matrix theory, with respect to the number of majonara fermions N... | {
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5f73d7fad32aaf46bd5fba3fd307ca87cc229ff1 | subsection | 53 | 57 | Out-of-time-ordered correlation functions | The spectral form factor of the random matrix theory could be related to out-of-time-ordered correlators of the physical models in an interesting way. Here we will discuss the unitary invariance case, where disordered physical models are invariant or nearly invariant under unitary transform. For Gaussian and Wishart-La... | {
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e6ca5af26d44dcb4401dcf3e7f9e35ad1fbe31d6 | subsection | 54 | 57 | Out-of-time-ordered correlation functions | For instance, in the two point case we have{{\cal F}^{(1)}}(t) = \frac{{\mathcal {R}_2^2(t) + {L^2} - 2{\mathcal {R}_2}(t)}}{{{L^2} - 1}}One can generate this type of connections to higher point cases and finite temperatures. | {
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07385ce2fa0b775c26d1f6acf64b73c17342a1f7 | subsection | 55 | 57 | Page states | A connection between Wishart-Laguerre ensembles and the Page states is used to be a modified criterion for quantum chaos in terms of wave functions . The page state, or alternatively called the random pure state, is defined as the following wavefunction in the Hilbert space \mathcal {H}=\mathcal {H}_A \otimes \mathcal ... | {
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d5869a6ede10c8d913fbc9f33d937e867b24cdfc | subsection | 56 | 57 | Conclusion and discussion | In this paper, we investigate in the very detail, to establish a generic framework on the computational technology of the spectral form factors. We hope that those technologies will give a systematic description of spectral form factors that are used in the field of quantum chaos, and will benefit people studying the c... | {
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"doi": "10.1070/sm1967v001n04abeh001994",
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... | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
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] | 2,018 | en | Physics | [
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a2bd87c5670efb1523aec2d93c3e9ba2fc2ef299 | abstract | 0 | 104 | Abstract | In a labeling scheme the vertices of a given graph from a particular class
are assigned short labels such that adjacency can be algorithmically determined
from these labels. A representation of a graph from that class is given by the
set of its vertex labels. Due to the shortness constraint on the labels such
schemes p... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
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18afe4e070d0172c76a5650830916f17d4eb6a53 | subsection | 1 | 104 | Introduction | Suppose you have a database and in one of its tables you need to store large interval graphs in such a way that adjacency can be determined quickly. This means generic data compression algorithms are not an option. A graph is an interval graph if each of its vertices can be mapped to a closed interval on the real line ... | {
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} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
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] | 2,018 | en | Computer Science | [
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07aedd53a73e609bfd16bd2b1fc607f579079f18 | subsection | 2 | 104 | Introduction | This restriction can be slightly weakened without affecting the previous question by considering all graph classes which are a subset of a small and hereditary graph class, the justification being that if a graph class \mathcal {C} has an implicit representation then obviously every subset of \mathcal {C} has an implic... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
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87d1ba34d8f539c45ee4607bc1422033a4b1fa80 | subsection | 3 | 104 | Introduction | The class of graphs with clique-width k seems to have a higher complexity than other hereditary graph classes that are known to have a labeling scheme, as we shall see later on. | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
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4a046071ff22013e8926da7903508cb23ac78f0a | subsection | 4 | 104 | Informative Labeling Schemes. | Labeling schemes are usually understood as a much broader concept than what we consider here. In the following we try to give a more accurate picture of this notion.
The idea of labeling schemes quickly evolved when Peleg recognized that instead of adjacency one can also infer other properties such as distance from the... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
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] | 2,018 | en | Computer Science | [
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a753c9d34b0a77fcfbab91ba89cc645785efa65b | subsection | 5 | 104 | Informative Labeling Schemes. | They also give labeling schemes for vertex- and edge-connectivity. The motivation for considering these properties stems from routing tasks in networks where it is useful to be aware of the capacity and connectivity between nodes. They note that for practical usability such a scheme has to be adapted to a dynamic setti... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
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] | 2,018 | en | Computer Science | [
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6ba63396bff4ea1c75f00b0d9921dd642c8fe3d0 | subsection | 6 | 104 | Informative Labeling Schemes. | For brevity we omit the qualifier `adjacency'. | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
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] | 2,018 | en | Computer Science | [
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d5996032a9e81db3aa3b70b0a0c2c9eeb3de62d7 | subsection | 7 | 104 | Overview of the Paper. | In Section we define how to interpret a set of languages (complexity class) such as ¶ as a set of labeling schemes. For instance, we write {G}_{}¶ to denote the set of graph classes that can be represented by a labeling scheme where the label decoder can be computed `in' ¶. Our definition generalizes the ones given in... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
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] | 2,018 | en | Computer Science | [
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1baa41e768230bd449927578023f60c146898f3a | subsection | 8 | 104 | Overview of the Paper. | Somewhat unexpectedly, this relative simplicity is still high enough to lead us to problems which seem to be situated on the frontier of algorithmic research.
We show that this kind of question naturally fits into the framework of parameterized complexity. For example, whether the Hamiltonian cycle problem is W[1]-hard... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
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] | 2,018 | en | Computer Science | [
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af619279554139f54e47e572ef1e098e26061141 | subsection | 9 | 104 | General Notation and Terminology. | Let \mathbb {N}= \lbrace 1,2,\dots \rbrace be the set of natural numbers and \mathbb {N}_0 is \mathbb {N}\cup \lbrace 0\rbrace . For n \in \mathbb {N} let [n] = \lbrace 1,2,\dots ,n\rbrace and let [n]_0 = [n] \cup \lbrace 0\rbrace . For a set A we write \mathcal {P}(A) to denote the power set of A.
When we say \log n w... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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36ebbc6a1473693367709de250e6d21e0b88803e | subsection | 10 | 104 | General Notation and Terminology. | For an undirected graph G and a vertex u of G we write N(u) to denote the vertices adjacent to u in G. We say two vertices u and v of an undirected graph G are twins if N(u) \setminus \lbrace v\rbrace = N(v) \setminus \lbrace u\rbrace . The twin relation is an equivalence relation.
The graph K_n denotes the complete gr... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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86f0419e15b71f8b2032de4faa79ba9ed5a254e1 | subsection | 11 | 104 | Complexity Classes. | We use the term complexity class informally to mean a countable set of languages with computational restrictions. Unless specified otherwise we consider languages over the binary alphabet \lbrace 0,1\rbrace . We will talk about the standard complexity classes {L} (logspace), ¶, , (polynomial-time hierarchy), and (set o... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
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] | 2,018 | en | Computer Science | [
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37c44012f65c20f3a8d77848361d5a6aac308c0e | subsection | 12 | 104 | First-Order Logic. | Let \mathcal {N} be the structure that has \mathbb {N}_0 as universe equipped with the order relation `\operatorname{<}' and addition `\operatorname{+}' and multiplication `\operatorname{\times }' as functions.
For n \ge 1 let \mathcal {N}_n be the structure that has [n]_0 = \lbrace 0,1,\dots ,n \rbrace as universe, th... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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315ac9a8f164f7bb34ca5fd670f3dd607b802d0a | subsection | 13 | 104 | Graph Class Properties. | Let \mathcal {C} be a graph class.
We call \mathcal {C} small if it has at most n^{\mathcal {O}(n)} graphs on n vertices. Stated differently, |\mathcal {C}_n| \in n^{\mathcal {O}(n)} = 2^{\mathcal {O}(n \log n)}; in the literature this is also called factorial speed of growth (, ).
\mathcal {C} is tiny if there exists ... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
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] | 2,018 | en | Computer Science | [
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77c114186060a0a0882c6194335b7ce2cc04fcae | subsection | 14 | 104 | Graph Class Properties. | For example, interval graphs are defined as the class of graphs \mathcal {C}_{\mathcal {F}} where \mathcal {F} is the set of (closed) intervals on the real line.An undirected graph H is called a minor of an undirected graph G if H can be obtained from G by deleting vertices and edges, and contracting edges (merging two... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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88a41a421489856880ccc91e2139b8c86be2fe3b | subsection | 15 | 104 | Graph Classes and Parameters. | We consider a graph parameter \lambda to be a total function which maps unlabeled graphs to natural numbers. We say a graph class \mathcal {C} is bounded by a graph parameter \lambda if there exists a c \in \mathbb {N} such that \lambda (G) \le c for all G \in \mathcal {C}. A graph parameter can be interpreted as the s... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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4af965eaea5c379e89d85a1967feedface92ed27 | subsection | 16 | 104 | Graph Classes and Parameters. | A kd-line segment graph is the intersection graph of line segments in \mathbb {R}^k. A graph G is a k-dot product graph if there exists a function f \colon V(G) \rightarrow \mathbb {R}^k such that two distinct vertices u,v in G are adjacent iff f(u) \cdot f(v) \ge 1 . | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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8894a9b774c46ad49c727c7c744cd3165a4edb33 | subsection | 17 | 104 | Labeling Schemes. | We use the terms implicit representation and labeling scheme interchangeably.Definition 2.1 A labeling scheme is a tuple S = (F,c) where F \subseteq \lbrace 0,1\rbrace ^* \times \lbrace 0,1\rbrace ^* is called label decoder and c \in \mathbb {N} is the label length. A graph G on n vertices is in the class of graphs spa... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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3ce11d69f39e65ac52dfacf72b4de89536a09c7e | subsection | 18 | 104 | Labeling Schemes. | A graph class has a polynomially sized sequence of universal graphs (polynomial universal graphs for short) iff it is in {G}_{}{ALL} (no computational complexity constraint).It is not difficult to see that there is a language L in ^0 such that F_L = F_{\mathrm {Intv}} and therefore interval graphs are in {G}_{}^0. It i... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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f1b504a29aee2686bbfdc20eddf0c97c81e0b29a | subsection | 19 | 104 | Hierarchy of Labeling Schemes | When labeling schemes were introduced by Muller in the label decoder was required to be computable. Clearly, this is a reasonable restriction since otherwise it would be impossible to query edges in a labeling scheme with an undecidable label decoder. Taking this consideration a step further, in order for a labeling sc... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
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] | 2,018 | en | Computer Science | [
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a0a546f320625076cca09b83c3f4f907e218d31f | subsection | 20 | 104 | Hierarchy of Labeling Schemes | However, \subseteq (\exp (1.5^n)) and {G}_{}(\exp (1.5^n)) \subsetneq {G}_{}{2EXP} does follow from Theorem REF and therefore {G}_{}\subsetneq {G}_{}{2EXP} holds. As a consequence {G}_{}k is a strict subset of {G}_{}{R} for every k \ge 0. That {G}_{}{R} is a strict susbet of {G}_{}{ALL} follows from the fact that its d... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
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8c8fe679d279c606523ef4e8c95be45b2be5378c | subsection | 21 | 104 | Hierarchy of Labeling Schemes | In the last step we construct a label decoder for \mathcal {C}_{(t(n))} and show that it can be computed in time \exp ^2(n) \cdot t(n).Definition 3.3 A surjective function \tau : \mathbb {N}\rightarrow \mathbb {N}^2 is an admissible pairing if all of the following holds:|\tau ^{-1}(y,z)| is infinite for all y,z \in \ma... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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e4924fe503debc5648670031a078304c8453fc04 | subsection | 22 | 104 | Hierarchy of Labeling Schemes | The diagonalization graph class of {A} is defined as:\mathcal {C}_{A} = \bigcup _{n \in \operatorname{dom}(\tau )} \left\lbrace G \in \mathcal {G}_n \: |
G \text{ is the smallest graph w.r.t.~$\prec $ not in } \text{gr}_{}(S_{\tau (n)})
\right\rbracewhere \mathcal {G}_n denotes the set of all graphs on n vertices.When ... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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6d01657c875c85cdd2d610e81bebe69669103d92 | subsection | 23 | 104 | Hierarchy of Labeling Schemes | For every m \in \mathbb {N} such that there exists G \in \mathcal {C} on 2^m vertices and for all x, y \in \lbrace 0,1\rbrace ^m let(x,y) \in F_{\mathcal {C}} \Leftrightarrow (x,y) \in E(G_0)Observe that the diagonalization graph class \mathcal {C}_{{A}} of some set of languages {A} satisfies the prerequisite of the pr... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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6fe940b984880dd1fd57b572f0b408d78f442032 | subsection | 24 | 104 | Hierarchy of Labeling Schemes | Due to the fact that t is time-constructible it is possible to run a counter during the simulation of M_y in order to not exceed the z \cdot t(|w|) steps. The input length is n := x + |w|. The simulation can be run in time n^{\mathcal {O}(1)} \cdot z \cdot t(|w|). Since z \le x \le n the desired time bound follows.Lemm... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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229c2ad90e8117dc349477df0f5c56bcd93b7c60 | subsection | 25 | 104 | Hierarchy of Labeling Schemes | In total this means the algorithm runs in time \mathcal {O}(\exp ^2(m^{\mathcal {O}(1)})\cdot t(2m) ) which is the required time bound since the input length is 2m.Lemma REF states that \mathcal {C}_{(t(n))} \notin {G}_{}(t(n)) and from Lemma REF it follows that \mathcal {C}_{(t(n))} \in {G}_{}(\exp ^2(n) \cdot t(n)) t... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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5df884ba61cdef9769622a5e81624bdf3fa68fa6 | subsection | 26 | 104 | Expressiveness of Primitive Labeling Schemes | To understand the limitations of labeling schemes it is reasonable to start with very simple ones first and then gradually increase the complexity. In this section we present two such simple families of labeling schemes and explain how they relate to other well-known sets of graph classes. These two families of labelin... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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1b601777e9b41083e3447d38de1ca8ee1a5b991f | subsection | 27 | 104 | Expressiveness of Primitive Labeling Schemes | It holds that G has degeneracy at most 2k. Every induced subgraph of G has bijective or-pointer number at most k. Additionally, every graph with bijective or-pointer number at most k can have at most kn edges which implies that such a graph must have a vertex with degree at most 2k.“\Leftarrow ”: Let G have degeneracy ... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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3d0a227fc24e7188d5b5e3af01131bac170e5a58 | subsection | 28 | 104 | Expressiveness of Primitive Labeling Schemes | There exists an induced subgraph of G with c vertices which has bijective or-pointer number at most k and therefore this subgraph can have at most kc edges. Informally, if a graph has many edges then in any k-or-pointer representation of this graph there cannot be many unique ids. As a consequence the structure of such... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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431be4fa814092af1b189a446de4cbb080d87a04 | subsection | 29 | 104 | Expressiveness of Primitive Labeling Schemes | To prove the above statement we show that (I) {G}_{}{R}\lnot \subseteq [\mathrm {Small} \cap \mathrm {Hereditary}]_{\subseteq } and (II) for every graph class \mathcal {C} \in {G}_{}{R} there exists a graph class \mathcal {D} in {G}_{}^0 with [\mathcal {C}]_{\subseteq } \subseteq [\mathcal {D}]_{\subseteq }.For (I) we ... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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fb51d85553ff150ce62d20b474a67d4b3801a5b3 | subsection | 30 | 104 | Expressiveness of Primitive Labeling Schemes | The partial labeling \ell ^{\prime } \colon V(G_0) \rightarrow \lbrace 0,1\rbrace ^{c (\log n + p(\log n))} with \ell ^{\prime }(u) = \ell (u) 0^{c p(\log n)} for all u \in V(G) shows that G is an induced subgraph of G_0. It remains to argue that for every computable label decoder F there exists a padding function p su... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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f81e10d5acd627614113ad705d50221f1717c036 | subsection | 31 | 104 | Expressiveness of Primitive Labeling Schemes | The first bit of v for the i-th level denotes whether v is in the left child of x_i. If v is in the left child of x_i then this means v is in the balanced k-module S^i and one also stores the index j such that v \in S^i_j for 1 \le j \le k. If v is in the right child of x_i then one stores the subset of X of \lbrace 1,... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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791624292dd6c3a8e4a17ae3f46887fad37d772b | subsection | 32 | 104 | Expressiveness of Primitive Labeling Schemes | Choose an unordered partition of n into i parts p_1, p_2, \dots , p_i \in [n], which means p_1 + \dots + p_i =n and p_1 \le p_2 \le \dots \le p_i. Let P_{n,i} denote the number of such partitions. This partition tells us that the first p_1 vertices of G are a twin class, the next p_2 vertices of G are a twin class and ... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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c43a7c4af460f56ca34b563fcb66002dab536e5f | subsection | 33 | 104 | Reductions Between Graph Classes | The concept of reduction is vital to complexity theory as it enables one to formally compare the complexity of problems as opposed to just treating each problem separately. In our context we want to say a graph class \mathcal {C} reduces to a graph class \mathcal {D} if the adjacency of graphs in \mathcal {C} can be ex... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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3ab33ff4002b3ccd3b0799bc7484ca6412eda62a | subsection | 34 | 104 | Reductions Between Graph Classes | While it is technically more tedious to define than the algebraic variant, the underlying intuition is just as simple. Instead of expressing the adjacency of a graph G using a sequence of graphs as before, we do this in terms of a single, larger graph H. Informally, every vertex of G is assigned to a constant-sized sub... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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7640bc02ed1c900d5bfcf766c66a5cfbe5aa5066 | subsection | 35 | 104 | Algebraic Reductions | Definition 5.1 Let f be a k-ary boolean function and H_1,\dots ,H_k are graphs with the same vertex set V and k \ge 0. We define f(H_1,\dots ,H_k) to be the graph with vertex set V and an edge (u,v) iff f( x_1, \dots , x_k) =1 where x_i = \llbracket (u,v) \in E(H_i) \rrbracket for all u, v \in V.The constant boolean fu... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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3cca0d5ac33305b5c7661a262c2518b894cd2742 | subsection | 36 | 104 | Algebraic Reductions | The following statement shows that f_1 = f_2 iff F_1 and F_2 are logically equivalent.Lemma 5.3 (Compositional Equivalence)
Given boolean functions f,g,h_1,\dots ,h_l where f,h_1,\dots ,h_l have arity k and g has arity l
such that f is the composition of g,h_1,\dots ,h_l, i.e. f(\vec{x}) = g(h_1(\vec{x}),\dots ,h_l(\v... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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dded317cdd156da9d42a5c17954af1489bb58029 | subsection | 37 | 104 | Algebraic Reductions | Boolean algebra is a special case of this algebra on graph classes if one restricts the universe to the class of complete graphs \left\lbrace K_n \mid n \in \mathbb {N} \right\rbrace and the edge-complement of it.Definition 5.4 (Algebraic Reduction)
Let F be a set of boolean functions and \mathcal {C}, \mathcal {D} ar... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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33f0946667bf8d5272258872c709c4c90ddd9a66 | subsection | 38 | 104 | Algebraic Reductions | Therefore G = h(\vec{I^1},\dots ,\vec{I^k}) = f(g(\vec{I^1}),\dots ,g(\vec{I^k})) with \vec{I^i} = (I^i_1,\dots ,I^i_l) for i \in [k].We say a set of graph classes \mathbb {A} is closed under a k-ary boolean function f if for all \mathcal {C}_1,\dots ,\mathcal {C}_k \in \mathbb {A} it holds that f(\mathcal {C}_1,\dots ... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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36f0a3a547254b98c4a3ffa4f132acc116c39f02 | subsection | 39 | 104 | Algebraic Reductions | Since every function in F can be expressed as composition of functions from B and projections and \mathbb {A} is closed under every function from B and projections it follows that it is closed under every function from F.If \mathbb {A} is closed under union then the implication in Lemma REF becomes an equivalence.Corol... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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e6de5439c6030341940dcf04cde28617f82b2f04 | subsection | 40 | 104 | Algebraic Reductions | Given two labeling schemes S_1 = (F_1,c_1) and S_2 = (F_2,c_2) let the labeling scheme S_3 = (F_3,c_1+c_2) with (x_1x_2,y_1y_2) \in F_3 \Leftrightarrow (x_1,y_1) \in F_1 \wedge (x_2,y_2) \in F_2 and \frac{|x_i|}{|x|} = \frac{|y_i|}{|y|} = \frac{c_i}{c} for i \in [2]. It holds that \text{gr}_{}(S_1) \wedge \text{gr}_{}(... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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012ca39783d18af180a8dae9fd469f9950ed1279 | subsection | 41 | 104 | Algebraic Reductions | Since classes such as {G}_{}¶ and {G}_{}^0 contain directed graph classes it trivially follows that no undirected graph class can be complete for them.
If we assume that only undirected graph classes can be small and hereditary then it trivially holds that {G}_{}^0 and its supersets do not have a \le _{\mathrm {BF}}-co... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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4f90997a651f5725c18547dba17e5398274462d2 | subsection | 42 | 104 | Subgraph Reductions | Given k \ge 0, a k^2-ary boolean function f and a (k \times k)-matrix A over \lbrace 0,1\rbrace . We write f(A) to denote the value of f when plugging in the entries of A from left to right and top to bottom. We say f is diagonal if the value of f only depends on the k entries on the main diagonal of A. Given k,l \ge 1... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.039852041751146317,
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0.009993525221943855,
0.03384067118167877,... | |
5f91daaeea64916622e59daf7ee1b35d602bdcaf | subsection | 43 | 104 | Subgraph Reductions | It holds that \mathcal {C}_Y \le _{\mathrm {sg}}\mathcal {C}_X if there exists a k \in \mathbb {N}, a k^2-ary boolean function f and a labeling \ell \colon Y \rightarrow X^k such that for all u \ne v \in Y it holds that u \cap v \ne \emptyset \Leftrightarrow f(A_{uv}^\ell ) = 1 with A_{u,v}^{\ell } = (\llbracket \ell (... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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0... | |
1e578a306d27d136063dda072274a7f8e2409ec5 | subsection | 44 | 104 | Subgraph Reductions | Let G be a graph in \mathcal {C} with n vertices. We show that there exists a graph H^{\prime \prime } on n^k vertices in \mathcal {D} such that G has an (H^{\prime \prime },f)-representation. There exists a graph H in \mathcal {D} such that G has an (H,f)-representation via a labeling \ell \colon V(G) \rightarrow V(H)... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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0... | |
b5d2d4a9ab1cb71376cbe2b462f8e38ead5eee3f | subsection | 45 | 104 | Subgraph Reductions | For u \in V(G) let \ell (u) = (u_1,\dots ,u_k) and let \ell ^{\prime }(u_i) = (u_{i,1},\dots ,u_{i,l}) for i \in [k]. We define \ell ^{\prime \prime }(u) as (u_{1,1},\dots ,u_{1,l},u_{2,1},\dots ,u_{2,l},\dots ,u_{k,1},\dots ,u_{k,l}).
The same argument shows that \le _{\mathrm {sg}}^{\mathrm {diag}} is transitive beca... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.... | |
7010c73b91d62f11de993a7509f3cdb445c9c6eb | subsection | 46 | 104 | Subgraph Reductions | Then a vertex u of G can be labeled with \ell _H(u_1) \dots \ell _H(u_k) where u_i is the i-th component of \ell _G(u).Corollary 5.16 The classes {G}_{}{L}, {G}_{}¶, {G}_{}, {G}_{}{R} and {G}_{}{ALL} are closed under \le _{\mathrm {sg}}.Observe that for all these complexity classes the same argument as the one given fo... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.0073... | |
9fde26cb657a65def3363b3b8ebdc14ab423920b | subsection | 47 | 104 | Subgraph Reductions | The class \mathcal {C}^{\prime } cannot be small since this would imply that \mathcal {C} is in [\mathrm {Small} \cap \mathrm {Hereditary}]_{\subseteq }. We argue that there exists a finite graph class \mathcal {E} such that \mathcal {C}^{\prime } \setminus \mathcal {E} \le _{\mathrm {sg}}\mathcal {D} via c,k and f. Fr... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.05800743028521538,
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0.021756600588560104,
-... | |
6caebce44291a946832e968f20adb82718101d6e | subsection | 48 | 104 | Subgraph Reductions | \le _{\mathrm {sg}} then this would imply that {G}_{}^0 is a subset of [\mathrm {Small} \cap \mathrm {Hereditary}]_{\subseteq } since [\mathrm {Small} \cap \mathrm {Hereditary}]_{\subseteq } is closed under \le _{\mathrm {sg}}.Recall that a graph class \mathcal {C} is called self-universal if for every finite subset of... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.05263811722397804,
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0.06402015686035156,
... | |
46a6a28b5df2230806ea0d1382d4b7efc1a2c084 | subsection | 49 | 104 | Subgraph Reductions | Therefore this direction follows from the previous lemma.“\Leftarrow ”: Let \mathcal {C} \le _{\mathrm {sg}}^{\mathrm {diag}}\mathcal {D} via c,k \in \mathbb {N} and a k^2-ary diagonal boolean function f. Let g be the k-ary boolean function which underlies f, i.e. f(A) = g(A_{1,1},\dots ,A_{k,k}). We claim that \mathca... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.04630561172962189,
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... | |
475f2814fe05960db4affece790df3fc4373ac6d | subsection | 50 | 104 | Logical Labeling Schemes | Many of the graph classes which are known to have an implicit representation can be represented by a labeling scheme where a vertex label is interpreted as a fixed number of non-negative, polynomially bounded integers and the label decoding algorithm performs basic arithmetic on these integers and compares the results.... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
0.002520818030461669,
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... | |
6c7a853721d42e487c27fcacfe4e0b513cc85db6 | subsection | 51 | 104 | Definition and Basic Properties | Definition 6.1 A (quantifier-free, atomic) logical labeling scheme is a tuple S=(\varphi ,c) with a (quantifier-free, atomic) formula \varphi \in _{2k} and c,k \in \mathbb {N}.
A graph G is in \text{gr}_{}(S) if there exists a labeling \ell \colon V(G) \rightarrow {[n^c]}_0^k such that (u,v) \in E(G) \Leftrightarrow \m... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.02910754084587097,
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0.01975100114941597,
0.0... | |
b0062e657885408b413efffb2e95d7f1ece742e9 | subsection | 52 | 104 | Definition and Basic Properties | For all u, v \in V(G) it holds that (u,v) \in E(G) \Leftrightarrow \mathcal {N}_{n^c},(\ell (u),\ell (v)) \models \varphi \Leftrightarrow (\ell ^{\prime }(u),\ell ^{\prime }(v),\mathrm {bin}(n^c)) is a positive instance of the bounded model checking problem for \varphi . This almost gives us a labeling scheme which sho... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.0698239877820015,
0.011093102395534515,
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0.017013493925333023,
0.... | |
51cc4baff84f76ef7b447cbb322870883c0b161d | subsection | 53 | 104 | Definition and Basic Properties | The naive approach to model-check a quantifier-free formula \varphi is as follows: evaluate the terms of \varphi (expressions involving addition and multiplication of the free variables (the input)), then evaluate the atomic formulas which means comparing numbers and finally compute the underlying boolean function of \... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.02590092457830906,
0.007806908339262009,
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0.02580934949219227,
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-0.0022817845456302166,
0.... | |
72a991bcae242cbda61bcb9eb62e481509ecb4db | subsection | 54 | 104 | Definition and Basic Properties | We claim that \mathcal {C} \subseteq \text{gr}_{}(S^{\prime }). Consider a graph G \in \mathcal {C} with vertex set V. There exist k graphs H_1,\dots ,H_k\in \mathcal {D} with vertex set V such that G = f(H_1,\dots ,H_k). Since H_i is in \mathcal {D} it is also in \text{gr}_{}(S) via a labeling \ell _i \colon V \righta... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.04728573560714722,
0.011348271742463112,
-0.012760127894580364,
-0.01361487340182066,
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0.0343424491584301,
0.003655562875792384,
0.003130886470898986,
0.06435012072324753,
0.018... | |
1f3b9f6e67725287564e10bd8a3bc038f442e439 | subsection | 55 | 104 | Definition and Basic Properties | We define \ell (u) as (u_{1,1},\dots ,u_{1,l},u_{2,1},\dots ,u_{2,l},\dots ,u_{k,1},\dots ,u_{k,l}).
It can be verified that G is in \text{gr}_{}(\psi ,cd) via the labeling \ell . No new atoms or quantifiers are introduced in \psi compared to \varphi and thus it remains in the same class of formulas.In a logical labeli... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.07830312103033066,
-0.0061641582287848,
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0.0456208735704422,
-0.0035... | |
101654044f8f03747003d0b6fc23fb001e699844 | subsection | 56 | 104 | Definition and Basic Properties | We construct a logical labeling scheme S^{\prime }=(\psi ,c) such that \mathcal {C} \subseteq \text{gr}_{\infty }(S^{\prime }) and S^{\prime } is in {G}_{}_{\mathrm {qf}}(\sigma ). We assume w.l.o.g. that we have access to the constants c_0=0 and c_1=n^c in \psi .
The constants can be realized by adding two variables t... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.026053467765450478,
0.012317017652094364,
0.016804257407784462,
0.013667767867445946,
0.012133864685893059,
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0.04123987630009651,
0.004819204565137625,
-0.0046932874247431755,
0.021917270496487617,
-0.00... | |
f186df4a81b5936dadcfa273c13ffd87081c108e | subsection | 57 | 104 | Definition and Basic Properties | Then A^{\prime } is the following formula (order of operation is implied by indentation and reading a propositional formula of the form \varphi \rightarrow \alpha \wedge \lnot \varphi \rightarrow \beta as “if \varphi then \alpha else \beta ”).& c_1 < +(x_1,y_2) \rightarrow \\&\hspace{28.45274pt} c_1 < \times (c_0,x_2) ... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.004654222633689642,
0.009659418836236,
-0.013230199925601482,
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0.02395780198276043,
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0.013588803820312023,
0.02050909772515297,
0.01... | |
c7aef0742e486edc1a63840a83dae23f5265f2ea | subsection | 58 | 104 | Definition and Basic Properties | Therefore \text{gr}_{\infty }(\varphi ,c) \subseteq \text{gr}_{}(\varphi ,cd) and \mathcal {C} \in {G}_{}_{\mathrm {qf}}(\sigma ).Fact 6.9 Let \sigma = \emptyset , or \sigma \subseteq \lbrace \operatorname{<},\operatorname{+},\operatorname{\times }\rbrace and `\operatorname{<}' is in \sigma .
{G}_{}(\sigma ) and {G}_{}... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.06425073742866516,
0.034016892313957214,
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0.01380506157875061,
0.021264372393488884,
-0.021676234900951385,
0.023933859542012215,
... | |
e4aaa71d61902ec76a492305cf6dffe43fa28495 | subsection | 59 | 104 | Definition and Basic Properties | The correctness relies on the fact that the labeling schemes S,S_1,S_2 are all interpreted over the same universe \mathcal {N}_{n^c} for all graphs with n vertices and n \in \mathbb {N}.Next, let us explain why (\star ) holds. If \sigma = \emptyset then (\star ) obviously holds for {G}_{}_{\mathrm {qf}}(=). Since {G}_{... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.040319379419088364,
0.028934724628925323,
-0.020144427195191383,
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0.0344897024333477,
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0.0005679927417077124,
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0.036687277257442474,
0.02189943566918373,
0.020831169560551643,
-0.0003004495520144701,
0.019564513117074966,
-... | |
b87c46e9d524e0135ca63cf1c2131aafb455aca2 | subsection | 60 | 104 | Definition and Basic Properties | Due to Lemma REF there exists a a logical labeling scheme S=(\varphi ,c) in {G}_{}_{\mathrm {qf}}(\sigma ) such that \mathcal {C} \subseteq \text{gr}_{\infty }(S).
Let A_1,\dots ,A_a be the atoms of \varphi and f is the underlying a-ary boolean function of \varphi .
Let \varphi have 2ak variables x_{i,j},y_{i,j} with i... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.04600847512483597,
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0.008679980412125587,
0.017176903784275055,
... |
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