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8596d6d1ba563dc3474f0679102df2fed616ed58 | subsection | 168 | 197 | Covariant canonical formalism | 1975) 199--220}.
}\bibitem {Carlip2014}
S.~Carlip, \emph {{Black hole thermodynamics}},
\href {http://dx.doi.org/10.1142/S0218271814300237}{\emph {Int. J. Mod. Phys.
D} {\bf 23} (Oct. 2014) 1430023}.
[\href {https://arxiv.org/abs/1410.1486}{{arXiv:1410.1486}}].
}\bibitem {Wald1993a}
R.~M. Wald, \emph {{Black hole entro... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.021010063588619232,
0.001216813689097762,
-0.025511128827929497,
0.03237716108560562,
0.009383578784763813,
-0.04037227854132652,
0.04418674111366272,
-0.0034711614716798067,
0.006339637562632561,
0.018675610423088074,
0.02126944623887539,
-0.0048939562402665615,
-0.030683541670441628,
... | |
18d66d9cdba4b01bbb4690915c84981b8b168474 | subsection | 169 | 197 | Covariant canonical formalism | Lett.}
{\bf 71} (1993) 666--669}.
[\href {https://arxiv.org/abs/hep-th/9303048}{{arXiv:hep-th/9303048}}].
}\bibitem {Frolov1993}
V.~Frolov and I.~Novikov, \emph {{Dynamical origin of the entropy of a black
hole}}, \href {http://dx.doi.org/10.1103/PhysRevD.48.4545}{\emph {Phys. Rev. D}
{\bf 48} (1993) 4545--4551}.
[\hre... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.02838810160756111,
-0.011364392936229706,
-0.03548131510615349,
0.013210154138505459,
-0.031072843819856644,
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0.04136944189667702,
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0.03621351718902588,
0.01754235289990902,
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0.0045076217502355576,
-0.03935588523745537,
... | |
6ead61e33b55ebab0a3b3f0707bb1ad024ae9885 | subsection | 170 | 197 | Covariant canonical formalism | A} {\bf 42} (May 2009) 504005}.
[\href {https://arxiv.org/abs/0905.4013}{{arXiv:0905.4013}}].
}\bibitem {Casini2017}
H.~Casini, E.~Test{é} and G.~Torroba, \emph {{Markov Property of the
Conformal Field Theory Vacuum and the a Theorem}},
\href {http://dx.doi.org/10.1103/PhysRevLett.118.261602}{\emph {Phys. Rev.
Lett.} {... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.013215902261435986,
-0.024051111191511154,
-0.02965184673666954,
0.02768319472670555,
0.00243792193941772,
-0.04273040220141411,
0.04447013884782791,
-0.05756395310163498,
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0.03549675643444061,
0.02467680722475052,
0.007752515841275454,
-0.007161157205700874,
-0.0... | |
7fe2e89f679db3852ebde3f753fe157937653dd3 | subsection | 171 | 197 | Covariant canonical formalism | Claudio~Teitelboim, ed.), pp.~118--131.
{World Scientific}, June 1998,
[\href {https://arxiv.org/abs/hep-th/9309145}{{arXiv:hep-th/9309145}}].
}\bibitem {Holzhey:1994we}
C.~Holzhey, F.~Larsen and F.~Wilczek, \emph {{Geometric and renormalized entropy
in conformal field theory}},
\href {http://dx.doi.org/10.1016/0550-32... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.01152724027633667,
0.024229327216744423,
-0.02779964543879032,
0.007033828645944595,
0.0025251598563045263,
-0.02239839732646942,
0.06084796041250229,
-0.05511103942990303,
0.021513445302844048,
0.04140957072377205,
0.011405178345739841,
-0.0027063456363976,
-0.01698189042508602,
0.0006... | |
85f6e8ff0327438427202f37306d691772285be2 | subsection | 172 | 197 | Covariant canonical formalism | 1995) 609--617}.
[\href {https://arxiv.org/abs/hep-th/9407001}{{arXiv:hep-th/9407001}}].
\end{equation}\bibitem {Fursaev1996}
D.~V. Fursaev and S.~N. Solodukhin, \emph {{On one-loop renormalization of
black-hole entropy}},
\href {http://dx.doi.org/10.1016/0370-2693(95)01290-7}{\emph {Phys. Lett. B}
{\bf 365} (Jan. 1996... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.05101463943719864,
0.007707878481596708,
-0.02710915543138981,
0.003308552550151944,
-0.012471431866288185,
0.0018792960327118635,
0.019816990941762924,
0.009885611943900585,
0.008382937870919704,
0.02308168262243271,
-0.02848215587437153,
0.008146476000547409,
-0.016033608466386795,
0.... | |
57f6a3d9cc7d8a858f18716eb8037095ddb7437e | subsection | 173 | 197 | Covariant canonical formalism | Wall, \emph {Quantum extremal surfaces: Holographic
entanglement entropy beyond the classical regime},
\href {http://dx.doi.org/10.1007/JHEP01(2015)073}{\emph {J. High Energy Phys.}
{\bf 1501} (2015) 73}.
[\href {https://arxiv.org/abs/1408.3203}{{arXiv:1408.3203}}].
}\bibitem {VanRaamsdonk2010}
M.~{Van Raamsdonk}, \emp... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03662562370300293,
0.026980876922607422,
-0.014383187517523766,
0.010217023082077503,
-0.03033822402358055,
-0.011201336979866028,
0.020281439647078514,
-0.04150903970003128,
0.021609118208289146,
0.025363244116306305,
0.02292153611779213,
0.01782447099685669,
-0.03381766006350517,
0.00... | |
541fb11a6cfad712c1e2ce2f873700d2de654284 | subsection | 174 | 197 | Covariant canonical formalism | Mod. Phys.}
{\bf 88} (Feb. 2016) 015002}.
[\href {https://arxiv.org/abs/1409.1231}{{arXiv:1409.1231}}].
}\bibitem {Witten:1998qj}
E.~Witten, \emph {{Anti-de Sitter space and holography}},
\href {http://dx.doi.org/10.4310/ATMP.1998.v2.n2.a2}{\emph {Adv. Theor. Math.
Phys.} {\bf 2} (1998) 253--291}.
[\href {https://arxiv... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.052644286304712296,
-0.013870625756680965,
-0.03579811379313469,
-0.00021207923418842256,
-0.005111836828291416,
-0.03823959082365036,
0.0028744544833898544,
-0.015427065081894398,
0.05157614126801491,
0.03567604348063469,
-0.0027962508611381054,
0.0037766555324196815,
-0.0077821989543735... | |
24bcb0952ea657c1332a6a1b38d375fa40b318fc | subsection | 175 | 197 | Covariant canonical formalism | High Energy
Phys.} {\bf 2006} (Aug. 2006) 045}.
\end{}\bibitem {Randall1999}
L.~Randall and R.~Sundrum, \emph {{An Alternative to Compactification}},
\href {http://dx.doi.org/10.1103/PhysRevLett.83.4690}{\emph {Phys. Rev. Lett.}
{\bf 83} (Dec. 1999) 4690--4693}.
[\href {https://arxiv.org/abs/hep-th/9906064}{{arXiv:hep-... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.033300213515758514,
-0.0028843905311077833,
-0.010057213716208935,
-0.004395261872559786,
-0.03217087686061859,
-0.03290341794490814,
0.039282653480768204,
-0.033880144357681274,
0.020618051290512085,
0.036932408809661865,
0.0011197997955605388,
0.0015585629735141993,
-0.00905759725719690... | |
7455ef2b5ae877103642995774cdc66a4cf93380 | subsection | 176 | 197 | Covariant canonical formalism | \end{}\bibitem {Jacobson1995a}
T.~Jacobson, \emph {{Thermodynamics of Spacetime: The Einstein Equation of
State}}, \href {http://dx.doi.org/10.1103/PhysRevLett.75.1260}{\emph {Phys.
Rev. Lett.} {\bf 75} (Aug. 1995) 1260--1263}.
[\href {https://arxiv.org/abs/gr-qc/9504004}{{arXiv:gr-qc/9504004}}].
}\bibitem {Verlinde201... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03200241178274155,
0.010837803594768047,
-0.019311273470520973,
0.0033558295108377934,
-0.010227653197944164,
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0.07297403365373611,
-0.03938523307442665,
0.03569382429122925,
0.016214758157730103,
0.012500464916229248,
0.015253770165145397,
-0.0279754139482975,
0.0... | |
2ba509c5b063a451b4f75b661ca0e142f8797792 | subsection | 177 | 197 | Covariant canonical formalism | High
Energy Phys.} {\bf 2017} (Aug. 2017) 57}.
[\href {https://arxiv.org/abs/1705.03026}{{arXiv:1705.03026}}].
}\bibitem {Haehl2017}
F.~M. Haehl, E.~Hijano, O.~Parrikar and C.~Rabideau, \emph {{Higher Curvature
Gravity from Entanglement in Conformal Field Theories}},
[\href {https://arxiv.org/abs/1712.06620}{{arXiv:171... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03551553562283516,
-0.025873860344290733,
-0.008055069483816624,
-0.030145488679409027,
-0.018901953473687172,
-0.029001301154494286,
0.06462362408638,
-0.027765581384301186,
-0.007910139858722687,
0.02958102338016033,
-0.0008753022993914783,
0.011029953137040138,
-0.024043163284659386,
... | |
5a0767f5e1cd5c7e83ca4c2b2ad7e90fbda490e5 | subsection | 178 | 197 | Covariant canonical formalism | Rev. D} {\bf 86} (2012) 064042}.
[\href {https://arxiv.org/abs/1206.5831}{{arXiv:1206.5831}}].
}\bibitem {Donnelly2015E}
W.~Donnelly and A.~C. Wall, \emph {Entanglement entropy of the electromagnetic
edge modes},
\href {http://dx.doi.org/10.1103/PhysRevLett.114.111603}{\emph {Phys. Rev.
Lett.} {\bf 114} (2015) 111603}.... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.05310896039009094,
0.01660417951643467,
-0.02565406821668148,
-0.02594403177499771,
-0.001113113365136087,
-0.018374478444457054,
0.02751593478024006,
-0.028507912531495094,
0.016085298731923103,
0.009782426059246063,
-0.007172761484980583,
0.00729485135525465,
-0.030659740790724754,
-0... | |
c311702995f85d7d097d625af69db66a3817e38b | subsection | 179 | 197 | Covariant canonical formalism | High Energy Phys.}
{\bf 26} (2016) 105}.
[\href {https://arxiv.org/abs/1602.01380}{{arXiv:1602.01380}}].
\end{}\bibitem {Carroll2016a}
S.~M. Carroll and G.~N. Remmen, \emph {{What is the entropy in entropic
gravity?}}, \href {http://dx.doi.org/10.1103/PhysRevD.93.124052}{\emph {Phys.
Rev. D} {\bf 93} (June 2016) 124052... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.036893393844366074,
-0.004459096118807793,
-0.00887241866439581,
0.00349976122379303,
-0.0017803962109610438,
-0.04293549060821533,
0.040768880397081375,
-0.0493437722325325,
0.042630333453416824,
0.01251140795648098,
-0.0015944415936246514,
-0.00018821944831870496,
-0.02683849446475506,
... | |
2fec9895f9e3bd4ba0dd9a9a00c4a8ccf2b8b5bc | subsection | 180 | 197 | Covariant canonical formalism | Leigh and O.~Parrikar, \emph {{Shape Dependence of
Entanglement Entropy in Conformal Field Theories}},
\href {http://arxiv.org/abs/1511.05179}{\emph {J. High Energy Phys.} {\bf 2016}
(Apr. 2016) 88}.
[\href {https://arxiv.org/abs/1511.05179}{{arXiv:1511.05179}}].
}\bibitem {Casini2016a}
H.~Casini, D.~A. Galante and R.~... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.013971739448606968,
-0.02029755711555481,
-0.036566123366355896,
-0.003288814565166831,
-0.0014631791273131967,
-0.033788565546274185,
0.039068982005119324,
-0.03925211727619171,
0.01929030939936638,
0.012872924096882343,
-0.023777136579155922,
-0.03272027149796486,
-0.0012972122058272362... | |
b6ce79bd0a282c6fd7ee4e2647539db311382973 | subsection | 181 | 197 | Covariant canonical formalism | B}
{\bf 556} (Sept. 1999) 89--114}.
[\href {https://arxiv.org/abs/hep-th/9905104}{{arXiv:hep-th/9905104}}].
}\bibitem {Balasubramanian:1998de}
V.~Balasubramanian, P.~Kraus, A.~Lawrence and S.~P. Trivedi, \emph {{Holographic
probes of anti--de Sitter spacetimes}},
\href {http://dx.doi.org/10.1103/PhysRevD.59.104021}{\em... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.06211771443486214,
-0.0112275630235672,
-0.05580220744013786,
0.008161339908838272,
-0.010419056750833988,
0.014385314658284187,
0.03536072000861168,
-0.043262727558612823,
0.03173007071018219,
0.01853463239967823,
-0.020716074854135513,
0.009702078998088837,
0.031485993415117264,
0.014... | |
e22b705110f3f6b82c2e824ae2c8166007f96b0b | subsection | 182 | 197 | Covariant canonical formalism | 2014) 086011}.
[\href {https://arxiv.org/abs/1406.4142}{{arXiv:1406.4142}}].
}\bibitem {Osborn:1991gm}
H.~Osborn, \emph {{Weyl consistency conditions and a local renormalisation group
equation for general renormalisable field theories}},
\href {http://dx.doi.org/10.1016/0550-3213(91)80030-P}{\emph {Nucl. Phys. B}
{\bf ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.014559048227965832,
0.027454640716314316,
-0.043097224086523056,
-0.01475744228810072,
-0.005623699631541967,
-0.026004841551184654,
0.044562287628650665,
-0.046668313443660736,
0.04627152532339096,
0.028126128017902374,
-0.005608438514173031,
-0.0019505538512021303,
-0.017733348533511162... | |
80405f12641d1bd6de73b53ee63df294041099fe | subsection | 183 | 197 | Covariant canonical formalism | Blanco and H.~Casini, \emph {{Entanglement entropy for non-coplanar
regions in quantum field theory}},
\href {http://dx.doi.org/10.1088/0264-9381/28/21/215015}{\emph {Classical
Quant. Grav.} {\bf 28} (Nov. 2011) 215015}.
[\href {https://arxiv.org/abs/1103.4400}{{arXiv:1103.4400}}].
}\bibitem {Shapiro2009}
I.~L. Shapiro... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.04529685527086258,
-0.017444172874093056,
-0.013544797897338867,
0.020237071439623833,
-0.022739998996257782,
-0.018985608592629433,
0.026326507329940796,
-0.03153076395392418,
0.014864938333630562,
0.0036628185771405697,
0.036231379956007004,
0.004734956193715334,
-0.01903139427304268,
... | |
ade446f4db962e7ce2ad09396ee098182869230b | subsection | 184 | 197 | Covariant canonical formalism | Clark, B.~R. Miller and B.~V. Saunders, eds. Release
1.0.10 of 2015-08-07,
\href {https://dlmf.nist.gov/10.25}{{https://dlmf.nist.gov/10.25}}.
}\bibitem {Bueno2017}
P.~Bueno, V.~S. Min, A.~J. Speranza and M.~R. Visser, \emph {{Entanglement
equilibrium for higher order gravity}},
\href {http://dx.doi.org/10.1103/PhysRev... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.05758141353726387,
-0.01235086191445589,
-0.03051479160785675,
-0.01418937835842371,
0.002826432464644313,
0.04211041331291199,
0.012663638219237328,
-0.005980899091809988,
0.033017005771398544,
0.0317964144051075,
0.017561262473464012,
0.007468495052307844,
-0.0584358274936676,
0.01145... | |
a8f718bc5839af1315d005a7a7ede71bd8fcaa3f | subsection | 185 | 197 | Covariant canonical formalism | 2013) 90}.
[\href {https://arxiv.org/abs/1304.4926}{{arXiv:1304.4926}}].
}\bibitem {Camps2013}
J.~Camps, \emph {{Generalized entropy and higher derivative Gravity}},
\href {http://dx.doi.org/10.1007/JHEP03(2014)070}{\emph {J. High Energy Phys.}
{\bf 1403} (2013) 70}.
[\href {https://arxiv.org/abs/1310.6659}{{arXiv:1310... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.05012083798646927,
-0.019550791010260582,
-0.008684152737259865,
-0.040444646030664444,
-0.026479797437787056,
-0.03073793277144432,
0.015010305680334568,
-0.015765782445669174,
-0.00990512315183878,
0.016009975224733353,
0.0013440214097499847,
0.00744028901681304,
-0.035988107323646545,
... | |
2ae88363789c346511503dd9d7b4c6fa3d6942fb | subsection | 186 | 197 | Covariant canonical formalism | Phys. B}
{\bf 486} (Feb. 1997) 339--352}.
[\href {https://arxiv.org/abs/hep-th/9607104}{{arXiv:hep-th/9607104}}].
}\bibitem {Myers2013}
R.~C. Myers, R.~Pourhasan and M.~Smolkin, \emph {{On Spacetime Entanglement}},
\href {http://dx.doi.org/10.1007/JHEP06(2013)013}{\emph {J. High Energy Phys.}
{\bf 2013} (Apr. 2013) 13}... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03219189867377281,
0.017758945003151894,
-0.019498221576213837,
0.007300390861928463,
-0.011030684225261211,
0.008520936593413353,
0.031520601361989975,
-0.025646721944212914,
0.01562298834323883,
0.023785389959812164,
0.01505848579108715,
0.009390575811266899,
-0.01401339378207922,
-0.... | |
051698dc58b2b7d3d9b1679ba5a9197985bd9c6f | subsection | 187 | 197 | Covariant canonical formalism | 2011) 024032}.
[\href {https://arxiv.org/abs/1011.1405}{{arXiv:1011.1405}}].
}\bibitem {Brustein2009}
R.~Brustein and M.~Hadad, \emph {{Einstein Equations for Generalized Theories of
Gravity and the Thermodynamic Relation \delta Q = T \delta S are
Equivalent}},
\href {http://dx.doi.org/10.1103/PhysRevLett.103.101301}{\... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.05132368206977844,
0.02187816984951496,
-0.01120607741177082,
-0.019650684669613838,
-0.02039826475083828,
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0.060782864689826965,
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0.01678241789340973,
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0.021984966471791267,
-0.025402477011084557,
0.... | |
f08490676ffae09b7536d4d671fe47be6c62f3e7 | subsection | 188 | 197 | Covariant canonical formalism | J. Mod. Phys.
D} {\bf 21} (Apr. 2012) 1242006}.
[\href {https://arxiv.org/abs/1204.6349}{{arXiv:1204.6349}}].
}\bibitem {Lashkari2016}
N.~Lashkari and M.~{Van Raamsdonk}, \emph {{Canonical Energy is Quantum Fisher
Information}}, \href {http://dx.doi.org/10.1007/JHEP04(2016)153}{\emph {J. High
Energy Phys.} {\bf 2016} (... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.02351430058479309,
0.005851871334016323,
-0.03939904272556305,
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-0.01284054759889841,
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-0.0168613251298666,
-0.0204... | |
ed93a7c8e1dd3bd2071e07b46937c6745c991abb | subsection | 189 | 197 | Covariant canonical formalism | Rev.
Lett.} {\bf 115} (July 2015) 261602}.
[\href {https://arxiv.org/abs/1507.07555}{{arXiv:1507.07555}}].
}\bibitem {Alishahiha2015}
M.~Alishahiha, \emph {{Holographic Complexity}},
\href {http://dx.doi.org/10.1103/PhysRevD.92.126009}{\emph {Phys. Rev. D} {\bf 92} (Sept. 2015) 126009}.
[\href {https://arxiv.org/abs/15... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.04263533651828766,
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... | |
9bdb672cee7c3457e6f63b29d278d91c87e3994d | subsection | 190 | 197 | Covariant canonical formalism | Wall, \emph {{State-Dependent Divergences in the
Entanglement Entropy}},
\href {http://dx.doi.org/10.1007/jhep10(2016)109}{\emph {J. High Energy Phys.}
{\bf 1610} (July 2016) 109}.
[\href {https://arxiv.org/abs/1607.01246}{{arXiv:1607.01246}}].
}\bibitem {Caceres2016}
E.~Caceres, P.~H. Nguyen and J.~F. Pedraza, \emph {... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.025426940992474556,
0.008195838890969753,
-0.03696521744132042,
-0.010027310810983181,
-0.022359224036335945,
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0.003510321956127882,
... | |
32589ad3b59ec5feb05234db4ca856dc1ece3c49 | subsection | 191 | 197 | Covariant canonical formalism | Rev.
Lett.} {\bf 117} (2016) 021601}.
[\href {https://arxiv.org/abs/1601.05416}{{arXiv:1601.05416}}].
}\bibitem {Jafferis2015}
D.~L. Jafferis, A.~Lewkowycz, J.~Maldacena and S.~J. Suh, \emph {{Relative
entropy equals bulk relative entropy}},
\href {http://dx.doi.org/10.1007/JHEP06(2016)004}{\emph {J. High Energy Phys.}... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.05643594637513161,
0.00971375871449709,
-0.00939327385276556,
-0.014940722845494747,
-0.026249276474118233,
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-0.024021141231060028,
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cf4bbc2245f9aa5f3a6d47f8921e2d0e376fff25 | subsection | 192 | 197 | Covariant canonical formalism | Hawking and W.~Israel,
eds.), ch.~16, pp.~676--684.
Cambridge University Press, 1987.
}\bibitem {Crnkovic:1987tz}
C.~Crnkovic, \emph {{Symplectic geometry of the convariant phase space}},
\href {http://dx.doi.org/10.1088/0264-9381/5/12/008}{\emph {Classical Quant.
Grav.} {\bf 5} (1988) 1557--1575}.
}\bibitem {Ashtekar1... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.0352855920791626,
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d80aaa7435f726675d3a04d3724955bfa34b0e7c | subsection | 193 | 197 | Covariant canonical formalism | 2017) 312--365}.
[\href {https://arxiv.org/abs/1703.04748}{{arXiv:1703.04748}}].
}\bibitem {Compere2007}
G.~Comp{è}re, \emph {{Symmetries and conservation laws in Lagrangian gauge
theories with applications to the mechanics of black holes and to gravity in
three dimensions}},
[\href {https://arxiv.org/abs/0708.3153}{{a... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03555481880903244,
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0.0... | |
286f91e619e1e1d126105ee9322f755aa17513d6 | subsection | 194 | 197 | Covariant canonical formalism | Grav.} {\bf 33} (2016) 015013}.
[\href {https://arxiv.org/abs/1508.02515}{{arXiv:1508.02515}}].
}\bibitem {Azeyanagi2009}
T.~Azeyanagi, G.~Compere, N.~Ogawa, Y.~Tachikawa and S.~Terashima,
\emph {{Higher-Derivative Corrections to the Asymptotic Virasoro Symmetry of
4d Extremal Black Holes}},
\href {http://dx.doi.org/10... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.041136469691991806,
0.01856939308345318,
-0.010139163583517075,
0.008628588169813156,
-0.005081025417894125,
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-0.03671155124902725,
0... | |
1e9ac9535a5edb9b9f2787c9453b13df52ee0060 | subsection | 195 | 197 | Covariant canonical formalism | \end{equation}\bibitem {Dong2017}
X.~Dong and A.~Lewkowycz, \emph {{Entropy, Extremality, Euclidean Variations,
and the Equations of Motion}},
\href {http://dx.doi.org/10.1007/jhep01(2018)081}{\emph {J. High Energy Phys.}
{\bf 2018} (Jan. 2017) 81}.
[\href {https://arxiv.org/abs/1705.08453}{{arXiv:1705.08453}}].
\end{a... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.015167257748544216,
0.00421524653211236,
-0.02008059434592724,
-0.044250551611185074,
-0.03707890957593918,
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-0.01654055155813694,
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a7ab9870159f01ccf1a236eda1ef60bf7e40910b | subsection | 196 | 197 | Covariant canonical formalism | Hoehn, T.~A. Koslowski and M.~I. Nelson, \emph {{Chaos, Dirac
observables and constraint quantization}},
[\href {https://arxiv.org/abs/1508.01947}{{arXiv:1508.01947}}].
}\bibitem {Szabados2009}
L.~B. Szabados, \emph {{Quasi-Local Energy-Momentum and Angular Momentum in
General Relativity}},
\href {http://dx.doi.org/10.... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03756149858236313,
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67d97d573f1b1fb133da36ff7c621e8b1e581839 | abstract | 0 | 24 | Abstract | 1. Advances in tracking technology have led to an exponential increase in
animal location data, greatly enhancing our ability to address interesting
questions in movement ecology, but also presenting new challenges related to
data manage- ment and analysis. 2. Step-Selection Functions (SSFs) are commonly
used to link e... | {
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Conducting Habitat Selection Analyses | [
"Johannes Signer",
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"Tal Avgar"
] | [
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1e7f6ff6a2560e710e26fcfd26cc05a814cbefa0 | subsection | 1 | 24 | Author's Affiliations | Johannes Signerjsigner@gwdg.de: Wildlife Sciences, University of Goettingen, Göttingen, Germany
John Fieberg: Department of Fisheries, Wildlife and Conservation Biology, University of Minnesota, St. Paul, MN, USA
Tal Avgar: Department of Integrative Biology, University of Guelph, Guelph, ON, Canada | {
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} | 1805.03227 | Animal Movement Tools (amt): R-Package for Managing Tracking Data and
Conducting Habitat Selection Analyses | [
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f33ecda1c71e892774482a4b11cbd5f96d32f453 | subsection | 2 | 24 | Summary | Advances in tracking technology have led to an exponential increase in animal location data, greatly enhancing our ability to address interesting questions in movement ecology, but also presenting new challenges related to data management and analysis.
Step-Selection Functions (SSFs) are commonly used to link environm... | {
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} | 1805.03227 | Animal Movement Tools (amt): R-Package for Managing Tracking Data and
Conducting Habitat Selection Analyses | [
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9d528a3b03b116531e894b62e4ba6b2748759eaf | subsection | 3 | 24 | Introduction | Advances in technology have led to large collections of fine-scale animal biotelemetry data , , fueling the development of new quantitative methods for studying animal movement . introduced the movement ecology paradigm, that conceptually connects different factors shaping the realized movement path of animals (e.g., t... | {
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Conducting Habitat Selection Analyses | [
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517d6e9c957f451ddf498c98fef7e7a6b82b3fa8 | subsection | 4 | 24 | Introduction | Unlike SSFs (that do not include an explicit movement component), a fitted iSSF is a fully-fledged biased random walk model that can be used to simulate animal space-use , , .Step-selection functions (SSFs and iSSFs) are usually straight forward to fit (using any conditional-logistic regression routine) once data are a... | {
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966ab6eb3240552918249535b56c7d85b9b701b5 | subsection | 5 | 24 | Functionality | A typical workflow to analyze animal tracking data can be divided into four main steps (described in detail below):Data preparation, inspection and management: Load and inspect gaps in the data, resample tracks if needed, and adjust coordinate reference systems.
Exploratory data analysis and descriptive analyses: Expl... | {
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9b7dac63ff4fd63ef65a27dd717231987bdefc23 | subsection | 6 | 24 | Data preparation, inspection, and management | After loading data into R, users should perform a variety of data quality checks and possibly remove fixes with missing coordinates (although this information could potentially be used to test if fixes are missing at random). We provide functions to quantify variability in sampling rates over time and among individuals... | {
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28063ea9f42a75332ce63d4929f34379160097a1 | subsection | 7 | 24 | Exploratory data analysis and descriptive analyses | Once data have been cleaned, the next logical step is to explore the data by looking at different movement-related statistics (e.g., distributions of turning angles or step lengths) and trajectory and space-use summaries (e.g., net squared displacement, path sinuosity, home range area). These summaries may be calculate... | {
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Conducting Habitat Selection Analyses | [
"Johannes Signer",
"John Fieberg",
"Tal Avgar"
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e7cc38433e634a6a07204b8144a0539922d588ec | subsection | 8 | 24 | Modeling | In the next step, we fit models to test hypotheses about animal movement and habitat selection. Importantly, amt provides functionality necessary for data development steps prior to fitting RSFs and (i)SSFs (e.g., methods for generating random points or steps, and extract environmental covariates for the observed and r... | {
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} | 1805.03227 | Animal Movement Tools (amt): R-Package for Managing Tracking Data and
Conducting Habitat Selection Analyses | [
"Johannes Signer",
"John Fieberg",
"Tal Avgar"
] | [
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aaea4d03fe89e60aed879f32ff9ba96b17432e8a | subsection | 9 | 24 | Simulation | As a final step, new data can be simulated from fitted models. Simulations can be used to obtain estimates of space use (i.e. the utilization distribution), identify corridors of high use, or asses the power of the model (testing how well parameters can be recovered as a function of sample size). Many packages that fit... | {
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Conducting Habitat Selection Analyses | [
"Johannes Signer",
"John Fieberg",
"Tal Avgar"
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9ffe8fef6fedf2866117687a120d96fb4e304980 | subsection | 10 | 24 | Case study | We illustrate a subset of the above steps using data from radio collard fishers available through movebank , . For details about the data and the capture of the animals, we refer to and . We begin by analyzing the space use of Ricky T (id 1016), and then illustrate how similar analyses can be extended to several animal... | {
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294c593ffa7ef2580ac969773b2c64a8a7e2625a | subsection | 11 | 24 | From data cleaning to simulated space use for one animal | We begin with loading the data of all fishers, remove observations with missing spatial coordinates (longitude, latitude), and subset relocations for Ricky T (id: 1016).library(raster)library(lubridate)library(amt)dat <- read_csv("data/Martes pennanti LaPoint New York.csv") filter(!is.na(`location-lat`)) select(x = `lo... | {
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dc64000ff936d752a588c9336dc7933ada48c060 | subsection | 12 | 24 | From data cleaning to simulated space use for one animal | If the argument include.crepuscule is set to TRUE, the function not only considers day and night, but also dawn and dusk.stps <- track_resample(dat_1, rate = minutes(10), tolerance = seconds(60)) filter_min_n_burst(min_n = 3) time_of_day(include.crepuscule = FALSE)We then use the str function to inspect the structure o... | {
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aedfcdb08f9b5803b4a3ee238e695ec2b1d34ba4 | subsection | 13 | 24 | From data cleaning to simulated space use for one animal | Depending on the target process under investigation (habitat selection or movement), covariates might be extracted at the end of the step (habitat selection process) or at the start of the step (movement process). If covariates are extracted at the end of the step, they are typically included in the model as main effec... | {
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6eef84a2130f28607acb9ce9fcac0332b6bbb899 | subsection | 14 | 24 | From data cleaning to simulated space use for one animal | 1) there is evidence to suggest that the animal prefers forested wetlands over other landuse classes, 2) there is no difference in habitat preference between day and night, 3) there is evidence to modify the shape of the underlying gamma distribution (through the log of the step length), and 4) the modification of the ... | {
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a721e078b545d9cf074e84a1269e1ec5b299a834 | subsection | 15 | 24 | From data cleaning to simulated space use for one animal | 2B).hk <- habitat_kernel(coef = list(forest = coef(m1)["wet"]), resources = wet)We then estimate the steady state UD (Fig. 2CE) with the function simulate_ud:ssud_day <- simulate_ud(movement_kernel = mk,habitat_kernel = hk,start = as.numeric(stps[1, c("x1_", "y1_")]),n = 1e7)In order to simulate the transient UD (Fig. ... | {
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08f9087df260be9e30344e58ddc3d9b51322ded2 | subsection | 16 | 24 | Many animals: quantifying population-level effects | We start again with the same data set (dat), containing data from 6 individual fishers. This time we are interested in quantifying among-animal variability in the selection coefficients. We proceed using nearly all the same steps as in the first example, but with a different data structure: data_frames with list-column... | {
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5712b513b23e5cdce2a93241b89addf59d634ab5 | subsection | 17 | 24 | Many animals: quantifying population-level effects | We then filter again bursts, keeping only those with at least three points (filter_min_n_burst), convert from a point to a step representation of the tracks (steps_by_burst), generate 9 random steps for each observed step (random_steps), extract the environmental covariates (extract_covariates), convert landuse to a fa... | {
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3f567b1a262ce4eb1808a80d7d2e2c47554c0a21 | subsection | 18 | 24 | Discussion and Outlook | We have illustrated how amt can be used to fit Step-Selection Functions (SSFs) and explore temporal movement and habitat selection patterns at the individual and population levels. We demonstrated how an iSSF, fit to a single fisher, can be used to simulate utilization distributions . The UD map (Fig. 2) can be thought... | {
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07207e01a0c7856b990e9aa45ef3d9cdc267810e | subsection | 19 | 24 | Authors contribution | JS, JF and TA conceived the ideas and designed the package; JS programmed the package; JS, JF and TA analysed the case study; JS, JF and TA wrote the manuscript. All authors contributed critically to the drafts and gave final approval for publication. | {
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b1349618bf2699eb398d19cceb9cde40e18214be | subsection | 20 | 24 | Tables | Table 1: Coefficients of fitted integrated Step Selection Function.
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f675ebaf83d8fa3f0ed9f834c3303e70e1dc3882 | subsection | 21 | 24 | Supplement S1 | Source code to reproduce the two examples. | {
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198f491211dcf8ceaf041f196b93f069cbe5e75f | subsection | 22 | 24 | Example 1: One animal | example1animalv3.R | {
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63f985308631d4c97eb37ca14ff5c09a299cf716 | subsection | 23 | 24 | Example 2: Many animals | exampleallanimalsv2.R | {
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2c8ad6f2140e1d99b82a6d4680fb5f67c315439f | abstract | 0 | 110 | Abstract | We show that numerous distinctive concepts of constructive mathematics arise
automatically from an "antithesis" translation of affine logic into
intuitionistic logic via a Chu/Dialectica construction. This includes apartness
relations, complemented subsets, anti-subgroups and anti-ideals, strict and
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1ef23dd91e5bea44dbc1c75b182104fe33d16047 | subsection | 1 | 110 | Introduction | One of the explicit motivations of Girard's linear logic was to recover an involutory “classical” negation while retaining “constructive content”:...the linear negation ...is a constructive and involutive negation; by the way, linear logic works in a classical framework, while being more constructive than intuitionist... | {
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99373c6ccf42c6cc2ccf4fc03dca36b4c7c26be1 | subsection | 2 | 110 | Introduction | In sec:realssec:topology we will see that a systematic use of par can solve various problems in intuitionistic constructive mathematics, such as defining a notion of “metric space” that includes the Hausdorff metric, or a union axiom for a “closure space” that is not unreasonably strong.Secondly, we can also translate ... | {
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e1a179f00d738395d574117647f60f281434896a | subsection | 3 | 110 | Outline | In sec:meaning we describe our viewpoint on linear logic informally, analogously to the BHK interpretation of intuitionistic logic; no prior familiarity with linear logic is required.
Then in sec:chu,sec:types we formalize it as a Chu construction for propositional and higher-order logic.
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56dc18fc736ae9897d7cac2e3ab946ce0a357e63 | subsection | 4 | 110 | A meaning explanation | Intuitionistic logic is often explained informally (e.g. in ) by the so-called Brouwer–Heyting–Kolmogorov (BHK) interpretation, which explains the meaning of the logical connectives and quantifiers “pragmatically” in terms of what counts as a proof of them.
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4790f293372084b5d69c5c24d90a45fad5b581d1 | subsection | 5 | 110 | A meaning explanation | In the words of Patterson :Once we take on the Brouwerian view that proofs should be constructions, both negation and “falsity” disappear because absurdity is not the same thing as demonstratively false. This is because a construction leading to a contradiction does not mean that we can provide a counterexample....In i... | {
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"arxiv_id": "",
"doi": "",
"end": 203,
"openalex_id": "https://openalex.org/W2320858954",
"raw": "Anna L. Patterson. Implicit Programming and the Logic of Constructible Duality. PhD thesis, University of Illinois at Urbana-Champaign, 1998.",
"source_ref_id":... | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0b2bb331569fda8bc95955948898e77b51861d55 | subsection | 6 | 110 | A meaning explanation | Not every intuitionistic proof (in the ordinary sense) of P\vee Q begins by deciding whether to prove P or Q, but intuitionistic logic satisfies the “disjunction property” that from any proof of P\vee Q we can extract either a proof of P or a proof of Q.
Similarly, any proof of P\multimap Q must contain enough informat... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.... |
2698175dd3c7e52f4f02568379406be4de4a0d53 | subsection | 7 | 110 | A meaning explanation | (For a discussion of notation, see rmk:lines.)
Its explanation is:A proof of P\mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}Q is a method converting any refutation of P into a proof of Q, together with a method converting any refutation of Q into a proof of P.
A refutation of P\mathbin {\raisebo... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... |
1e7188c8293540416eab022e692bdf86e6142f66 | subsection | 8 | 110 | A meaning explanation | The most nontrivial part of this is the refutations of P\boxtimes \top (or dually the proofs of \smash{{P}^\perp }\mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}\bot ), which by definition consist of a method transforming any proof of \top into a refutation of P, together with a method transformin... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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60659fab2b3d6c87320eaa0592f6a1861230b80e | subsection | 9 | 110 | A meaning explanation | If we define(x=y) \mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}{\textstyle \sqcap }n. |x_n-y_n| \le \textstyle {\textstyle \frac{2}{n}}then we find that (assuming that \smash{{(p<q)}^\perp } \equiv (p\ge q) for p,q\in )\smash{{(x=y)}^\perp } \equiv {\textstyle \bigsqcup }n. |x_n-y_n| > \textstyle {\t... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.021... |
92d297a04a7035ccf3c70870e68006bda323df40 | subsection | 10 | 110 | A meaning explanation | By contrast:A proof of (P \sqcap Q) \multimap R consists of:
A method converting any proofs of P and Q into a proof of R, and
A method converting any refutation of R into either a refutation of P or a refutation of Q.
A refutation of (P \sqcap Q) \multimap R consists of a proof of P, a proof of Q, and a refutation ... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.... |
70d71a6138cb92c2451b3ef694e61da27f162c69 | subsection | 11 | 110 | A meaning explanation | But of course the distinction between \boxtimes /\mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}} and \sqcap /\sqcup would still have to be maintained.)The main exceptions to this notational principle are the linear implication \multimap , the exponentials , (see below), and equality.
The symbols \... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... |
ce16fce17d73c4804a4c8cba1691dd3de0b56a45 | subsection | 12 | 110 | The Chu construction | Like the BHK interpretation, the explanation of the linear connectives and quantifiers in sec:meaning is informal, and nonspecific about what constitutes a “method”.
However, the relationship between the two interpretations is an instance of a precise category-theoretic operation known as the Chu construction , .The Ch... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 314,
"openalex_id": "",
"raw": "Po-Hsiang Chu. Constructing *-autonomous categories. M. Sc. thesis, McGill University, 1978.",
"source_ref_id": "4cc50943916af4bd4a1e915085470edf1ef323b8",
"start": 166
},
{
... | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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b1135734b2854a406f9320f7c0e6104691ca4dd6 | subsection | 13 | 110 | The Chu construction | \end{array}We then define\smash{{P}^\perp } &\mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}P \multimap \bot &\hspace{56.9055pt}
P \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}Q &\mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}\smash{{(\smash{{P}^\perp } \boxtime... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.2307/2314383",
"end": 779,
"openalex_id": "https://openalex.org/W2067914884",
"raw": "Errett Bishop and Douglas Bridges. Constructive analysis. Springer–Verlag, 1985.",
"source_ref_id": "8d9377810aa39e6511c9f6bb4ced2acc0672c2f4",
... | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.0... |
3d32bad74b699489757badff510336f1ca34a9ce | subsection | 14 | 110 | The Chu construction | The induced comonad, and its dual monad, are defined byP \mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}({P}^+, \lnot {P}^+) \hspace{56.9055pt}
P \mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}(\lnot {P}^-, {P}^-).For the Seely condition, we compute(P\sqcap Q) &= ({P}^+ \wedge {Q}^+, \... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0... |
76156c9d52bfce333b574b466474f04832e554c1 | subsection | 15 | 110 | The Chu construction | A refutation of P \multimap Q is a proof of P together with a refutation of Q.Unlike \smash{{(P\multimap Q)}^\perp } \equiv (\smash{{Q}^\perp } \multimap \smash{{P}^\perp }), we only have \smash{{(P \multimap Q)}^\perp } \equiv (\smash{{Q}^\perp } \multimap (\smash{{P}^\perp })).
Thus P is “usable multiple times” (sinc... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/0304-3975(87)90045-4",
"end": 467,
"openalex_id": "https://openalex.org/W2911865844",
"raw": "Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50(1):1– 101, 1987.",
"source_ref_id": "e4d0310fdf3f5fa47108bb5aa6021f... | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.0278... |
ee4275714ad54f06a0f84b85fe5ff863cc92addb | subsection | 16 | 110 | The Chu construction | Decidable implies affirmative: (P \sqcup \smash{{P}^\perp }) \vdash (P\multimap P).
Refutative affirmativity: (P\multimap P).
Distributive exponentials: P \vdash P.
Exact exponentials: (P \sqcup Q) \equiv P \sqcup Q and (P \sqcap Q) \equiv P \sqcap Q.
Continuous exponentials: ({\textstyle \bigsqcup }x. P(x)) \equiv... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.0... |
30e641e4141c5ccb5be7af22b58894910af5f947 | subsection | 17 | 110 | The Chu construction | For REF , we haveP\boxtimes P \equiv ({P}^+ \wedge {P}^+ \;,\; ({P}^+ \rightarrow {P}^-)\wedge ({P}^+\rightarrow {P}^-))
\equiv ({P}^+, \lnot {P}^+)
\equiv P(since \lnot ({P}^+\wedge {P}^-), we have ({P}^+ \rightarrow {P}^-)\equiv \lnot {P}^+).
For REF , we haveP &\equiv ({P}^+ ,\lnot {P}^+)
&\qquad P &\equiv (\lnot {P... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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76397c7c537bf4a5757e5d0d1cc1b40299e3d38c | subsection | 18 | 110 | The Chu construction | Then we have an adjunction\mathrm {Afr}: \rightleftarrows : \mathrm {Chu}(-,\mathbf {0})in which the left adjoint \mathrm {Afr} takes the Heyting algebra of affirmative elements as in thm:seely-heyt, and the right adjoint is fully faithful.If \psi : H\rightarrow H^{\prime } is a lattice homomorphism of Heyting algebras... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.014761744998395443,
0.02... |
d03245788d92e9680e73bb394f9ba47bc3984b9d | subsection | 19 | 110 | The Chu construction | The counit of the adjunction will be the isomorphism \mathrm {Afr}(\mathrm {Chu}(H,\mathbf {0})) \cong H noted in {thm:seely-heyt}.
For the unit, we define \eta : L \rightarrow \mathrm {Chu}(\mathrm {Afr}(L),\mathbf {0}) by \eta (P) \mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}(P, (\smash{{P}^\perp }... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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9305b4f63e6411cc6b35a392b966196515b42c11 | subsection | 20 | 110 | The Chu construction | \end{align*}
Thus to show \eta P \boxtimes \eta Q\le \eta (P \boxtimes Q) it suffices without loss of generality to show (\smash{{P}^\perp } \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}\smash{{Q}^\perp }) \le (P \multimap (\smash{{Q}^\perp })), as follows:
{
{
{
{
{
\smash{{P}^\perp } \mathbin... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... |
57694fa1ccd31ed1956d78e6f48da5957dbcdf8c | subsection | 21 | 110 | The Chu construction | \end{multline*}
}\begin{}
In purely linear constructive mathematics, we could assume any of these as additional axioms; but in this paper, we will use only exactness of exponentials~\ref {item:exact-exp}.
However, before leaving {thm:axioms} we make the following additional remarks:
\begin{}
\item Additive distributivi... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.00... |
44a6498c1f75802f50351c6181a7295e68712a63 | subsection | 22 | 110 | The Chu construction | For a dramatic counterexample, let H = () be the open-set lattice of the real numbers, with x: and {P(x)}^+ \mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}\setminus \lbrace x\rbrace and {P(x)}^- \mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}\mathbf {0}; then \lnot (\lnot {P(x)}^+ \wed... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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6d3309d53ea9bd325f07ee123f0e7367e349722d | subsection | 23 | 110 | The Chu construction | For instance, \mathrm {Chu}(\lbrace \mathbf {0},\mathbf {1}\rbrace ,\mathbf {0})= \lbrace (\mathbf {0},\mathbf {1}) \le (\mathbf {0},\mathbf {0}) \le (\mathbf {1},\mathbf {0}) \rbrace coincides with
three-valued Łukasiewicz logic, where (\mathbf {0},\mathbf {0}) is called “unknown” or “undefined”.Remark Dan Licata has ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 566,
"openalex_id": "",
"raw": "William Lovas and Karl Crary. Structural normalization for classical natural deduction. http://www.cs.cmu.edu/~wlovas/papers/clnorm.pdf, 2006.",
"source_ref_id": "3b70bf7a20100dcc664d0084d577a... | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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c491c4245c1147a56ca28d67ec352207773a162a | subsection | 24 | 110 | Types and predicates | The Chu construction for Heyting algebras gives a precise way to translate between linear and intuitionistic propositional logic.
However, to do substantial mathematics we require not just a propositional logic but a first-order and even higher-order logic.
While attempting not to get bogged down by too much detail, in... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 584,
"openalex_id": "https://openalex.org/W1520893111",
"raw": "Bart Jacobs. Categorical Logic and Type Theory. Number 141 in Studies in Logic and the Foundations of Mathematics. North Holland, Amsterdam, 1999.",
"source_ref... | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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23653545b08ab840bdfca899dd71e01490e47ea7 | subsection | 25 | 110 | Types and predicates | Non-pointwise examples include realizability; see .Our higher-order linear logic is exactly the same, except that now the propositions are instead equipped with the linear operations\boxtimes , \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}, \sqcap , \sqcup , \top , \bot , \multimap , \smash{{(-)... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1017/s0305004100057534",
"end": 51,
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"raw": "J. M. E. Hyland, Peter T. Johnstone, and A. M. Pitts. Tripos theory. Math. Proc. Cambridge Philos. Soc., 88(2):205–231, 1980.",
"source_... | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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7c285bec662ac4b424dd50621e5448a244d4358a | subsection | 26 | 110 | Types and predicates | There may be no such subobject, but fortunately (by ) we can describe an equivalent category to \mathrm {Chu}((A,\Omega ^),\mathbf {0}) whose objects are arbitrary pairs ({P}^+, {P}^-).Theorem 4.1 Any tripos gives rise to an affine tripos over the same , with \Omega ^\mathrel {\smash{\overset{\scriptscriptstyle \mathsf... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 186,
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"raw": "Michael Barr. The Chu construction. Theory and Applications of Categories, 2(2):17–35, 1996.",
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"start": 0
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]
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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92f23a5f5cdcb1b2aefa4d5cc352fc8defd699c9 | subsection | 27 | 110 | Types and predicates | That is, from a closed proof of P\sqcup Q we can extract either a proof of P or Q, and from a closed proof of {\textstyle \bigsqcup }x. P(x) we can extract a t such that P(t).Remark
Our logics do not allow quantification over types (i.e. “for all types A” internally to the logic).
Thus, theorems about structured types... | {
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{
"arxiv_id": "",
"doi": "10.1016/0304-3975(87)90045-4",
"end": 1246,
"openalex_id": "https://openalex.org/W2911865844",
"raw": "Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50(1):1– 101, 1987.",
"source_ref_id": "e4d0310fdf3f5fa47108bb5aa6021... | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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010646e0929c50eb5265c2917b9f75878b71a4a0 | subsection | 28 | 110 | Types and predicates | (Intuitionistically, proof by contradiction implies excluded middle P\vee \lnot P since \lnot (P\vee \lnot P) \equiv (\lnot P\wedge P) is a contradiction; but linearly \smash{{(P\sqcup \smash{{P}^\perp })}^\perp } \equiv (\smash{{P}^\perp } \sqcap P) is no contradiction since we can't use both \smash{{P}^\perp } and P.... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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3b39bfb326a7c1f8d563f75f44867addb11e375a | subsection | 29 | 110 | Intuitionistic sets and functions | For most of the paper, we will define concepts “naturally” in the linear context and then pass across the standard interpretation to recover well-known intuitionistic notions.
But for sets and equality, we begin with the intuitionistic context, to fix conventions.
Thus, in this section we work in our intuitionistic hig... | {
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... | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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82d8b2180373d43f2bee76f3e81bb4b679946c32 | subsection | 30 | 110 | Intuitionistic sets and functions | With reflexivity, this implies \mathsf {E}x \equiv (x=x); so we could define \mathsf {E}x \mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}(x=x) and dispense with both reflexivity and the new axiom, arriving at a partial equivalence relation.
This is common in tripos theory and realizability; but divorci... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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9b09e27fe2160b6653f8ac7c6c470699b78a7133 | subsection | 31 | 110 | Intuitionistic sets and functions | We write U\cap V \mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}{x:A | (x\in U) \wedge (x\in V)} and so on.Definition
A function between two sets is an operation f:B^A such that\begin{array}{rll}
&\vdash _{x\in A} & f(x) \in B\\
(x_1=x_2) & \vdash _{x_1,x_2\in A}& (f(x_1)=f(x_2)).
\end{array}The funct... | {
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"doi": "10.1016/0022-4049(95)00029-1",
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"openalex_id": "https://openalex.org/W2038364868",
"raw": "M. Makkai. Avoiding the axiom of choice in general category theory. J. Pure Appl. Algebra, 108(2):109–173, 1996.",
"source_ref_id":... | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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9df47e306528e08dfbfe2af78dc2cacfa56762b7 | subsection | 32 | 110 | Intuitionistic sets and functions | For instance, this is how one builds the topos represented by a tripos (such as a realizability topos), and in particular how one recovers the correct internal logic of a topos from its tripos of subobjects. | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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fa1933a6a701666fe5d1b0776e73000199032ebe | subsection | 33 | 110 | Linear sets and functions | We now switch to the linear context, for this section and the rest of the paper, except when discussing the standard interpretation.Definition
A pre-set is an (-)type A equipped with an existence predicate \mathsf {E}, which is affirmative:\mathsf {E}x \vdash _{x:A} \mathsf {E}x.As in notn:ipreset, for an -pre-set A w... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... |
488634af3f90de2b727b42d76acc4b27e3fb4f0f | subsection | 34 | 110 | Linear sets and functions | (\mathsf {E}x \multimap P(x)).The symbol \mathrel {{\sqsubset }{\mathord {-}}} reminds us of linearity, and the reminds us that x\mathchoice{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\raisebox {... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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8e01c8f1b5a01aa7b54f786850c0d2f8ceaa7d3b | subsection | 35 | 110 | Linear sets and functions | For instance, if P(x) \vdash _{x\mathchoice{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\raisebox {-1pt}{\scriptstyle }\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\raisebox {-1pt}... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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50ec82b23ab09c494b028e1f3a94db5423813a1c | subsection | 36 | 110 | Linear sets and functions | But this is not the same as P(x) \vdash _{x\mathchoice{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\raisebox {-1pt}{\scriptstyle }\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\rais... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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9aca87c23cb58da75b2a0a7db45a8df271940620 | subsection | 37 | 110 | Linear sets and functions | If \mathsf {E}x were not affirmative, then we would also have to include {P(x)}^+ \wedge {Q(x)}^- \vdash _{x:A} {\mathsf {E}} x, where {\mathsf {E}} x would be a “nonexistence predicate”.Definition
A set is a pre-set with a predicate \circeq on A\times A such that&\;\vdash _{x\mathchoice{\mathrel {\raisebox {-1pt}{}\m... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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9719b4a1dfeb53c99884af5a662b77690a245e03 | subsection | 38 | 110 | Linear sets and functions | If A is a pre-set, \protect \!\left.\:{x\mathrel {{\sqsubset }{\mathord {-}}}A P(x)}\:\right. denotes the pre-set with \mathsf {E}x \mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}(x\mathchoice{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\raisebox {-1pt}{}\ma... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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dc727139cb3d94f07ee392f810ee83ebcd94d4e6 | subsection | 39 | 110 | Linear sets and functions | (x\ne z) \rightarrow (x\ne y) \vee (y\ne z).
It is affirmative if and only if \ne is denial, (x\ne y) \equiv \lnot (x=y).
It is refutative if and only if \ne is tight: \lnot (x\ne y) \equiv (x=y).Example
If A and B are sets, their cartesian product set is the cartesian product type A\times B with \mathsf {E}(x,y) \m... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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1ded2a3dd03db51f88a41a838e7125924eed67fa | subsection | 40 | 110 | Linear sets and functions | For transitivity:\begin{array}[b]{ccccc}
(x\circeq y) \boxtimes (y\circeq z) &\equiv & ((x\circeq y) \boxtimes (y\circeq z)) &\vdash & (x\circeq z)\\
(x\circeq y) \sqcap (y\circeq z) &\equiv & ((x\circeq y) \sqcap (y\circeq z)) &\vdash & (x\circeq z).
\end{array}Example
The type \Omega is a set with \mathsf {E}x \math... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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2996ac4e0e2fa4cfb74db2bde9fff301a7633f0c | subsection | 41 | 110 | Linear sets and functions | In neither case is the equality strong, nor is it affirmative nor refutative even if P and Q are both one or the other.Definition
A relation on a set A is a predicate P such that(x\circeq y) \boxtimes P(x) \vdash _{x,y\mathchoice{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {... | {
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{
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... | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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760680b4aa17bcb4f7821e8a9c279592f03414d9 | subsection | 42 | 110 | Linear sets and functions | \end{array}The subset condition (x\circeq y) \boxtimes (x\mathrel {{\sqsubset }{\mathord {-}}}U) \vdash _{x,y\mathchoice{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\raisebox {-1pt}{\scriptstyle }... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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202e757edfb48ac2cb839e47485ca6871c8bebfd | subsection | 43 | 110 | Linear sets and functions | The “strong extensionality” condition in REF is exactly the contrapositive information arising from the strong subset condition.Definition
A function between two sets is an operation f:B^A such that\begin{array}{rll}
&\vdash _{x\mathchoice{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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1291c9aaa74fe9f12ce5ce5f47525e9db652fac1 | subsection | 44 | 110 | Linear sets and functions | ((x_1\circeq x_2) \multimap (f(x_1)\circeq f(x_2))) }\:\right.\\
(f \circeq g) &\mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}{\textstyle \sqcap }x^A. (f(x)\circeq g(x)).Theorem 6.4
In the standard interpretation, an -function f:A\rightarrow B is an -function that is strongly extensional, i.e. (f(x_1... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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15711783474de03f65818977e18c160c43f1a395 | subsection | 45 | 110 | Linear sets and functions | Similarly, the set-operation from thm:oc-sets is a reflection from the subcategory of strong -sets into its subcategory of strong refutative -sets.
We can also construct disjoint unions and an empty set; thus \mathbf {Set}_{} is itself a (non-posetal) model of an “intuitionistic affine logic” with the connectives \boxt... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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14b12371984a45516bb8475dc37cd8d443a716fa | subsection | 46 | 110 | Linear sets and functions | (x\mathrel {{\sqsubset }{\mathord {-}}}U_i) }\:\right. \\
{U}^\perp &= \protect \left.\:{x \smash{\smash{{(x\mathrel {{\sqsubset }{\mathord {-}}}U)}^\perp }}}\:\right. &\qquad \mathord {\lnot \hspace{-2.0pt}{\scriptstyle \Box }}&= \protect \left.\:{x \bot }\:\right..We write U\sqsubseteq V to mean {\textstyle \sqcap }x... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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