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# zbMATH — the first resource for mathematics

## Nunez, Manuel A.

Compute Distance To:
 Author ID: nunez.manuel-a Published as: Nunez, Manuel; Nunez, Manuel A.; Nuñez, Manuel; Núñez, M.; Núñez, Manuel External Links: ORCID · dblp
 Documents Indexed: 143 Publications since 1988, including 6 Books
all top 5

#### Co-Authors

 64 single-authored 27 Rodríguez, Ismael 20 Merayo, Mercedes G. 15 López, Natalia 9 Rubio, Fernando 8 Hierons, Robert Mark 4 de Frutos-Escrig, David 4 Llana Díaz, Luis Fernando 3 Cavalli, Ana R. 3 Rojo, Jesús 2 Cruz, José B. jun. 2 Freund, Robert M. 2 Gopal, Ram D. 2 Lastra, Alberto 2 López Pelayo, Fernando 2 Sanz, Javier 2 Schneider, Mark A. 2 Tristán, Luis A. 1 Alvarez, Jorge 1 Andova, Suzana 1 Andrés, César 1 Arnedo, José Antonio 1 Bai, Xue 1 Baker, Paul W. 1 Bayse, Emmanuel 1 Díaz, Luis Fdo. Llana 1 Ferriz-Mas, Antonio 1 Frantzen, Lars 1 Galindo, Félix 1 Garfinkel, Robert S. 1 Gostanza, Pedro Palao 1 Gregorio-Rodríguez, Carlos 1 Lee, Hsiao-Hui 1 Llana, Luis 1 Maamar, Zakaria 1 McIver, Annabelle K. 1 Montes de Oca, Edgardo 1 Nguyen Thanh Ngoc 1 Peña, Ricardo 1 Pousttchi, Key 1 Rupérez, David 1 Trawiński, Bogdan 1 Zaïdi, Fatiha 1 Zhdanov, Dmitry V.
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#### Serials

 12 Journal of Physics A: Mathematical and General 11 Physics Letters. A 8 Journal of Mathematical Physics 5 European Journal of Mechanics. B. Fluids 5 The Journal of Logic and Algebraic Programming 4 Mathematical Methods in the Applied Sciences 3 Geophysical and Astrophysical Fluid Dynamics 3 Physica D 3 Distributed Computing 3 Lecture Notes in Computer Science 2 Journal of Fluid Mechanics 2 Journal of Mathematical Economics 2 Formal Aspects of Computing 2 Mathematical Programming. Series A. Series B 2 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 2 Communications in Nonlinear Science and Numerical Simulation 2 Physica Scripta 2 Computer Networks 1 Applicable Analysis 1 IMA Journal of Applied Mathematics 1 Letters in Mathematical Physics 1 Fluid Dynamics Research 1 IEEE Transactions on Computers 1 International Journal of Mathematics and Mathematical Sciences 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Operations Research 1 Theoretical Computer Science 1 Science of Computer Programming 1 International Journal of Production Research 1 Optimization 1 Journal of Computer Science and Technology 1 Applications of Mathematics 1 Applied Mathematical Modelling 1 Communications in Partial Differential Equations 1 European Journal of Operational Research 1 SIAM Journal on Applied Mathematics 1 SIAM Review 1 SIAM Journal on Optimization 1 Discrete and Continuous Dynamical Systems 1 INFORMS Journal on Computing 1 ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik 1 Chaos 1 Fundamenta Informaticae 1 Discrete and Continuous Dynamical Systems. Series B 1 Journal of Intelligent and Fuzzy Systems 1 Journal of Logical and Algebraic Methods in Programming
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#### Fields

 66 Fluid mechanics (76-XX) 61 Computer science (68-XX) 26 Partial differential equations (35-XX) 8 Statistical mechanics, structure of matter (82-XX) 7 General and overarching topics; collections (00-XX) 6 Operations research, mathematical programming (90-XX) 5 Dynamical systems and ergodic theory (37-XX) 5 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 4 Optics, electromagnetic theory (78-XX) 4 Astronomy and astrophysics (85-XX) 3 Mathematical logic and foundations (03-XX) 3 Relativity and gravitational theory (83-XX) 2 Operator theory (47-XX) 2 Geophysics (86-XX) 1 Ordinary differential equations (34-XX) 1 Approximations and expansions (41-XX) 1 Functional analysis (46-XX) 1 Convex and discrete geometry (52-XX) 1 Probability theory and stochastic processes (60-XX) 1 Mechanics of deformable solids (74-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Systems theory; control (93-XX) 1 Mathematics education (97-XX)

#### Citations contained in zbMATH

64 Publications have been cited 171 times in 111 Documents Cited by Year
Algebraic theory of probabilistic processes. Zbl 1048.68057
Núñez, Manuel
2003
An overview of probabilistic process algebras and their equivalences. Zbl 1203.68115
López, Natalia; Núñez, Manuel
2004
A characterization of ill-posed data instances for convex programming. Zbl 1049.90059
Nunez, Manuel A.
2002
Specification, testing and implementation relations for symbolic-probabilistic systems. Zbl 1088.68021
López, Natalia; Núñez, Manuel; Rodríguez, Ismael
2006
Condition measures and properties of the central trajectory of a linear program. Zbl 0920.90097
Nunez, Manuel A.; Freund, Robert M.
1998
Formal testing from timed finite state machines. Zbl 1131.68029
Merayo, Mercedes G.; Núñez, Manuel; Rodríguez, Ismael
2008
$$\mathcal {HOTL}$$: Hypotheses and observations testing logic. Zbl 1131.68064
Rodríguez, Ismael; Merayo, Mercedes G.; Núñez, Manuel
2008
Regularity criteria for the Navier-Stokes equations involving the ratio pressure-gradient of velocity. Zbl 1183.35221
Núñez, Manuel
2010
A passive testing approach based on invariants: application to the WAP. Zbl 1080.68534
Bayse, Emmanuel; Cavalli, Ana; Núñez, Manuel; Zaïdi, Fatiha
2005
Advances in nonlinear dynamos. Zbl 1084.85500
Ferriz-Mas, Antonio (ed.); Núñez, Manuel (ed.)
2003
Estimates on hyperdiffusive magnetohydrodynamics. Zbl 1044.76072
Núñez, Manuel
2003
Encoding PAMR into (timed) EFSMs. Zbl 1037.68557
Núñez, Manuel; Rodríguez, Ismael
2002
Controllable test cases for the distributed test architecture. Zbl 1183.68098
Hierons, Robert M.; Merayo, Mercedes G.; Núñez, Manuel
2008
Towards testing stochastic timed systems. Zbl 1279.68029
Núñez, Manuel; Rodríguez, Ismael
2003
Condition-measure bounds on the behavior of the central trajectory of a semidefinite program. Zbl 1010.90054
Nunez, Manuel A.; Freund, Robert M.
2001
Extending EFSMs to specify and test timed systems with action duration and time-outs. Zbl 1373.68141
Merayo, Mercedes G.; Núñez, Manuel; Rodríguez, Ismael
2008
A theorem of existence for the equations of magnetohydrodynamics of partially ionized plasmas. Zbl 1136.76059
Núñez, Manuel
2008
Customized testing for probabilistic systems. Zbl 1185.68094
Llana-Díaz, Luis F.; Núñez, Manuel; Rodríguez, Ismael
2006
A logic for assessing sets of heterogeneous testing hypotheses. Zbl 1185.68422
Rodríguez, Ismael; Merayo, Mercedes G.; Núñez, Manuel
2006
Existence theorems for two-fluid magnetohydrodynamics. Zbl 1110.76057
Núñez, Manuel
2005
Fair testing through probabilistic testing. Zbl 0952.68091
Núñez, Manuel; Rupérez, David
1999
Some rigorous results for the kinematic dynamo problem with general boundary conditions. Zbl 0871.76099
Núñez, Manuel
1997
Some properties of the formation of fast magnetosonic shocks. Zbl 1364.76077
Núñez, Manuel
2015
MHD shear flows with non-constant transverse magnetic field. Zbl 1260.76042
Núñez, Manuel
2012
Using schedulers to test probabilistic distributed systems. Zbl 1259.68033
Hierons, Robert M.; Núñez, Manuel
2012
Implementation relations and test generation for systems with distributed interfaces. Zbl 1277.68190
Hierons, Robert M.; Merayo, Mercedes G.; Núñez, Manuel
2012
The maximum and the addition of assignment games. Zbl 1219.91061
Miquel, S.; Núñez, M.
2011
Testing from a stochastic timed system with a fault model. Zbl 1161.68026
Hierons, Robert M.; Merayo, Mercedes G.; Núñez, Manuel
2009
The limit states of magnetic relaxation. Zbl 1113.76100
Núñez, Manuel
2007
Ion-neutral friction as a model for magnetic relaxation. Zbl 1181.76159
Núñez, Manuel
2006
Conformance testing relations for timed systems. Zbl 1183.68180
Núñez, Manuel; Rodríguez, Ismael
2006
Fast testing of critical properties through passive testing. Zbl 1029.68646
Arnedo, José Antonio; Cavalli, Ana; Núñez, Manuel
2003
Uniform boundedness of the magnetic field in a resistive plasma. Zbl 1013.82021
Núñez, Manuel
2003
A testing theory for generally distributed stochastic processes (extended abstract). Zbl 1006.68539
López, Natalia; Núñez, Manuel
2001
Spectral analysis of viscous static compressible fluid equilibria. Zbl 1031.76045
Núñez, Manuel
2001
Denotational semantics for probabilistic refusal testing. Zbl 0920.68066
Gregorio-Rodríguez, Carlos; Nuñez, Manuel
1998
The decay of axisymmetric magnetic fields: A review of Cowling’s theorem. Zbl 0865.76098
Núñez, Manuel
1996
Passive testing with asynchronous communications and timestamps. Zbl 06916554
Merayo, Mercedes G.; Hierons, Robert M.; Núñez, M.
2018
Guest editorial: Collective intelligent information and database systems. Zbl 1366.00066
Nguyen, Ngoc-Thanh (ed.); Núñez, Manuel (ed.); Trawiński, Bogdan (ed.)
2017
Generation of sheet currents by high frequency fast MHD waves. Zbl 1360.78006
Núñez, Manuel
2016
A simple mean-dispersion model of ambiguity attitudes. Zbl 1319.91060
Schneider, Mark A.; Nunez, Manuel A.
2015
Timed implementation relations for the distributed test architecture. Zbl 1291.68048
Hierons, Robert M.; Merayo, Mercedes G.; Núñez, Manuel
2014
On the gravitational potential of modified Newtonian dynamics. Zbl 1287.85001
Núñez, Manuel
2013
On the prevention of fraud and privacy exposure in process information flow. Zbl 06599279
Bai, Xue; Gopal, Ram; Nunez, Manuel; Zhdanov, Dmitry
2012
Using time to add order to distributed testing. Zbl 1372.68186
Hierons, Robert M.; Merayo, Mercedes G.; Núñez, Manuel
2012
Boundary layer separation of hydromagnetic flows. Zbl 1396.76094
Núñez, Manuel
2012
Uniformization of the density in reacting magnetohydrodynamic flows. Zbl 1167.76386
Núñez, Manuel
2009
On the scarcity of solutions of the equations of magnetohydrodynamic equilibria with flow. Zbl 1221.76234
Núñez, Manuel
2008
Passive testing of timed systems. Zbl 1183.68094
Andrés, César; Merayo, Mercedes G.; Núñez, Manuel
2008
Testing finite state machines presenting stochastic time and timeouts. Zbl 1149.68313
Merayo, Mercedes G.; Núñez, Manuel; Rodríguez, Ismael
2007
Formal specification of multi-agent systems by using EUSMs. Zbl 1141.68478
Merayo, Mercedes G.; Núñez, Manuel; Rodríguez, Ismael
2007
Formal testing of systems presenting soft and hard deadlines. Zbl 1141.68477
Merayo, Mercedes G.; Núñez, Manuel; Rodriguez, Ismael
2007
Implementation relations for stochastic finite state machines. Zbl 1185.68393
Merayo, Mercedes G.; Núñez, Manuel; Rodríguez, Ismael
2006
Derivation of a suitable finite test suite for customized probabilistic systems. Zbl 1225.68046
Llana-Díaz, Luis F.; Núñez, Manuel; Rodríguez, Ismael
2006
Extending EFSMs to specify and test timed systems with action durations and timeouts. Zbl 1225.68121
Merayo, Mercedes G.; Núñez, Manuel; Rodríguez, Ismael
2006
Testing of symbolic-probabilistic systems. Zbl 1081.68575
López, Natalia; Núñez, Manuel; Rodríguez, Ismael
2005
Formal specification of symbolic-probabilistic systems. Zbl 1111.68511
López, Natalia; Núñez, Manuel; Rodríguez, Ismael
2004
Growth of the magnetic field in Hall magnetohydrodynamics. Zbl 1067.76099
Núñez, Manuel
2004
Dynamic effects on the stretching of the magnetic field by a plasma flow. Zbl 1161.76612
Núñez, Manuel
2003
Rigorous bounds on intermittent bursts for turbulent flows. Zbl 1036.76022
Núñez, Manuel
2003
Plasma velocity in hydromagnetic dynamos. Zbl 1059.76079
Núñez, Manuel
2002
On the regularity of the magnetic field in a diffusive plasma. Zbl 0982.82024
Núñez, Manuel
2001
Global timed bisimulation: An introduction. Zbl 0954.68113
de Frutos-Escrig, David; López, Natalia; Núñez, Manuel
1999
On a class of partial differential equations with singular coefficients. Zbl 0637.35022
Núñez, Manuel
1988
Passive testing with asynchronous communications and timestamps. Zbl 06916554
Merayo, Mercedes G.; Hierons, Robert M.; Núñez, M.
2018
Guest editorial: Collective intelligent information and database systems. Zbl 1366.00066
Nguyen, Ngoc-Thanh (ed.); Núñez, Manuel (ed.); Trawiński, Bogdan (ed.)
2017
Generation of sheet currents by high frequency fast MHD waves. Zbl 1360.78006
Núñez, Manuel
2016
Some properties of the formation of fast magnetosonic shocks. Zbl 1364.76077
Núñez, Manuel
2015
A simple mean-dispersion model of ambiguity attitudes. Zbl 1319.91060
Schneider, Mark A.; Nunez, Manuel A.
2015
Timed implementation relations for the distributed test architecture. Zbl 1291.68048
Hierons, Robert M.; Merayo, Mercedes G.; Núñez, Manuel
2014
On the gravitational potential of modified Newtonian dynamics. Zbl 1287.85001
Núñez, Manuel
2013
MHD shear flows with non-constant transverse magnetic field. Zbl 1260.76042
Núñez, Manuel
2012
Using schedulers to test probabilistic distributed systems. Zbl 1259.68033
Hierons, Robert M.; Núñez, Manuel
2012
Implementation relations and test generation for systems with distributed interfaces. Zbl 1277.68190
Hierons, Robert M.; Merayo, Mercedes G.; Núñez, Manuel
2012
On the prevention of fraud and privacy exposure in process information flow. Zbl 06599279
Bai, Xue; Gopal, Ram; Nunez, Manuel; Zhdanov, Dmitry
2012
Using time to add order to distributed testing. Zbl 1372.68186
Hierons, Robert M.; Merayo, Mercedes G.; Núñez, Manuel
2012
Boundary layer separation of hydromagnetic flows. Zbl 1396.76094
Núñez, Manuel
2012
The maximum and the addition of assignment games. Zbl 1219.91061
Miquel, S.; Núñez, M.
2011
Regularity criteria for the Navier-Stokes equations involving the ratio pressure-gradient of velocity. Zbl 1183.35221
Núñez, Manuel
2010
Testing from a stochastic timed system with a fault model. Zbl 1161.68026
Hierons, Robert M.; Merayo, Mercedes G.; Núñez, Manuel
2009
Uniformization of the density in reacting magnetohydrodynamic flows. Zbl 1167.76386
Núñez, Manuel
2009
Formal testing from timed finite state machines. Zbl 1131.68029
Merayo, Mercedes G.; Núñez, Manuel; Rodríguez, Ismael
2008
$$\mathcal {HOTL}$$: Hypotheses and observations testing logic. Zbl 1131.68064
Rodríguez, Ismael; Merayo, Mercedes G.; Núñez, Manuel
2008
Controllable test cases for the distributed test architecture. Zbl 1183.68098
Hierons, Robert M.; Merayo, Mercedes G.; Núñez, Manuel
2008
Extending EFSMs to specify and test timed systems with action duration and time-outs. Zbl 1373.68141
Merayo, Mercedes G.; Núñez, Manuel; Rodríguez, Ismael
2008
A theorem of existence for the equations of magnetohydrodynamics of partially ionized plasmas. Zbl 1136.76059
Núñez, Manuel
2008
On the scarcity of solutions of the equations of magnetohydrodynamic equilibria with flow. Zbl 1221.76234
Núñez, Manuel
2008
Passive testing of timed systems. Zbl 1183.68094
Andrés, César; Merayo, Mercedes G.; Núñez, Manuel
2008
The limit states of magnetic relaxation. Zbl 1113.76100
Núñez, Manuel
2007
Testing finite state machines presenting stochastic time and timeouts. Zbl 1149.68313
Merayo, Mercedes G.; Núñez, Manuel; Rodríguez, Ismael
2007
Formal specification of multi-agent systems by using EUSMs. Zbl 1141.68478
Merayo, Mercedes G.; Núñez, Manuel; Rodríguez, Ismael
2007
Formal testing of systems presenting soft and hard deadlines. Zbl 1141.68477
Merayo, Mercedes G.; Núñez, Manuel; Rodriguez, Ismael
2007
Specification, testing and implementation relations for symbolic-probabilistic systems. Zbl 1088.68021
López, Natalia; Núñez, Manuel; Rodríguez, Ismael
2006
Customized testing for probabilistic systems. Zbl 1185.68094
Llana-Díaz, Luis F.; Núñez, Manuel; Rodríguez, Ismael
2006
A logic for assessing sets of heterogeneous testing hypotheses. Zbl 1185.68422
Rodríguez, Ismael; Merayo, Mercedes G.; Núñez, Manuel
2006
Ion-neutral friction as a model for magnetic relaxation. Zbl 1181.76159
Núñez, Manuel
2006
Conformance testing relations for timed systems. Zbl 1183.68180
Núñez, Manuel; Rodríguez, Ismael
2006
Implementation relations for stochastic finite state machines. Zbl 1185.68393
Merayo, Mercedes G.; Núñez, Manuel; Rodríguez, Ismael
2006
Derivation of a suitable finite test suite for customized probabilistic systems. Zbl 1225.68046
Llana-Díaz, Luis F.; Núñez, Manuel; Rodríguez, Ismael
2006
Extending EFSMs to specify and test timed systems with action durations and timeouts. Zbl 1225.68121
Merayo, Mercedes G.; Núñez, Manuel; Rodríguez, Ismael
2006
A passive testing approach based on invariants: application to the WAP. Zbl 1080.68534
Bayse, Emmanuel; Cavalli, Ana; Núñez, Manuel; Zaïdi, Fatiha
2005
Existence theorems for two-fluid magnetohydrodynamics. Zbl 1110.76057
Núñez, Manuel
2005
Testing of symbolic-probabilistic systems. Zbl 1081.68575
López, Natalia; Núñez, Manuel; Rodríguez, Ismael
2005
An overview of probabilistic process algebras and their equivalences. Zbl 1203.68115
López, Natalia; Núñez, Manuel
2004
Formal specification of symbolic-probabilistic systems. Zbl 1111.68511
López, Natalia; Núñez, Manuel; Rodríguez, Ismael
2004
Growth of the magnetic field in Hall magnetohydrodynamics. Zbl 1067.76099
Núñez, Manuel
2004
Algebraic theory of probabilistic processes. Zbl 1048.68057
Núñez, Manuel
2003
Advances in nonlinear dynamos. Zbl 1084.85500
Ferriz-Mas, Antonio (ed.); Núñez, Manuel (ed.)
2003
Estimates on hyperdiffusive magnetohydrodynamics. Zbl 1044.76072
Núñez, Manuel
2003
Towards testing stochastic timed systems. Zbl 1279.68029
Núñez, Manuel; Rodríguez, Ismael
2003
Fast testing of critical properties through passive testing. Zbl 1029.68646
Arnedo, José Antonio; Cavalli, Ana; Núñez, Manuel
2003
Uniform boundedness of the magnetic field in a resistive plasma. Zbl 1013.82021
Núñez, Manuel
2003
Dynamic effects on the stretching of the magnetic field by a plasma flow. Zbl 1161.76612
Núñez, Manuel
2003
Rigorous bounds on intermittent bursts for turbulent flows. Zbl 1036.76022
Núñez, Manuel
2003
A characterization of ill-posed data instances for convex programming. Zbl 1049.90059
Nunez, Manuel A.
2002
Encoding PAMR into (timed) EFSMs. Zbl 1037.68557
Núñez, Manuel; Rodríguez, Ismael
2002
Plasma velocity in hydromagnetic dynamos. Zbl 1059.76079
Núñez, Manuel
2002
Condition-measure bounds on the behavior of the central trajectory of a semidefinite program. Zbl 1010.90054
Nunez, Manuel A.; Freund, Robert M.
2001
A testing theory for generally distributed stochastic processes (extended abstract). Zbl 1006.68539
López, Natalia; Núñez, Manuel
2001
Spectral analysis of viscous static compressible fluid equilibria. Zbl 1031.76045
Núñez, Manuel
2001
On the regularity of the magnetic field in a diffusive plasma. Zbl 0982.82024
Núñez, Manuel
2001
Fair testing through probabilistic testing. Zbl 0952.68091
Núñez, Manuel; Rupérez, David
1999
Global timed bisimulation: An introduction. Zbl 0954.68113
de Frutos-Escrig, David; López, Natalia; Núñez, Manuel
1999
Condition measures and properties of the central trajectory of a linear program. Zbl 0920.90097
Nunez, Manuel A.; Freund, Robert M.
1998
Denotational semantics for probabilistic refusal testing. Zbl 0920.68066
Gregorio-Rodríguez, Carlos; Nuñez, Manuel
1998
Some rigorous results for the kinematic dynamo problem with general boundary conditions. Zbl 0871.76099
Núñez, Manuel
1997
The decay of axisymmetric magnetic fields: A review of Cowling’s theorem. Zbl 0865.76098
Núñez, Manuel
1996
On a class of partial differential equations with singular coefficients. Zbl 0637.35022
Núñez, Manuel
1988
all top 5

#### Cited by 154 Authors

 29 Nunez, Manuel A. 10 Rodríguez, Ismael 9 Hierons, Robert Mark 9 Merayo, Mercedes G. 4 Bernardo, Marco 4 Cánovas, María Josefa 4 Parra, Juan 3 Delahaye, Benoît 3 Legay, Axel 3 López-Cerdá, Marco Antonio 3 Toledo, Fco. Javier 2 Andrés, César 2 Caillaud, Benoît 2 Cao, Yongzhi 2 Cuartero, Fernando 2 De Nicola, Rocco 2 Goberna, Miguel Angel 2 Guo, Zhengguang 2 He, Jifeng 2 Ivanov Todorov, Maxim 2 Larsen, Kim Guldstrand 2 Liu, Qiao 2 Llana, Luis 2 Llerena, Francesc 2 Loreti, Michele 2 McCormick, David S. 2 Núñez, Marina 2 Pedersen, Mikkel Lykke 2 Pineau, Benjamin 2 Pribyl’, M. A. 2 Qin, Shengchao 2 Rabanal, Pablo 2 Robinson, James Cooper 2 Rodrigo, José Luis 2 Rubio, Fernando 2 Sanders, Jeff W. 2 Valero, Valentín 2 Wang, Hanpin 2 Wąsowski, Andrzej 2 Wu, Jiahong 2 Yu, Xinwei 2 Zhu, Huibiao 1 Andova, Suzana 1 Arbab, Farhad 1 Arnold, Florian 1 Atay, Ata 1 Bădică, Amelia 1 Bădică, Costin 1 Baier, Christel 1 Bowen, Jonathan P. 1 Brandenburg, Axel 1 Bravetti, Mario 1 Bueno, Antonio 1 Cacciagrano, Diletta Romana 1 Camacho, Carlos J. 1 Cao, Chongsheng 1 Cazorla, Diego 1 Chen, Qionglei 1 Chen, Weien 1 Chen, Zengqiang 1 Cheung, Dennis 1 Colombo, Marco 1 Cook, William John 1 Corradini, Flavio 1 Cucker, Felipe 1 de Bakker, Jacobus W. 1 de Frutos-Escrig, David 1 de Vink, Erik P. 1 den Hartog, Jerry I. 1 Descombes, Stéphane 1 Diaz, Gregorio 1 Djamel, Benterki 1 Dormy, Emmanuel 1 Dowling, Timothy E. 1 El Haj Ben Ali, Safae 1 El Yassini, Khalid 1 Eyink, Gregory L. 1 Fan, Shaokun 1 Fecher, Harald 1 Fefferman, Charles Louis 1 Gala, Sadek 1 Galimullin, R. F. 1 Gebler, Daniel E. 1 Georgievska, Sonja 1 Gérard-Varet, David 1 Gerhold, Marcus 1 Gómez-Senent, Francisco José 1 Gondzio, Jacek 1 González-Brevis, Pablo 1 Gregorio-Rodríguez, Carlos 1 Gromov, Maksim Leonidovich 1 Grothey, Andreas 1 Guck, Dennis 1 Halická, Margaréta 1 Hatefi, Hassan 1 Hermanns, Holger 1 Huang, Shuqin 1 Ipate, Florentin 1 Işik, Erman 1 Iskakov, Alekseĭ Borisovich ...and 54 more Authors
all top 5

#### Cited in 54 Serials

 10 The Journal of Logic and Algebraic Programming 5 Theoretical Computer Science 5 Optimization 5 Formal Aspects of Computing 4 Distributed Computing 3 Astronomische Nachrichten 3 Journal of Mathematical Physics 3 Physics Letters. A 3 Information and Computation 3 Mathematical Programming. Series A. Series B 3 Journal of Mathematical Fluid Mechanics 3 Journal of Logical and Algebraic Methods in Programming 2 Geophysical and Astrophysical Fluid Dynamics 2 Journal of Differential Equations 2 Programming and Computer Software 2 European Journal of Operational Research 2 European Journal of Mechanics. B. Fluids 1 Acta Informatica 1 Archive for Rational Mechanics and Analysis 1 Computers & Mathematics with Applications 1 Journal of Computational Physics 1 Mathematical Methods in the Applied Sciences 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 International Journal of Mathematics and Mathematical Sciences 1 International Journal of Game Theory 1 Journal of Functional Analysis 1 Journal of Mathematical Economics 1 Kybernetika 1 Mechanics Research Communications 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Acta Applicandae Mathematicae 1 Physica D 1 Applied Numerical Mathematics 1 Journal of Complexity 1 Applied Mathematics Letters 1 Linear Algebra and its Applications 1 SIAM Journal on Applied Mathematics 1 Computational Optimization and Applications 1 Formal Methods in System Design 1 International Journal of Modern Physics D 1 St. Petersburg Mathematical Journal 1 Top 1 INFORMS Journal on Computing 1 Mathematical Physics, Analysis and Geometry 1 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 1 International Journal of Applied Mathematics and Computer Science 1 Journal of Dynamical and Control Systems 1 RAIRO. Operations Research 1 Bulletin of the Brazilian Mathematical Society. New Series 1 Optimization Letters 1 Discrete and Continuous Dynamical Systems. Series S 1 Kinetic and Related Models 1 Mathematical Programming Computation 1 Science China. Information Sciences
all top 5

#### Cited in 22 Fields

 50 Computer science (68-XX) 32 Fluid mechanics (76-XX) 22 Partial differential equations (35-XX) 20 Operations research, mathematical programming (90-XX) 6 Numerical analysis (65-XX) 6 Astronomy and astrophysics (85-XX) 5 Probability theory and stochastic processes (60-XX) 4 Operator theory (47-XX) 4 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 3 Calculus of variations and optimal control; optimization (49-XX) 3 Systems theory; control (93-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Convex and discrete geometry (52-XX) 2 Optics, electromagnetic theory (78-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Functional analysis (46-XX) 1 Statistics (62-XX) 1 Mechanics of particles and systems (70-XX) 1 Mechanics of deformable solids (74-XX) 1 Geophysics (86-XX) 1 Information and communication theory, circuits (94-XX)


text:
at Comenius Program, Harley Mason Room, Corpus Christi College on Nov 11, 2021 [reveal]
Neil D. Lawrence, Amazon and University of Sheffield

#### Abstract

What is the nature of machine intelligence and how does it differ from humans? In this talk we introduce embodiment factors. They represent the extent to which our intelligence is locked inside us. The locked in nature of our intelligence makes us fundamentally different from the machine intelligences we are creating around us. Having summarized these differences we consider the Three Ds of machine learning system design: a set of considerations to take into acount when building machine intelligences.

## The Diving Bell and the Butterfly

The Diving Bell and the Butterfly is the autobiography of Jean-Dominique Bauby.

In 1995, when he was editor-in-chief of the French Elle magazine, he suffered a stroke, which destroyed his brainstem. He became almost totally physically paralyzed, but was still mentally active. He acquired what is known as locked-in syndrome.

Incredibly, Bauby wrote his memoir after he became paralyzed.

His left eye was the only muscle he could voluntarily move, and he wrote the entire book by winking it.

How could he do that? Well, first, they set up a mechanism where he could scan across letters and blink at the letter he wanted to use. In this way, he was able to write each letter.

It took him 10 months of four hours a day to write the book. Each word took two minutes to write.

Imagine doing all that thinking, but so little speaking, having all those thoughts and so little ability to communicate.

The idea behind this talk is that we are all in that situation. While not as extreme as for Bauby, we all have somewhat of a locked in intelligence.

## Embodiment Factors

Let me explain what I mean. Claude Shannon introduced a mathematical concept of information for the purposes of understanding telephone exchanges.

Information has many meanings, but mathematically, Shannon defined a bit of information to be the amount of information you get from tossing a coin.

If I toss a coin, and look at it, I know the answer. You don’t. But if I now tell you the answer I communicate to you 1 bit of information. Shannon defined this as the fundamental unit of information.

If I toss the coin twice, and tell you the result of both tosses, I give you two bits of information. Information is additive.

Shannon also estimated the average information associated with the English language. He estimated that the average information in any word is 12 bits, equivalent to twelve coin tosses.

So every two minutes Bauby was able to communicate 12 bits, or six bits per minute.

This is the information transfer rate he was limited to, the rate at which he could communicate.

Compare this to me, talking now. The average speaker for TEDX speaks around 160 words per minute. That’s 320 times faster than Bauby or around a 2000 bits per minute. 2000 coin tosses per minute.

But, just think how much thought Bauby was putting into every sentence. Imagine how carefully chosen each of his words was. Because he was communication constrained he could put more thought into each of his words. Into thinking about his audience.

So, his intelligence became locked in. He thinks as fast as any of us, but can communicate slower. Like the tree falling in the woods with no one there to hear it, his intelligence is embedded inside him.

Two thousand coin tosses per minute sounds pretty impressive, but this talk is not just about us, it’s about our computers, and the type of intelligence we are creating within them.

So how does two thousand compare to our digital companions? When computers talk to each other, they do so with billions of coin tosses per minute.

Let’s imagine for a moment, that instead of talking about communication of information, we are actually talking about money. Bauby would have 6 dollars. I would have 2000 dollars, and my computer has billions of dollars.

The internet has interconnected computers and equipped them with extremely high transfer rates.

However, by our very best estimates, computers actually think slower than us.

How can that be? You might ask, computers calculate much faster than me. That’s true, but underlying your conscious thoughts there are a lot of calculations going on.

Each thought involves many thousands, millions or billions of calculations. How many exactly, we don’t know yet, because we don’t know how the brain turns calculations into thoughts.

Our best estimates suggest that to simulate your brain a computer would have to be as large as the UK Met Office machine here in Exeter. That’s a 250 million pound machine, the fastest in the UK. It can do 16 billion billon calculations per second.

It simulates the weather across the word every day, that’s how much power we think we need to simulate our brains.

So, in terms of our computational power we are extraordinary, but in terms of our ability to explain ourselves, just like Bauby, we are locked in.

For a typical computer, to communicate everything it computes in one second, it would only take it a couple of minutes. For us to do the same would take 15 billion years.

If intelligence is fundamentally about processing and sharing of information. This gives us a fundamental constraint on human intelligence that dictates its nature.

I call this ratio between the time it takes to compute something, and the time it takes to say it, the embodiment factor (Lawrence, 2017). Because it reflects how embodied our cognition is.

If it takes you two minutes to say the thing you have thought in a second, then you are a computer. If it takes you 15 billion years, then you are a human.

So when it comes to our ability to compute we are extraordinary, not compute in our conscious mind, but the underlying neuron firings that underpin both our consciousness, our subconsciousness as well as our motor control etc.

If we think of ourselves as vehicles, then we are massively overpowered. Our ability to generate derived information from raw fuel is extraordinary. Intellectually we have formula one engines.

But in terms of our ability to deploy that computation in actual use, to share the results of what we have inferred, we are very limited. So when you imagine the F1 car that represents a psyche, think of an F1 car with bicycle wheels.

Just think of the control a driver would have to have to deploy such power through such a narrow channel of traction. That is the beauty and the skill of the human mind.

In contrast, our computers are more like go-karts. Underpowered, but with well-matched tires. They can communicate far more fluidly. They are more efficient, but somehow less extraordinary, less beautiful.

## Human Communication

For human conversation to work, we require an internal model of who we are speaking to. We model each other, and combine our sense of who they are, who they think we are, and what has been said. This is our approach to dealing with the limited bandwidth connection we have. Empathy and understanding of intent. Mental dispositional concepts are used to augment our limited communication bandwidth.

Fritz Heider referred to the important point of a conversation as being that they are happenings that are “psychologically represented in each of the participants” (his emphasis) (Heider, 1958).

### Bandwidth Constrained Conversations

Embodiment factors imply that, in our communication between humans, what is not said is, perhaps, more important than what is said. To communicate with each other we need to have a model of who each of us are.

To aid this, in society, we are required to perform roles. Whether as a parent, a teacher, an employee or a boss. Each of these roles requires that we conform to certain standards of behaviour to facilitate communication between ourselves.

Control of self is vitally important to these communications.

The high availability of data available to humans undermines human-to-human communication channels by providing new routes to undermining our control of self.

### A Six Word Novel

But this is a very different kind of intelligence than ours. A computer cannot understand the depth of the Ernest Hemingway’s apocryphal six word novel: “For Sale, Baby Shoes, Never worn,” because it isn’t equipped with that ability to model the complexity of humanity that underlies that statement.

## Computer Conversations

Similarly, we find it difficult to comprehend how computers are making decisions. Because they do so with more data than we can possibly imagine.

In many respects, this is not a problem, it’s a good thing. Computers and us are good at different things. But when we interact with a computer, when it acts in a different way to us, we need to remember why.

Just as the first step to getting along with other humans is understanding other humans, so it needs to be with getting along with our computers.

Embodiment factors explain why, at the same time, computers are so impressive in simulating our weather, but so poor at predicting our moods. Our complexity is greater than that of our weather, and each of us is tuned to read and respond to one another.

Their intelligence is different. It is based on very large quantities of data that we cannot absorb. Our computers don’t have a complex internal model of who we are. They don’t understand the human condition. They are not tuned to respond to us as we are to each other.

Embodiment factors encapsulate a profound thing about the nature of humans. Our locked in intelligence means that we are striving to communicate, so we put a lot of thought into what we’re communicating with. And if we’re communicating with something complex, we naturally anthropomorphize them.

We give our dogs, our cats and our cars human motivations. We do the same with our computers. We anthropomorphize them. We assume that they have the same objectives as us and the same constraints. They don’t.

This means, that when we worry about artificial intelligence, we worry about the wrong things. We fear computers that behave like more powerful versions of ourselves that will struggle to outcompete us.

In reality, the challenge is that our computers cannot be human enough. They cannot understand us with the depth we understand one another. They drop below our cognitive radar and operate outside our mental models.

The real danger is that computers don’t anthropomorphize. They’ll make decisions in isolation from us without our supervision, because they can’t communicate truly and deeply with us.

# Evolved Relationship with Information

The high bandwidth of computers has resulted in a close relationship between the computer and data. Large amounts of information can flow between the two. The degree to which the computer is mediating our relationship with data means that we should consider it an intermediary.

Originaly our low bandwith relationship with data was affected by two characteristics. Firstly, our tendency to over-interpret driven by our need to extract as much knowledge from our low bandwidth information channel as possible. Secondly, by our improved understanding of the domain of mathematical statistics and how our cognitive biases can mislead us.

With this new set up there is a potential for assimilating far more information via the computer, but the computer can present this to us in various ways. If it’s motives are not aligned with ours then it can misrepresent the information. This needn’t be nefarious it can be simply as a result of the computer pursuing a different objective from us. For example, if the computer is aiming to maximize our interaction time that may be a different objective from ours which may be to summarize information in a representative manner in the shortest possible length of time.

For example, for me, it was a common experience to pick up my telephone with the intention of checking when my next appointment was, but to soon find myself distracted by another application on the phone, and end up reading something on the internet. By the time I’d finished reading, I would often have forgotten the reason I picked up my phone in the first place.

There are great benefits to be had from the huge amount of information we can unlock from this evolved relationship between us and data. In biology, large scale data sharing has been driven by a revolution in genomic, transcriptomic and epigenomic measurement. The improved inferences that can be drawn through summarizing data by computer have fundamentally changed the nature of biological science, now this phenomenon is also infuencing us in our daily lives as data measured by happenstance is increasingly used to characterize us.

Better mediation of this flow actually requires a better understanding of human-computer interaction. This in turn involves understanding our own intelligence better, what its cognitive biases are and how these might mislead us.

For further thoughts see Guardian article on marketing in the internet era from 2015.

You can also check my blog post on System Zero. also from 2015.

## New Flow of Information

Classically the field of statistics focussed on mediating the relationship between the machine and the human. Our limited bandwidth of communication means we tend to over-interpret the limited information that we are given, in the extreme we assign motives and desires to inanimate objects (a process known as anthropomorphizing). Much of mathematical statistics was developed to help temper this tendency and understand when we are valid in drawing conclusions from data.

Data science brings new challenges. In particular, there is a very large bandwidth connection between the machine and data. This means that our relationship with data is now commonly being mediated by the machine. Whether this is in the acquisition of new data, which now happens by happenstance rather than with purpose, or the interpretation of that data where we are increasingly relying on machines to summarise what the data contains. This is leading to the emerging field of data science, which must not only deal with the same challenges that mathematical statistics faced in tempering our tendency to over interpret data, but must also deal with the possibility that the machine has either inadvertently or malisciously misrepresented the underlying data.

## Boulton and Watt’s Steam Engine

James Watt’s steam engine contained an early machine learning device. In the same way that modern systems are component based, his engine was composed of components. One of which is a speed regulator sometimes known as Watt’s governor. The two balls in the center of the image, when spun fast, rise, and through a linkage mechanism.

The centrifugal governor was made famous by Boulton and Watt when it was deployed in the steam engine. Studying stability in the governor is the main subject of James Clerk Maxwell’s paper on the theoretical analysis of governors (Maxwell, 1867). This paper is a founding paper of control theory. In an acknowledgment of its influence, Wiener used the name cybernetics to describe the field of control and communication in animals and the machine (Wiener, 1948). Cybernetics is the Greek word for governor, which comes from the latin for helmsman.

A governor is one of the simplest artificial intelligence systems. It senses the speed of an engine, and acts to change the position of the valve on the engine to slow it down.

Although it’s a mechanical system a governor can be seen as automating a role that a human would have traditionally played. It is an early example of artificial intelligence.

The centrifugal governor has several parameters, the weight of the balls used, the length of the linkages and the limits on the balls movement.

Two principle differences exist between the centrifugal governor and artificial intelligence systems of today.

1. The centrifugal governor is a physical system and it is an integral part of a wider physical system that it regulates (the engine).
2. The parameters of the governor were set by hand, our modern artificial intelligence systems have their parameters set by data.

This has the basic components of sense and act that we expect in an intelligent system, and this system saved the need for a human operator to manually adjust the system in the case of overspeed. Overspeed has the potential to destroy an engine, so the governor operates as a safety device.

The first wave of automation did bring about sabotoage as a worker’s response. But if machinery was sabotaged, for example, if the linkage between sensor (the spinning balls) and action (the valve closure) was broken, this would be obvious to the engine operator at start up time. The machine could be repaired before operation.

## What is Machine Learning?

Machine learning allows us to extract knowledge from data to form a prediction.

$\text{data} + \text{model} \stackrel{\text{compute}}{\rightarrow} \text{prediction}$

A machine learning prediction is made by combining a model with data to form the prediction. The manner in which this is done gives us the machine learning algorithm.

Machine learning models are mathematical models which make weak assumptions about data, e.g. smoothness assumptions. By combining these assumptions with the data, we observe we can interpolate between data points or, occasionally, extrapolate into the future.

Machine learning is a technology which strongly overlaps with the methodology of statistics. From a historical/philosophical view point, machine learning differs from statistics in that the focus in the machine learning community has been primarily on accuracy of prediction, whereas the focus in statistics is typically on the interpretability of a model and/or validating a hypothesis through data collection.

The rapid increase in the availability of compute and data has led to the increased prominence of machine learning. This prominence is surfacing in two different but overlapping domains: data science and artificial intelligence.

## From Model to Decision

The real challenge, however, is end-to-end decision making. Taking information from the environment and using it to drive decision making to achieve goals.

The big data paradox is the modern phenomenon of “as we collect more data, we understand less.” It is emerging in several domains, political polling, characterization of patients for trials data, monitoring twitter for political sentiment.

I like to think of the phenomenon as relating to the notion of “can’t see the wood for the trees.” Classical statistics, with randomized controlled trials, improved society’s understanding of data. It improved our ability to monitor the forest, to consider population health, voting patterns etc. It is critically dependent on active approaches to data collection that deal with confounders. This data collection can be very expensive.

In business today, it is still the gold standard, A/B tests are used to understand the effect of an intervention on revenue or customer capture or supply chain costs.

The new phenomenon is happenstance data. Data that is not actively collected with a question in mind. As a result, it can mislead us. For example, if we assume the politics of active users of twitter is reflective of the wider population’s politics, then we may be misled.

However, this happenstance data often allows us to characterise a particular individual to a high degree of accuracy. Classical statistics was all about the forest, but big data can often become about the individual tree. As a result we are misled about the situation.

The phenomenon is more dangerous, because our perception is that we are characterizing the wider scenario with ever increasing accuracy. Whereas we are just becoming distracted by detail that may or may not be pertinent to the wider situation.

This is related to our limited bandwidth as humans, and the ease with which we are distracted by detail. The data-inattention-cognitive-bias.

## Complexity in Action

As an exercise in understanding complexity, watch the following video. You will see the basketball being bounced around, and the players moving. Your job is to count the passes of those dressed in white and ignore those of the individuals dressed in black.

In a classic study Simons and Chabris (1999) ask subjects to count the number of passes of the basketball between players on the team wearing white shirts. Fifty percent of the time, these subjects don’t notice the gorilla moving across the scene.

The phenomenon of inattentional blindness is well known, e.g in their paper Simons and Charbris quote the Hungarian neurologist, Rezsö Bálint,

It is a well-known phenomenon that we do not notice anything happening in our surroundings while being absorbed in the inspection of something; focusing our attention on a certain object may happen to such an extent that we cannot perceive other objects placed in the peripheral parts of our visual field, although the light rays they emit arrive completely at the visual sphere of the cerebral cortex.

Rezsö Bálint 1907 (translated in Husain and Stein 1988, page 91)

When we combine the complexity of the world with our relatively low bandwidth for information, problems can arise. Our focus on what we perceive to be the most important problem can cause us to miss other (potentially vital) contextual information.

This phenomenon is known as selective attention or ‘inattentional blindness.’

## Data Selective Attention Bias

We are going to see how inattention biases can play out in data analysis by going through a simple example. The analysis involves body mass index and activity information.

## BMI Steps Data

The BMI Steps example is taken from Yanai and Lercher (2020). We are given a data set of body-mass index measurements against step counts. For convenience we have packaged the data so that it can be easily downloaded.

import pods
data = pods.datasets.bmi_steps()
X = data['X']
y = data['Y']

It is good practice to give our variables interpretable names so that the analysis may be clearly understood by others. Here the steps count is the first dimension of the covariate, the bmi is the second dimension and the gender is stored in y with 1 for female and 0 for male.

steps = X[:, 0]
bmi = X[:, 1]
gender = y[:, 0]

We can check the mean steps and the mean of the BMI.

print('Steps mean is {mean}.'.format(mean=steps.mean()))
print('BMI mean is {mean}.'.format(mean=bmi.mean()))

## BMI Steps Data Analysis

We can also separate out the means from the male and female populations. In python this can be done by setting male and female indices as follows.

male_ind = (gender==0)
female_ind = (gender==1)

And now we can extract the variables for the two populations.

male_steps = steps[male_ind]
male_bmi = bmi[male_ind]

And as before we compute the mean.

print('Male steps mean is {mean}.'.format(mean=male_steps.mean()))
print('Male BMI mean is {mean}.'.format(mean=male_bmi.mean()))

Similarly, we can get the same result for the female portion of the populaton.

female_steps = steps[female_ind]
female_bmi = bmi[female_ind]
print('Female steps mean is {mean}.'.format(mean=female_steps.mean()))
print('Female BMI mean is {mean}.'.format(mean=female_bmi.mean()))

Interesting, the female BMI average is slightly higher than the male BMI average. The number of steps in the male group is higher than that in the female group. Perhaps the steps and the BMI are anti-correlated. The more steps, the lower the BMI.

Python provides a statistics package. We’ll import this in python so that we can try and understand the correlation between the steps and the BMI.

from scipy.stats import pearsonr
corr, _ = pearsonr(steps, bmi)
print("Pearson's overall correlation: {corr}".format(corr=corr))

male_corr, _ = pearsonr(male_steps, male_bmi)
print("Pearson's correlation for males: {corr}".format(corr=male_corr))

female_corr, _ = pearsonr(female_steps, female_bmi)
print("Pearson's correlation for females: {corr}".format(corr=female_corr))

## A Hypothesis as a Liability

This analysis is from an article titled “A Hypothesis as a Liability” (Yanai and Lercher, 2020), they start their article with the following quite from Herman Hesse.

“ ‘When someone seeks,’ said Siddhartha, ‘then it easily happens that his eyes see only the thing that he seeks, and he is able to find nothing, to take in nothing. […] Seeking means: having a goal. But finding means: being free, being open, having no goal.’ ”

Hermann Hesse

Their idea is that having a hypothesis can constrain our thinking. However, in answer to their paper Felin et al. (2021) argue that some form of hypothesis is always necessary, suggesting that a hypothesis can be a liability

My view is captured in the introductory chapter to an edited volume on computational systems biology that I worked on with Mark Girolami, Magnus Rattray and Guido Sanguinetti.

Popper nicely captures the interaction between hypothesis and data by relating it to the chicken and the egg. The important thing is that these two co-evolve.

## Number Theatre

Unfortunately, we don’t always have time to wait for this process to converge to an answer we can all rely on before a decision is required.

Not only can we be misled by data before a decision is made, but sometimes we can be misled by data to justify the making of a decision. David Spiegelhalter refers to the phenomenon of “Number Theatre” in a conversation with Andrew Marr from May 2020 on the presentation of data.

## Data Theatre

Data Theatre exploits data inattention bias to present a particular view on events that may misrepresents through selective presentation. Statisticians are one of the few groups that are trained with a sufficient degree of data skepticism. But it can also be combatted through ensuring there are domain experts present, and that they can speak freely.

The best book I have found for teaching the skeptical sense of data that underlies the statisticians craft is David Spiegelhalter’s Art of Statistics.

# The Art of Statistics

David’s (Spiegelhalter, 2019) book brings important examples from statistics to life in an intelligent and entertaining way. It is highly readable and gives an opportunity to fast-track towards the important skill of data-skepticism that is the mark of a professional statistician.

Felin, T., Koenderink, J., Krueger, J.I., Noble, D., Ellis, G.F.R., 2021. The data-hypothesis relationship. Genome Biology 22. https://doi.org/10.1186/s13059-021-02276-4
Heider, F., 1958. The psychology of interpersonal relations. John Wiley.
Hilbert, M., López, P., 2011. The world’s technological capcity to store, communicate and compute information. Science 332, 60–65. https://doi.org/10.1126/science.1200970
Lawrence, N.D., 2017. Living together: Mind and machine intelligence. arXiv.
Lawrence, N.D., 2010. Introduction to learning and inference in computational systems biology.
Maxwell, J.C., 1867. On governors. Proceedings of the Royal Society of London 16, 270–283.
Simons, D.J., Chabris, C.F., 1999. Gorillas in our midst: Sustained inattentional blindness for dynamic events. Perception 28, 1059–1074. https://doi.org/10.1068/p281059
Spiegelhalter, D.J., 2019. The art of statistics. Pelican.
Wiener, N., 1948. Cybernetics: Control and communication in the animal and the machine. MIT Press, Cambridge, MA.
Yanai, I., Lercher, M., 2020. A hypothesis is a liability. Genome Biology 21.

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## Meetup : Sydney Rationality Dojo - Urge Propagation

1 28 September 2014 01:56AM

## Discussion article for the meetup : Sydney Rationality Dojo - Urge Propagation

WHEN: 05 October 2014 03:00:00PM (+1000)

WHERE: Humanist House, 10 Shepherd St Chippendale

We'll be examining how to connect your desire for goals or outcomes to specific emotional urges to perform the actions to bring about that outcome.

After the session is over, there will also be an optional group dinner.

## Meetup : October Rationality Dojo - Non-Violent Communication

1 28 September 2014 12:31AM

## Discussion article for the meetup : October Rationality Dojo - Non-Violent Communication

WHEN: 05 October 2014 03:30:00PM (+0800)

WHERE: Ross House Association, 247-251 Flinders Lane, Melbourne

[ATTN: Please remember the new location for the dojos: the Jenny Florence Room, Level 3, Ross House at 247 Flinders Lane, Melbourne. 3:30pm start / arrival - formal dojo activities will commence at 4:00pm.]

The Less Wrong Sunday Rationality Dojos are crafted to be serious self-improvement sessions for those committed to the Art of Rationality and personal growth. Each month a community member will run a session involving a presentation of content, discussion, and exercises. Continuing the succession of immensely successful dojos, Chris will run a session on Non-Violent Communication.

As always, we will review the personal goals we committed to at the previous Dojo (I will have done X by the next Dojo). Our goals are now being recorded via Google Forms here - https://docs.google.com/forms/d/1MCHH4MpbW0SI_2JyMSDlKnnGP4A0qxojQEZoMZIdopk/viewform, and Melbourne Less Wrong organisers have access to the form results if you wish to review the goals you set last month.

This month, we are also seeking 2-3 lightning talks from members. Speakers will be limited to 5 minutes with room for questions. We will be asking for talks from attendees present, but if you already have a talk topic in mind, please contact Louise at lvalmoria@gmail.com The Dojo is likely to run for 2-3 hours, after which some people will get dinner together.

If you have any trouble finding the venue or getting in, call Louise on 0419 192 367.

If you would like to present at a future Dojo or suggest a topic, please fill it in on the Rationality Dojo Roster: http://is.gd/dojoroster

To organise similar events, please send an email to melbournelw@gmail.com

## The Future of Humanity Institute could make use of your money

42 26 September 2014 10:53PM

Many people have an incorrect view of the Future of Humanity Institute's funding situation, so this is a brief note to correct that; think of it as a spiritual successor to this post. As John Maxwell puts it, FHI is "one of the three organizations co-sponsoring LW [and] a group within the University of Oxford's philosophy department that tackles important, large-scale problems for humanity like how to go about reducing existential risk." (If you're not familiar with our work, this article is a nice, readable introduction, and our director, Nick Bostrom, wrote Superintelligence.) Though we are a research institute in an ancient and venerable institution, this does not guarantee funding or long-term stability.

Academic research is generally funded through grants, but because the FHI is researching important but unusual problems, and because this research is multi-disciplinary, we've found it difficult to attract funding from the usual grant bodies. This has meant that we’ve had to prioritise a certain number of projects that are not perfect for existential risk reduction, but that allow us to attract funding from interested institutions.

With more assets, we could both liberate our long-term researchers to do more "pure Xrisk" research, and hire or commission new experts when needed to look into particular issues (such as synthetic biology, the future of politics, and the likelihood of recovery after a civilization collapse).

We are not in any immediate funding crunch, nor are we arguing that the FHI would be a better donation target than MIRI, CSER, or the FLI. But any donations would be both gratefully received and put to effective use. If you'd like to, you can donate to FHI here. Thank you!

## New LW Meetups: Prague, Hasselt

2 26 September 2014 10:53PM

New meetups (or meetups with a hiatus of more than a year) are happening in:

Irregularly scheduled Less Wrong meetups are taking place in:

The remaining meetups take place in cities with regular scheduling, but involve a change in time or location, special meeting content, or simply a helpful reminder about the meetup:

Locations with regularly scheduled meetups: Austin, Berkeley, Berlin, Boston, Brussels, Buffalo, Cambridge UK, Canberra, Columbus, London, Madison WI, Melbourne, Moscow, Mountain View, New York, Philadelphia, Research Triangle NC, Seattle, Sydney, Toronto, Vienna, Washington DC, Waterloo, and West Los Angeles. There's also a 24/7 online study hall for coworking LWers.

## Meetup : West LA—The Worst Argument in the World

1 26 September 2014 04:24AM

## Discussion article for the meetup : West LA—The Worst Argument in the World

WHEN: 01 October 2014 07:00:00PM (-0700)

WHERE: 11066 Santa Monica Blvd, Los Angeles, CA

How to Find Us: Go into this Del Taco. We will be in the back room if possible.

Parking is free in the lot out front or on the street nearby.

Discussion: Last week, I included a link to the Worst Argument in the World essay, not because it was relevant to last week's topic, but because I thought it was something that people should read. This week, I realized what I should have done instead. The three recommended readings are all rewrites of the same essay, but since the first one is the best, you don't need to read the others unless you didn't understand it or something.

No prior exposure to Less Wrong is required; this will be generally accessible.

## Meetup : Washington, D.C.: Book Swap

1 26 September 2014 03:51AM

## Discussion article for the meetup : Washington, D.C.: Book Swap

WHEN: 28 September 2014 03:00:00PM (-0400)

WHERE: National Portrait Gallery

We will be meeting in the Kogod Courtyard of the National Portrait Gallery (8th and F Sts or 8th and G Sts NW, go straight past the information desk from either entrance) to lend, borrow, and discuss whatever random books people bring. As usual, folk will congregate in the courtyard between 3:00 and 3:30; the meeting will begin with people taking turns to suggest topics of discussion based on the books they brought, returning to the usual unstructured conversation when people are done discussing books.

A couple notes:

• There is no obligation to lend, borrow, or return books at book swap meetups. You are not required to bring any given book, or any books at all; you are not required to lend the books that you do bring; if you wish to lend out a given book, you are not required to lend it out to the first person to ask (or to the next person to ask, or to anyone at all); you are not required to borrow a book that is offered to you; and if you have already borrowed a book, you are not required to return it at this meetup. This is a Schelling point for book swaps, no more.
• Books belong to their owners - keep any book you borrow in good condition, and return it in a timely fashion when you finish reading it or when the owner asks for it back.
• In the past, the owners of some books have given permission for borrowers to lend them directly to anyone at the meetup who is interested. If you do not have permission from the owner of the book, do not do this.

Upcoming meetups:

• Oct. 5: Fun & Games
• Oct. 12: TBA
• Oct. 19: Mini Talks
• Oct. 26: TBA

## Meetup : DC EA meetup / Petrov day dinner

1 25 September 2014 10:50PM

## Discussion article for the meetup : DC EA meetup / Petrov day dinner

WHEN: 26 September 2014 07:00:00PM (-0400)

WHERE: 3001 Veazey Ter NW # 1005, Washington, DC 20008

On this day in 1983, in an unparalleled feat of Effective Altruism, Stanislav Petrov declined to destroy the world: http://lesswrong.com/lw/jq/926_is_petrov_day/

We'll celebrate his achievement by getting together for food, drinks, and not destroying the world.

Food and drinks will be provided, though please feel free to help.

Ben Hoffman will give a brief talks about the test run of the project to comment on proposed regulations. Most of the night will be free discussion on anything we want.

Schedule:

7:00 - 7:30 PM Arrive 7:30 - 8:00 PM Talk on DC EA projects 8:00 - 9:00 PM Dinner, drinks, discussion 9:00 - 9:30 PM Petrov day ritual 9:30 - Late: Free discussion

## Logics for Mind-Building Should Have Computational Meaning

15 25 September 2014 09:17PM

The Workshop

Late in July I organized and held MIRIx Tel-Aviv with the goal of investigating the currently-open (to my knowledge) Friendly AI problem called "logical probability": the issue of assigning probabilities to formulas in a first-order proof system, in order to use the reflective consistency of the probability predicate to get past the Loebian Obstacle to building a self-modifying reasoning agent that will trust itself and its successors.  Vadim Kosoy, Benjamin and Joshua Fox, and myself met at the Tel-Aviv Makers' Insurgence for six hours, and each presented our ideas.  I spent most of it sneezing due to my allergies to TAMI's resident cats.

My idea was to go with the proof-theoretic semantics of logic and attack computational construction of logical probability via the Curry-Howard Isomorphism between programs and proofs: this yields a rather direct translation between computational constructions of logical probability and the learning/construction of an optimal function from sensory inputs to actions required by Updateless Decision Theory.

The best I can give as a mathematical result is as follows:

$P(\Gamma \vdash a:A \mid \Gamma \vdash b:B) = P(\Gamma,x:B \vdash [\forall y:B, x/y]a:A)$

$P(\Gamma \vdash (a, b): A \wedge B) = P(\Gamma \vdash a:A \mid \Gamma \vdash b:B) * P(\Gamma \vdash b:B)$

$\frac{x:A \notin \Gamma}{P(\Gamma \vdash x:A) = \mathcal{M}_{\lambda\mu} (A)}$

$\frac{x:A \in \Gamma}{P(\Gamma \vdash x:A) = 1.0}$

The capital $\Gamma$ is a set of hypotheses/axioms/assumptions, and the English letters are metasyntactic variables (like "foo" and "bar" in programming lessons).  The lower-case letters denote proofs/programs, and the upper-case letters denote propositions/types.  The turnstile $\vdash$ just means "deduces": the judgement $\Gamma \vdash a:A$ can be read here as "an agent whose set of beliefs is denoted $\Gamma$ will believe that the evidence a proves the proposition A."  The $[\forall y:B, x/y]a$ performs a "reversed" substitution, with the result reading: "for all y proving/of-type B, substitute x for y in a".  This means that we algorithmically build a new proof/construction/program from a in which any and all constructions proving the proposition B are replaced with the logically-equivalent hypothesis x, which we have added to our hypothesis-set $\Gamma$.

Thus the first equation reads, "the probability of a proving A conditioned on b proving B equals the probability of a proving A when we assume the truth of B as a hypothesis."  The second equation then uses this definition of conditional probability to give the normal Product Rule of probabilities for the logical product (the $\wedge$ operator), defined proof-theoretically.  I strongly believe I could give a similar equation for the normal Sum Rule of probabilities for the logical sum (the $\vee$ operator) if I could only access the relevant paywalled paper, in which the λμ-calculus acting as an algorithmic interpretation of the natural-deduction system for classical propositional logic (rather than intuitionistic) is given.

The third item given there is an inference rule, which reads, "if x is a free variable/hypothesis imputed to have type/prove proposition A, not bound in the hypothesis-set $\Gamma$, then the probability with which we believe x proves A is given by the Solomonoff Measure of type A in the λμ-calculus".  We can define that measure simply as the summed Solomonoff Measure of every program/proof possessing the relevant type, and I don't think going into the details of its construction here would be particularly productive.  Free variables in λ-calculus are isomorphic to unproven hypotheses in natural deduction, and so a probabilistic proof system could learn how much to believe in some free-standing hypothesis via Bayesian evidence rather than algorithmic proof.

The final item given here is trivial: anything assumed has probability 1.0, that of a logical tautology.

The upside to invoking the strange, alien λμ-calculus instead of the more normal, friendly λ-calculus is that we thus reason inside classical logic rather than intuitionistic, which means we can use the classical axioms of probability rather than intuitionistic Bayesianism.  We need classical logic here: if we switch to intuitionistic logics (Heyting algebras rather than Boolean algebras) we do get to make computational decidability a first-class citizen of our logic, but the cost is that we can then believe only computationally provable propositions. As Benjamin Fox pointed out to me at the workshop, Loeb's Theorem then becomes a triviality, with real self-trust rendered no easier.

The Apologia

My motivation and core idea for all this was very simple: I am a devout computational trinitarian, believing that logic must be set on foundations which describe reasoning, truth, and evidence in a non-mystical, non-Platonic way.  The study of first-order logic and especially of incompleteness results in metamathematics, from Goedel on up to Chaitin, aggravates me in its relentless Platonism, and especially in the way Platonic mysticism about logical incompleteness so often leads to the belief that minds are mystical.  (It aggravates other people, too!)

The slight problem which I ran into is that there's a shit-ton I don't know about logic.  I am now working to remedy this grievous hole in my previous education.  Also, this problem is really deep, actually.

I thus apologize for ending the rigorous portion of this write-up here.  Everyone expecting proper rigor, you may now pack up and go home, if you were ever paying attention at all.  Ritual seppuku will duly be committed, followed by hors d'oeuvre.  My corpse will be duly recycled to make paper-clips, in the proper fashion of a failed LessWrongian.

The Parts I'm Not Very Sure About

With any luck, that previous paragraph got rid of all the serious people.

I do, however, still think that the (beautiful) equivalence between computation and logic can yield some insights here.  After all, the whole reason for the strange incompleteness results in first-order logic (shown by Boolos in his textbook, I'm told) is that first-order logic, as a reasoning system, contains sufficient computational machinery to encode a Universal Turing Machine.  The bidirectionality of this reduction (Hilbert and Gentzen both have given computational descriptions of first-order proof systems) is just another demonstration of the equivalence.

In fact, it seems to me (right now) to yield a rather intuitively satisfying explanation of why the Gaifman-Carnot Condition (that every instance we see of $P(x_i)$ provides Bayesian evidence in favor of $\forall x.P(x)$) for logical probabilities is not computably approximable.  What would we need to interpret the Gaifman Condition from an algorithmic, type-theoretic viewpoint?  From this interpretation, we would need a proof of our universal generalization.  This would have to be a dependent product of form $\Pi(x:A).P(x)$, a function taking any construction $x:A$ to a construction of type $P(x)$, which itself has type Prop.  To learn such a dependent function from the examples would be to search for an optimal (simple, probable) construction (program) constituting the relevant proof object: effectively, an individual act of Solomonoff Induction.  Solomonoff Induction, however, is already only semicomputable, which would then make a Gaifman-Hutter distribution (is there another term for these?) doubly semicomputable, since even generating it involves a semiprocedure.

The benefit of using the constructive approach to probabilistic logic here is that we know perfectly well that however incomputable Solomonoff Induction and Gaifman-Hutter distributions might be, both existing humans and existing proof systems succeed in building proof-constructions for quantified sentences all the time, even in higher-order logics such as Coquand's Calculus of Constructions (the core of a popular constructive proof assistant) or Luo's Logic-Enriched Type Theory (the core of a popular dependently-typed programming language and proof engine based on classical logic).  Such logics and their proof-checking algorithms constitute, going all the way back to Automath, the first examples of computational "agents" which acquire specific "beliefs" in a mathematically rigorous way, subject to human-proved theorems of soundness, consistency, and programming-language-theoretic completeness (rather than meaning that every true proposition has a proof, this means that every program which does not become operationally stuck has a type and is thus the proof of some proposition).  If we want our AIs to believe in accordance with soundness and consistency properties we can prove before running them, while being composed of computational artifacts, I personally consider this the foundation from which to build.

Where we can acquire probabilistic evidence in a sound and computable way, as noted above in the section on free variables/hypotheses, we can do so for propositions which we cannot algorithmically prove.  This would bring us closer to our actual goals of using logical probability in Updateless Decision Theory or of getting around the Loebian Obstacle.

Some of the Background Material I'm Reading

Another reason why we should use a Curry-Howard approach to logical probability is one of the simplest possible reasons: the burgeoning field of probabilistic programming is already being built on it.  The Computational Cognitive Science lab at MIT is publishing papers showing that their languages are universal for computable and semicomputable probability distributions, and getting strong results in the study of human general intelligence.  Specifically: they are hypothesizing that we can dissolve "learning" into "inducing probabilistic programs via hierarchical Bayesian inference", "thinking" into "simulation" into "conditional sampling from probabilistic programs", and "uncertain inference" into "approximate inference over the distributions represented by probabilistic programs, conditioned on some fixed quantity of sampling that has been done."

In fact, one might even look at these ideas and think that, perhaps, an agent which could find some way to sample quickly and more accurately, or to learn probabilistic programs more efficiently (in terms of training data), than was built into its original "belief engine" could then rewrite its belief engine to use these new algorithms to perform strictly better inference and learning.  Unless I'm as completely wrong as I usually am about these things (that is, very extremely completely wrong based on an utterly unfounded misunderstanding of the whole topic), it's a potential engine for recursive self-improvement.

They also have been studying how to implement statistical inference techniques for their generate modeling languages which do not obey Bayesian soundness.  While most of machine learning/perception works according to error-rate minimization rather than Bayesian soundness (exactly because Bayesian methods are often too computationally expensive for real-world use), I would prefer someone at least study the implications of employing unsound inference techniques for more general AI and cognitive-science applications in terms of how often such a system would "misbehave".

Many of MIT's models are currently dynamically typed and appear to leave type soundness (the logical rigor with which agents come to believe things by deduction) to future research.  And yet: they got to this problem first, so to speak.  We really ought to be collaborating with them, with the full-time grant-funded academic researchers, rather than trying to armchair-reason our way to a full theory of logical probability as a large group of amateurs or part-timers and only a small core cohort of full-time MIRI and FHI staff investigating AI safety issues.

(I admit to having a nerd crush, and I am actually planning to go visit the Cocosci Lab this coming week, and want/intend to apply to their PhD program.)

They have also uncovered something else I find highly interesting: human learning of both concepts and causal frameworks seems to take place via hierarchical Bayesian inference, gaining a "blessing of abstraction" to countermand the "curse of dimensionality".  The natural interpretation of these abstractions in terms of constructions and types would be that, as in dependently-typed programming languages, constructions have types, and types are constructions, but for hierarchical-learning purposes, it would be useful to suppose that types have specific, structured types more informative than Prop or Typen (for some universe level n).  Inference can then proceed from giving constructions or type-judgements as evidence at the bottom level, up the hierarchy of types and meta-types to give probabilistic belief-assignments to very general knowledge.  Even very different objects could have similar meta-types at some level of the hierarchy, allowing hierarchical inference to help transfer Bayesian evidence between seemingly different domains, giving insight into how efficient general intelligence can work.

Just-for-fun Postscript

If we really buy into the model of thinking as conditional simulation, we can use that to dissolve the modalities "possible" and "impossible".  We arrive at (by my count) three different ways of considering the issue computationally:

1. Conceivable/imaginable: the generative models which constitute my current beliefs do or do not yield a path to make some logical proposition true or to make some causal event happen (planning can be done as inference, after all), with or without some specified level of probability.
2. Sensibility/absurdity: the generative models which constitute my current beliefs place a desirably high or undesirably low probability on the known path(s) by which a proposition might be true or by which an event might happen.  The level which constitutes "desirable" could be set as the $\alpha$ value for a hypothesis test, or some other value determined decision-theoretically.  This could relate to Pascal's Mugging: how probable must something be before I consider it real rather than an artifact of my own hypothesis space?
3. Consistency or Contradiction: the generative models which constitute my current beliefs, plus the hypothesis that some proposition is true or some event can come about, do or do not yield a logical contradiction with some probability (that is, we should believe the contradiction exists only to the degree we believe in our existing models in the first place!).

I mostly find this fun because it lets us talk rigorously about when we should "shut up and do the 1,2!impossible" and when something is very definitely 3!impossible.

## Meetup : London social meetup

1 25 September 2014 04:35PM

## Discussion article for the meetup : London social meetup

WHEN: 28 September 2014 02:00:00PM (+0100)

WHERE: Shakespeare's Head, Holborn, WC2B 6BG

The next LW London meetup will be on September 28th. Join us from 2pm to talk about the sorts of things that your other friends will look funny at you for talking about.

I likely won't be there myself this week, but if you have trouble finding us, you can contact Leon on 07860466862.

We run this meetup approximately every other week; these days we tend to get in the region of 5-15 people in attendance. By default, meetups are just unstructured social discussion about whatever strikes our fancy: books we're reading, recent posts on LW/related blogs, logic puzzles, toilet usage statistics....

Sometimes we play The Resistance or other games. We usually finish around 7pm, give or take an hour, but people arrive and leave whenever suits them.

Related discussion happens on both our google group and our facebook group.

## Newcomblike problems are the norm

29 24 September 2014 06:41PM

This is crossposted from my blog. In this post, I discuss how Newcomblike situations are common among humans in the real world. The intended audience of my blog is wider than the readerbase of LW, so the tone might seem a bit off. Nevertheless, the points made here are likely new to many.

# 1

Last time we looked at Newcomblike problems, which cause trouble for Causal Decision Theory (CDT), the standard decision theory used in economics, statistics, narrow AI, and many other academic fields.

These Newcomblike problems may seem like strange edge case scenarios. In the Token Trade, a deterministic agent faces a perfect copy of themself, guaranteed to take the same action as they do. In Newcomb's original problem there is a perfect predictor Ω which knows exactly what the agent will do.

Both of these examples involve some form of "mind-reading" and assume that the agent can be perfectly copied or perfectly predicted. In a chaotic universe, these scenarios may seem unrealistic and even downright crazy. What does it matter that CDT fails when there are perfect mind-readers? There aren't perfect mind-readers. Why do we care?

The reason that we care is this: Newcomblike problems are the norm. Most problems that humans face in real life are "Newcomblike".

These problems aren't limited to the domain of perfect mind-readers; rather, problems with perfect mind-readers are the domain where these problems are easiest to see. However, they arise naturally whenever an agent is in a situation where others have knowledge about its decision process via some mechanism that is not under its direct control.


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# Black Hole Essay

Discussion in 'Physics & Math' started by Reiku, Nov 23, 2011.

Not open for further replies.
1. ### ReikuBannedBanned

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In my class we have been asked to write an essay on Black Holes. I have chosen the Flat Approximation of the Local Neighbourhood of a Black Hole as my topic. Would the members here be so kind to read over it and express their criticisms?

Close Encounters of the Black Hole Kind

The idea as you get closer to a Black Hole, the hyperbolic acceleration increases. The Hyperbolic angle is $\omega$. You can model this as being increased by $\omega = 1, \omega = 2, \omega = 3...$ and so on as you check reference the grid. The Hyperbolic coordinates are then $\omega$ and $r$ where $r$ is the distance from the center of the Black Hole.

If $x$ and $t$ are your spatial and time coordinates, then you can state

$x = \rho cos h \omega$

and

$t = \rho sin h \omega$

They are similar to polar coordinates. Our metric is then:

$d\tau^2 = \rho^2 d\omega^2 - d\rho^2$

Here our $\omega$ term is playing the role of time and $\rho$ is playing the role of space, so you may want to add the extra coordinates of space to this

$d\tau^2 = \rho^2 d\omega^2 - d\rho^2 - dy^2 - dz^2$

$d\tau$ is called the proper time.

We are going to take an imaginary journey to near the neighbourhood of a Black Hole. You must first imagine that we can get so close to the black hole, that you are tangent to the Horizon. The Horizon is given as $2MG = r$.

To do this, we need to use the Schwarzchild Metric (or Schwarzchild Vacuum) which is used in physics to describe a perfectly spherical distrubuted mass of a non-rotating body. The surface of such an object is called the Event Horizon. In our case, we will bw considering the exotic body of a Black Hole, a situation in physics where matter has been compressed to such a high density that the escape velocity exceeds that of what could be acheived by currently the fastest known particle; a photon. This does however include any kind of massless radiation. The Schwartzchild Metric is given as:

$d \tau^2 = (1 - \frac{2GM}{r}) dt^2 - \frac{dr^2}{(1-\frac{2GM}{r})} - r^2 d \Omega^2$

Here $d\Omega^2 = d\phi^2 + (sin \phi)^2 d \theta$ and this suffices a 2-dimensional coordinate sphere. This is all there really is to a metric of a black hole. It can give you as much information as you are likely to require to describe them. This is the general solution ''outside'' of where the mass is associated - outside the boundary of a black hole, or indeed, any object you might consider, like Earth or a star.

There are two singular regions which can be considered in the metric for a Black Hole. One is a coordinate artefact where our coordinate do something rather odd and the other one is located inside the Black Hole which will be encountered if someone was to pass through the Event Horizon.

When $r=0$, $dt^2$ blows up to infinity and even worse, blows up with a negative sign. $dr$ and $dt^2$ interchange signs so the first $(1 - \frac{2GM}{r})$ is negative and the second $(1 - \frac{2GM}{r})$ becomes positive.

This shows that spacelike events become timelike and vice versa; This doesn't mean that space literally becomes time, or that time literally becomes space, it is just a strange coordinate artefact... the behaviour of moving in space will become restricted and moving in time will become more free. In the second bit $(\frac{dr^2}{(1 - \frac{2MG}{r})})$ when $r=2MG$ the $(\frac{2MG}{r})=1$ and if you follow through the rest of the expression, you simply find that $1-1 = 0$ (no surprise), but the cooefficient of $dr^2$ becomes infinite.

Similar things can be found in our original example of the hyperbolic metric

$d\tau^2 = \rho^2 d\omega^2 - d\rho^2$

This is actually the same as

$= Rd\omega^2 - \frac{dR^2}{4\rho^2}$

Here $\rho^2 = R$. When you apply the same conditions as before, you will see that $= Rd\omega^2$ and $\frac{dR^2}{4\rho^2}$ swap signs for when $R=0$ when $d\omega \rightarrow 0$.

It might be strange to think that $\rho^2 < 0$. This is just changing the way you are looking at the sign of $R$. Positive $R$ is where we began assuming the equations of the metric. Negative $-R$ is now in the region above this. (See Diagram).

Now we can ask whether there is an approximation of the Schwarzchild Metric which will suit the metric we began with:

$d\tau^2 = \rho^2 d\omega^2 - d\rho^2$

We should begin to rearrange terms.

$d\tau^2 = (\frac{r - 2MG}{r}) dt^2 - \frac{dr^2r}{r-2MG}$

So, what have I done? Well keep in mind this equation is incomplete. We are taking this step-by-step. The term $(\frac{r - 2MG}{r})$ is simply been rearranged from $(1 - \frac{2GM}{r})$. That's nothing special, just a bit of algebra. Here we must remember not to set the numerator $r=2MG$. That assume we are touching the Horizon. What we are assuming is that we are just by the horizon and no more.

However, the $r$ in the denominator does not change as drastically as the $r$ in the numerator, so we can actually set this equal to $2MG$. Continue for the rest of the equation will give you:

$d\tau^2 = (\frac{r - 2MG}{2MG}) dt^2 - \frac{dr^22MG}{r-2MG} - (2MG)^2\Omega$

Is now a complete equation again (but realize this is not a metric of a spherical Black Hole any more), modified slightly nonetheless. You can actually modify this further by replacing the last term with $(dy^2 + dz^2)$.

$d\tau^2 = (\frac{r - 2MG}{2MG}) dt^2 - \frac{dr^22MG}{r-2MG} - dz^2 - dy^2$

This is very good for a Flat Space Approximation. It does work well for the Neighbourhood of the local surface of a Black Hole which may appear to calculations as pretty much as a flat metric.

Now, reminding ourselves that $d\tau^2 = \rho^2d\omega^2$, the proper distance $\rho^2$ from the Horizon $r=2MG$ then this without the sign is

$ds^2 = dr^2 \frac{2MG}{r - 2MG}$

Take the square root of this

$ds = dr \sqrt{\frac{2MG}{r - 2MG}}$

Now we need to take the integral in respect to $r=2MG$ to $r$ (the point in which we are sitting just by the Horizon) then

$\rho = \int_{r=2MG}^{r} dr' \frac{\sqrt{2MG}}{\sqrt{r'-2MG}}$

The $\sqrt{2MG}$ is a constant so we can bring that out

$\sqrt{2MG} \int_{r=2MG}^{r} dr'{\frac{1}{\sqrt{r - 2MG}}$

Which yields

$\rho = 2\sqrt{2MG}\sqrt{r - 2MG}$

Now we need to rewrite the metric in harmony of this new equation. This is of course, the proper distance from the Horizon.

Rearranging gives

$\frac{\rho}{2\sqrt{2MG}} = \sqrt{r - 2MG}$

Simplifying makes

$\frac{\rho}{\sqrt{8MG}} = \sqrt{r - 2MG}$

and now we are going to square it

$\frac{\rho^2}{8MG} = r - 2MG$

Knowing there is another $2MG$ when we compare this to the metric gives

$\frac{\rho^2}{16(MG)^2}dt^2 - d\rho^2 - dz^2 - dy^2$

Now this is not quite the same as

$\rho^2d\omega^2 - d\rho^2 - dz^2 - dy^2$

To sort this problem, we need to know that

$\omega = \frac{t}{4MG}$

$d\omega = \frac{1}{4MG}dt$

In light of this we easily realize that $d\omega$ is really just $\frac{dt^2}{16(MG)^2}$ so we finally have

$\rho^2(d\frac{t}{4MG})^2 - d\rho^2 - dz^2 - dy^2$

which now makes a perfect approximation of the Schwarztchild Metric. So we have just taken a journey very close to a black hole in a Flat Approximated Spacetime.

Last edited: Nov 23, 2011

3. ### TachBannedBanned

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5,265
Err, no. This is not even wrong.

This is not even wrong either, it borders on hilarious. I stopped reading but there is enough nonsense to fill several pages pointing out your goofs. I am sure that many others will have a field day with your essay. It is so reminescent of Reiku, it is surreal.

ETA: Sorry, I couldn't resist this "pearl"

...and this one:

F-

Last edited: Nov 23, 2011

5. ### ReikuBannedBanned

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Oh I am reporting you. If you are going to call my by any name, you can call me by my real name here. Learn it.

It's Mister.

(as a side note, how do you think the Schwartzchild Metric is written. This is how we have learned it in class??)

7. ### ReikuBannedBanned

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Read the ******* work. I explained that that equation was incomplete you twat. :bugeye:

8. ### TachBannedBanned

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Based on what you are writing , I don't think that you go to any class.

9. ### TachBannedBanned

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It is not that it is "incomplete", it is downright hilarious.

10. ### ReikuBannedBanned

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And why is it hilarious? These are serious equations, nothing about them is wrong or incomplete. I hope they ban your ass soon.

Hopefully sooner than later.

11. ### TachBannedBanned

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For someone with basic knowledge of the subject, your errors are glaring. And I did not spend more than three minutes looking over your work. I am sure that many others will have a field day with it.

12. ### ReikuBannedBanned

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I asked for criticism. Shut up or put up. You are simply trolling tach. I know nothing is wrong with them!!!!

13. ### ReikuBannedBanned

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I see I missed out a squared term in that object. I've written a lot of equations, that's a lot of tex tach. Just shut up and move along if you aren't going to contribute properly. I see you do it in so many threads here.

14. ### ReikuBannedBanned

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Big deal, missed another squared sign. That's because I copied and pasted. Anything more.. harder to offer?

15. ### ReikuBannedBanned

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There fixed. Breath tach, breath!!!!

16. ### TachBannedBanned

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Have fun with another couple of hilarious ones, they are harder to spot (and to fix) :

Bye.

Last edited: Nov 23, 2011
17. ### ReikuBannedBanned

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How is it hilarious, when I copied and pasted from basically the same equation, and I missed out on one squared sign??????

You must be pleased very easily.

Right good, **** off back to your doppler dissitation, and arsing up your integrals.

18. ### ReikuBannedBanned

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11,238
oh I see what tach is getting his knickers in a twist about. The denominator in the second term of the Schwarzchild metric has a ''c'' instead of r, which, was of course carried out to other parts of the work because I copied and pasted.

See, tach is just a troll. He couldn't have came in here, been a gentleman and pointed the fact out. Instead he posted abusive posts one after the other as if it was all one big joke. The difference between me and tach, is if I notice a mistake, I won't try and go back and edit the work without making a notification about it. NOT only that, but I doubt tach even recognizes his mistakes even after he has need pointed them out.

Thanks though tach, as useless as your ability is to have a conversation about a subject, I needed to get this in by tomorrow. Unfortunately for you, the part where I hadn't sqaured two of my terms and the ''c'' doesn't even appear in my work. These are typo's. So that is good for me. I am quite happy now that you couldn't even spot a proper mistake.

19. ### TachBannedBanned

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Nah, you have a lot more errors. Many of them you can't pass as typos.

Don't worry about it, you have plenty of "proper" mistakes. I just pointed a couple, I don't want to spoil the fun for the others.

20. ### ReikuBannedBanned

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Don't hold back tach. Typo's are one thing.

And what mistakes are you so clearly seeing? Common... I want to know. Afterall, I need to present this.

21. ### ReikuBannedBanned

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This is confusing me I will add though. Maybe tach has a problem with the terminology? Maybe, just maybe he is confused with the part of me saying

''Now, reminding ourselves that [eq] , the proper distance [eq] from the Horizon [eq] then this without the sign is''

I am guessing but he might have thought the $d\tau$ part is strictly proper time. Is he aware of the following?

You can calculate from the Schwarzschild metric

$ds^2=(1-\frac{2GM}{r}) dt^2 - r^2d\phi^2$,

where $ds$ means the proper time and length intervals as a function of the coordinate intervals.

But then who knows with Tach. He comes into peoples threads and acts as conspicuous and as mysterious as he can without ever giving a direct answer.

edit: removed a c^2 here. Didn't see why I should have it in after neglecting it using natural units throughout my work.

22. ### TachBannedBanned

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What you put together ain't fixable. Besides, I doubt that there is any class where you are going to present any of this "stuff".

23. ### ReikuBannedBanned

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LOL... that comment made my night. It really did tach. So It's not wrong? Why all the hoolah?

Mod note : Language!

Last edited by a moderator: Nov 23, 2011


text:
For many quantities, you need units which you get by combining the basic units. These units are called derived units. For example, combination of the units of length and time gives the derived unit of speed or velocity, m s–1.

One of the advantages of the SI system of units is that they form a coherent set in the sense that the product or division of the SI units gives a unit which is also the SI unit of some other derived quantity. For example, product of the SI units of force and length gives directly the SI unit of work, namely, newton-metre (Nm) which has been given a special name joule.


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# How to understand co-$\mathcal{L}$ where $\mathcal{L}$ is a class of languages

I think this is a basic topic in complexity, but I would like to ask how to understand co-$$\mathcal{L}$$ where $$\mathcal{L}$$ is a class of languages. From the definition of my textbook, $$co-\mathcal{L} = \{ \overline{L} \mid L \in \mathcal{L} \}$$

and where $$\overline{L}$$ is the complement. From what I read in an earlier part of my textbook, the complement of $$L$$ is equal to $$\Sigma^* - L$$.

However, say that $$\mathcal{L}$$ is NP. An instance of a language $$L$$ that is in $$\mathcal{L}$$ is the set of graphs with Hamiltonian paths. However, in this case, its complement $$\bar{L} \in$$co-$$\mathcal{L}$$ is the set of graphs without Hamiltonian paths, i.e. $$\bar{L} \in$$co-NP.

But is the set of graphs without Hamiltonian paths equal to $$\Sigma^* - L$$ (following the definition of complement) ? In this case, we would be including some strings $$\in \Sigma^* - L$$ that do not represent graphs.

Another example is $$A_{TM}$$, which represents the language

$$\{\langle M,w\rangle \mid M \text{ accepts input } w \}$$

In this case, does $$\overline{A_{TM}}$$ represent $$\Sigma^* - A_{TM}$$?. If this is the case, we would be including in $$\overline{A_{TM}}$$ several strings that do not represent TM's, or that refer to other input that is not equal to $$w$$. Or rather, does $$\overline{A_{TM}}$$ represent the language

$$\{\langle M,w \rangle \mid M \text{ diverges on input } w \}$$

We typically think of instances to problems as being in some format. There are several ways to think of this. Consider for example $$A_{TM}$$, in which the input is a pair $$\langle M,w \rangle$$. The three most obvious are:

1. Every input string can be decoded into an input pair $$\langle M,w \rangle$$.
2. Inputs not of the form $$\langle M,w \rangle$$ don't belong to the language.
3. We think of $$A_{TM}$$ as a promise problem: inputs $$\langle M,w \rangle$$ where $$M$$ accepts $$w$$ are Yes instances, inputs $$\langle M,w \rangle$$ where $$M$$ doesn't accept $$w$$ are No instances, and we don't care about other inputs.

If you take complementation, here is what you get under each interpretation:

1. $$\overline{A_{TM}}$$ consists of all inputs strings which, when decoded to $$\langle M,w \rangle$$, are such that $$M$$ doesn't accept $$w$$.
2. $$\overline{A_{TM}}$$ consists of inputs not of the form $$\langle M,w \rangle$$, and of those of the form $$\langle M,w \rangle$$ such that $$M$$ doesn't accept $$w$$.
3. $$\overline{A_{TM}}$$ is the promise problem corresponding to option 1.

While these different interpretations look different, in practice the difference is very slight, given that it is easy to recognize that the input is of the correct format. For example, given interpretation 2, consider the following two languages: $$\overline{A_{TM}}$$, and $$\widetilde{A_{TM}}$$, the language of all pairs $$\langle M,w \rangle$$ such that $$M$$ doesn't accept $$w$$. These two languages differ in malformed inputs, that is, outputs not of the form $$\langle M,w \rangle$$. Since such inputs are easy to detect, we have an algorithm for $$\overline{A_{TM}}$$ iff we have one for $$\widetilde{A_{TM}}$$, and furthermore, the complexity of both algorithms is very similar.

For this reason, we typically ignore such issues, and implicitly work under the "promise problem" interpretation: the input is assumed to be of the form $$\langle M,w \rangle$$. Purists can think of the first interpretation instead, which from this point of view behaves identically.

More generally, whatever interpretation you choose, you still formally have to describe what encoding is used for $$\langle M,w \rangle$$. We typically don't bother, since all reasonable interpretations are inter-reducible, and so don't change the computability or complexity of the problem. That said, the difference between weak NP-hardness and strong NP-hardness lies exactly in which input representation is being used.