Computers for SETI , Kurzweils SI NGULARI TY and Evo-SETI Claudio - - PowerPoint PPT Presentation

Computers for SETI , Kurzweil’s SI NGULARI TY and Evo-SETI

Claudio Maccone

IAA Director for Scientific Space Exploration, IAA SETI Permanent Committee Chair, Associate, Istituto Nazionale di Astrofisica (INAF, Italy) E-mail : clmaccon@libero.it Home Page : www.maccone.com UK SETI Research Network, Manchester, UK, March 2-3, 2017

700-pages BOOK about “Mathematical SETI”

Part 1: GEOMETRI C BROWNI AN MOTI ON (GBM) Part 2: Darwinian EXPONENTI AL GROWTH Part 3: 2006: «THE SI NGULARI TY I S NEAR» Part 4: Merging GBM, SI NGULARI TY & Evo-SETI Part 5: Peak-Locus Theorem & EvoENTROPY

TALK’s SCHEME

Part 1: GEOMETRI C BROWNI AN MOTI ON (GBM)

GEOMETRIC BROWNIAN MOTION (GBM): exponential mean value

WARNING !!! GEOMETRIC BROWNIAN MOTION is a WRONG NAME :

This stochastic process in NOT a Brownian Motion since its probability density function is a LOGNORMAL, and NOT A GAUSSIAN ! So, the pdf ranges between ZERO and INFINITY, and NOT between minus infinity and infinity !

GEOMETRIC BROWNIAN MOTION (GBM): exponential mean value :

( )

( )

for .

B t ts

L t e t ts

−

= ≥

( )

( ) ( ) ( ) ( )

2 2 2

ln 2 2

GBM_pdf ; , , . 2

t ts n B t ts t ts

e n B t ts t ts n

σ σ

σ π σ

    −     − − −         − −

− = −

GEOMETRIC BROWNIAN MOTION b_lognormal probability density :

Part 2: Darwinian EXPONENTI AL as a GBM in the number of LI VI NG SPECI ES

TWO REFERENCE PAPERS

► A Mathematical Model for Evolution and SETI ► Origins of Life and Evolution of Biospheres

(OLEB), Vol. 41 (2011), pages 609-619.

► SETI, Evolution and Human History Merged

into a Mathematical Model. ADVISED

► International Journal of ASTROBIOLOGY,

• Vol. 12, issue 3 (2013), pages 218-245.

Darwinian EXPONENTIAL GROWTH

► Life on Earth evolved since 3.5 billion years ago. ► The AVERAGE number of Species GROWS

EXPONENTIALLY: we assume that today 50 million species live on Earth.

► Then:

Darwinian EXPONENTIAL GROWTH

( ) ( )

( )

exponential mean value curve in time starting at with a value of 1 (RNA?) :

B t ts L

t ts L t m t e

−

= ≡ =

( ) ( )

( )

( )

9 7 7 16 9

3.5 10 years Origin of Life on Earth, 50 million species=5 10 speciesTODAY, ln 5 10 ln 1.605 10 . 3.5 10 year sec

L L

ts m m B ts

−

  = − ⋅   = ⋅   ⋅ ⋅  = = =  − ⋅ 

DARWINIAN EVOLUTION is a GBM in the increasing number of Species

3.5 − 3 − 2.5 − 2 − 1.5 − 1 − 0.5 − 1 107 × 2 107 × 3 107 × 4 107 × 5 107 × 6 107 × 7 107 × 8 107 × 9 107 × 1 108 ×

Evolution as INCREASING NUMBER OF SPECIES

Time in billions of years Number of LIVING SPECIES on Earth

Part 3: 2006 «THE SI NGULARI TY I S NEAR» book by Ray Kurzweil

THE SINGULARITY IS THE KNEE OF THE GBM EXPONENTIAL

10 − 9 − 8 − 7 − 6 − 5 − 4 − 3 − 2 − 1 − 1 107 × 2 107 × 3 107 × 4 107 × 5 107 × 6 107 × 7 107 × 8 107 × 9 107 × 1 108 ×

Number of CIVILIZATIONS since 10 billion years ago

Time in billions of years Number of CIVILIZATIONS in the Universe

THE SINGULARITY IS THE KNEE OF THE GBM EXPONENTIAL

In other words: BEFORE the SINGULARITY , the Darwinian Evolution is very SLOW. AFTER the SINGULARITY , the Computer Evolution is very FAST.

THE SINGULARITY IS THE KNEE OF THE GBM EXPONENTIAL

In other words still: REPRODUCTION in Darwinian Evolution is very SLOW. REPRODUCTION, among computers is very FAST.

THE SINGULARITY IS THE KNEE OF THE GBM EXPONENTIAL

In other words still: REPRODUCTION in Darwinian Evolution is very SLOW. REPRODUCTION, among computers is very FAST.

THE SINGULARITY TIME IS NOW

• r just a few decades from NOW,

i.e. the same when compared to 3.5 billion years of Darwinian Evolution

GBM_knee SINGULARITY

t t = =

Part 4:

Mer erging GBM,

SI NGULARI TY, & Evo-SETI

KNEE EQUATION : relates the time along the GBM exponential when the knee occurs t_GBM_knee , to the time of the Origin of Life ts , and the exponential increase rate B

( )

GBM_knee SINGULARITY

ln 2B t t ts B = = −

KNEE EQUATION : is derived in the paper by finding the radius of the osculating circle to the GBM exponential and setting to zero its derivative.

MERGING THE TWO EQUATIONS YIELDS:

( )

GBM_knee SINGULARITY GBM_knee

ln 2 t t B t ts B = =     = − 

( )

ln 2B ts B =

FINDING B FROM ts AND THE KNEE-CENTERED GBM EXPONENTIAL :

( )

ln 2B ts B =

( ) ( )

2

Bt L

e L t m t B ≡ =

THE AVERAGE NUMBER OF LIVING SPECIES TODAY IS BUT BIOLOGISTS ARE VERY UNCERTAIN ABOUT THIS NUMBER: MILLIONS OR BILLIONS ?

( ) ( )

1 2

L

L m m B ≡ ≡ =

IMPORTANT NEW EQUATION relating the time ts of the Origin of Life on Earth to the current number m0 of Species living on Earth. 1) if ts = -3.5 Gy then m0= 132.4 M 2) if ts = -3.8 Gy then m0= 143.1 M

( )

0 log 2 ts m m − = ⋅

Part 5: Peak-Locus Theorem, b-Lognormals ENTROPY & COMPUTERS AFTER THE SI NGULARI TY

PEAK-LOCUS THEOREM (PLT):

► The Peak-Locus Theorem (PLT) shows that the

“Running b-lognormal” always has its PEAK ON

THE DARWINIAN EXPONENTIAL, and is more and more peaked as long the time increases.

PEAK-LOCUS THEOREM (PLT):

► The Peak-Locus Theorem (PLT) shows that the

“Running b-lognormal” always has its PEAK ON

THE DARWINIAN EXPONENTIAL, and is more and more peaked as long the time increases.

ENTROPY(p) of the Running bLog

► The Shannon ENTROPY of any probability density is ► The Shannon ENTROPY of the Running b-

lognormal is a function of its peak time p and of its µ and σ :

( ) ( )

( )

1 log . ln 2

X X

H f x f x dx

∞ −∞

  = − ⋅  

∫

( ) ( )

( )

( )

1 1 ( ) ln 2 . ln 2 2 H p p p π σ µ   = − + +    

EvoENTROPY(p) of the Running bLog

► Because of the Peak-Locus Theorem, ENTROPY reads ► Then EvoENTROPY of the Running b-lognormal

is a function of its peak time p and of B and ts :

( )

2

1 1 ( ) ln 2 . ln(2) 2 2 B H p B B p π   = + − +    

( ) ( ) ( ) ( ) ( )

EvoEntropy . ln 2 B p ts p H p H ts −   = − − =  

EvoENTROPY of the Running bLogn. is the MOLECULAR CLOCK

3.5 − 3 − 2.5 − 2 − 1.5 − 1 − 0.5 − 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

EvoEntropy of the LATEST SPECIES in bits/individual

Time in billions of years before present (t=0) EvoEntropy of the LATEST SPECIES in bits/individual

1)

We developed here a new mathematical model embracing all

• f Darwinian Evolution (RNA to Humans) and SINGULARITY

(i.e. computers taking over humans) happening NOWADAYS. 2) Our mathematical model is based on the properties of lognormal probability distributions. It also is fully compatible with the Statistical Drake Equation, i.e. the foundational equation of SETI, the Search for Extra-Terrestrial Intelligence. 3) Merging all these apparently different topics into the larger but single topic called “Big History” is the achievement of this

• paper. As such, our statistical theory would be crucial to

estimate how much more advanced than Humans the Aliens would be when SETI scientists will succeed in finding the first ET Civilization.

CONCLUSIONS

Thank you very m uch !