AI: Conway Creatures!

Recently I’ve been reading up on Cellular Evolutionary Algorithms and on the use of Genetic Algorithms (GAs) to evolve Cellular Automata (CA). I want to try out all sorts of different things so I will be doing a series of posts where I explore different interpretations of a concept – Conway Creatures.

Get Involved!
I have a few programmers, logicians and AI enthusiasts that follow my blog so I wanted to see if I could make something of the concept. I’d like to see all your interpretations of what a Conway creature is. Given a rough description, which is essentially: creatures (2D,3D, etc) of which the properties are evolved in some way using a mix of CA and GAs. If I get any i’ll do posts dedicated to the entries. I would also like to see varied definitions of the concept such as the use of other works by John Conway (e.g., the growth rate of the Look and Say sequence – Conway’s Constant, the Surreal Numbers, Conway notation for polyhedra, etc), other CA rules and other Evolutionary methods.

The First Detour – Longevity in Game of Life:
I began to write the program and found myself wondering “What are the characteristics of longevity in CAs?” and I’m still not sure. I’ve been trying different takes on mutation and crossover where I take chunks of the board and flip them or turn them all on/off. It doesn’t seem to make much difference, I was hoping it would conserve locality. I also tried out Boltzmann selection (Simulated Annealing) but tournament selection (dueling) worked much better. Any ideas?

This is my program so far, I am taking a little detour to look into longevity, but in Issue 1 there will be a creature where the 3D terrain is. Probably not very complicated, I was thinking of making the creature some sort of shape, like an octahedron.

Numeral System Automata

Cellular Automata
Cellular automata Is made up of a grid of cells which sit in a finite number of states (such as on/off or blue/green/red). Cellular automaton (singular) adhear to the rules of the system and evolve over periodic time intervals. One successful application of CA is Conway’s Game of Life, the universe is an infinite 2D grid and the rules are as follows:

A single Gosper's Glider Gun creating "gliders"

  1. Any live cell with fewer than two live neighbours dies, as if by needs caused by underpopulation.
  2. Any live cell with more than three live neighbours dies, as if by overcrowding.
  3. Any live cell with two or three live neighbours lives, unchanged, to the next generation.
  4. Any dead cell with exactly three live neighbours cells will come to life.

Positional Number Systems
We use these all the time, notably the decimal system in which when a symbol exceeds 9 it “carries” one to the left and negates ten. 9+1=A, then A goes to 10.

a_n \dots a_2a_1a_0 = (a_n {\times} b^n) + \dots + (a_2 {\times} b^2) + (a_1 {\times} b^1) + (a_0 {\times} b^0)
Example: 1250 = (1 \times 10^3) + (2 \times 10^2) + (5 \times 10^1) + (0 \times 10^0)

The Idea
To take the properties of a number system and create rules to govern individual cells. For our numeral universe we are going to need something different to what Conway used, ours is going to have an intrinsic property, gravity – and thus a floor to gravitate to. The center of our universe will be the radix point (decimal point), you could say the numeral universe is radixocentric. The real-life universe is made up of multiple dimensions, the numeral one is made up of infinite column dimensions.

Although initially I will only being dealing with addition and thus carry, I have put on paper negation in phinary which was quite fun. I talk about the arithmetic of phinary in my research (section 4). The Numeral Universe rules and an example are below:

  1. Positional System – Each dimensional element (cell) is the value of the previous dimension’s capacity.
  2. Gravity – Any cell above an empty cell moves to it.
  3. Carry – When a dimension is full, it moves to the next higher one.

Example in base 3 (ternary). The green and red blocks are the two numbers being added to the universe, the grey block is the carry.

But what does this all mean?
Well, I’m not really sure – it’s nice to show number systems in a different way, it could be useful for teaching. But as for me, I’m using the rules to govern an alternative tetris game, I’ll upload it when I’m finnished.

Also if anyone is wondering how I made those fractal looking blocks, I spammed the hell out of the filters on Pixlr the free online image editor.

Movie Idea

The Vienna Circle was a group of mathematicians, physicists, economists and philosophers alike who met to disscus epistemology and philosophy of science. This lead to a powerful movement called Logical Positivism which dispelled metaphysics, aesthetics and ethics from science as niether right nor wrong – of no value what so ever.

Based on the Vienna Circle, but taken well out of context and set in ancient Japan. A collection of yojimbo (bodygaurd), ninja, ronin (masterless samurai) and farmers – analog to the real members, who meet in secret to talk skeptically about spirituality. Portrayed as pro-industrial natural philosophers.

Imagine these guys discussing Hilbert's Decision Problem, Samurai comming to attack them, then them being a bunch of badass logic-ninja defending science!

Marrying Democracy and Technocracy

Hopefully in this post, much like the Guzzle Puzzle Helpline, I will find the right words to explain my idea. The idea is something that I tackled before but since then my views have evolved somewhat. The idea – finding a middle ground between representative democracy and technocracy.

Constitutional Monarchy
For over 900 years, the British unitary parliamentary democracy-constitutional monarchy system has held as a steady form of government. The Queen acts as head of state and whilst still retaining certain powers (calling elections, dissolving parlement, etc) must adhire to the law. This is called Rule of Law and it governs every single person, parlement and military, everyone.

Senate of Rome
Surviving the Kingdon, Republic and Empire of Rome, the senate is another clearly stable political structure of over 900 years. At it’s core the senate was always a collection of ‘wise men’ giving advice (senatus consultum) to those in power. During republic domain the senate was at it’s most influencial and although their advice had no real authority, it was usually obeyed.

The Idea (Constitutional Technocracy)
To take the three-tier structure of British Parliament, or more directly, Canadian Parliament, change the nature of the House of Lords (Senate) and increase their influence:

  1. Head of State (Monarch)
  2. Senate (Technocrats)
  3. House of Commons (Democrats)

The Senate is voted in by anyone with a PhD, those standing for position (representives of fields) must submit their papers and works. This is because, although technocrats have been burocratically designated like in Greece and Italy, real technocrats should be subject to peer review – just like good science. Putting technocrats in influential positions will encourage an intellectual community, bringing foreign science to us. Our science is already on the up-and-up but it would increase enthusiasm in knowledge and reason.
One out there idea is to give control to the Senate in states of emergancy – be it medical (epidemic), military (assassination), economic (crash), etc..

As for the House of Commons, anyone over the age of 18 can vote as long as they pass a standardised political party test. This test is on the policies of parties running in their area. One of the issues I see with democracy is missguided votes squewing the ballot, this precaution would help voters understand the bigger picture. Originally I concieved an Academocracy, but as the comments suggested a large problem with it was disfranchising people. Although again I’m saying not everyone should vote – what I’m really saying is everyone should know why they are voting and what they are voting for. Which is surely reasonable? Hopefully it will also cut down on the tribalism of chosing who your parents or friends vote for, because you will know if you correlate with the party you choose.

House of Lords Reform
My above idea is basically making the House of Lords [seem] more important. The reason I came back to thinking about this issue is a recent proposal in British politics to elect members of the House of Lords democratically. I oppose this notion because of how useful an outside view is, when someone doesn’t have to please constituents or try to stay in power, they are more likely to give an actual view without tribalism.

“The great strength of the Lords is that it contains not just a bunch of experienced retired MPs but a whole raft of individuals with specialist knowledge and experience from the worlds of commerce, medicine, the services, the civil service, academia, the unions – the list is endless – none of whom would be likely to be available to stand for election.”
– Lord Steel, former Alliance leader

Academocracy – Putting knowledge first.

“Elitists tend to favor systems such as meritocracy, technocracy and plutocracy as opposed to radical democracy, political egalitarianism and populism.”

My Idea – (Knowledge is Power)
An Academic Republic is an idea I’ve been thinking about for a while, it fits into a grand scheme of mine. The title is just academic and democracy squeezed together, but the idea falls under an elitist view: Representative democracy like we already have, but only those with a degree can vote. It sounds harsh and excluding, but it would only be acceptable alonside the rest of my set up:

  • Education up to and including University is free.
  • Re-entering unfinnished education, at any point, is free.
  • Only those with a degree/HND can vote. (Academocracy)
  • Bicameral system (HoC/HoL) of the type proposed in this previous post.

The plan is to encorrage the learning and persuing of ideas. A degree in the arts, sciences or any of the ones that already exist can get you a vote so people will just pick something they enjoy. You could do a degree in neurobiology then be a pastry chef if that’s the plan!!

An effect of delaying the vote until people have experienced university would hopefully stop the unnecessary sway of easily lead voters who just vote for whoever isn’t in power at the time. Rationality and the scientific method would be key in the new curriculums.

The outcome would be a paradigm shift in the global view of said nation as an intellectual icon. More importantly the produce of a highly educated society would be increased in quality if not quantity aswell. Simply by knowing more methods people will find more solutions for everyday problems. A sort of like reciprocal altruism between the nation and science.

Education wouldn’t be compulsory, if someone left school at 16 then at 20 decided they want to weigh in on politics they could continue education in what ever subjects they feel (maybe politics if that’s caught their eye) and after completing their degree – vote. Goverment funded “Catch-up Courses” would be introduced to get from GCSE level to A level in less than a year for people over the age of 18.

The curriculum would need a change, certain subject ideas i’ve had are Big Picture lessons, where the realworld applications of what you are learning at the time are shown to you by university lecturers. Another is Free Thinking lessons, where creativity (no matter how against the grain) is encoraged and you go to teachers with ideas for projects and they help point you in the right direction for learning. Marked solely on atttendance and something like once a month.

The Russell Equation (R tea R)

This post is quite nonsensical, read at your own peril.

Bertrand Russell is a fantastically interesting character, a passive-activist against the first world war, mathematician, logician, philosopher and social critic. Alongside being one of the co-founders of Analytical Philosophy, co-authering Principia Mathematica and adding to fields such as logic, mathematics, set theory, epistemology – Bertrand came up with two important arguments. They are Russell’s Teapot (the burden of proof lies with the person making scientifically unfalsifiable claim) and Russell’s Paradox (the set R contains all sets that are not members of themselves, is R in the set?). I enjoyed reading up on these two, and simply because they both start with “Russell’s” I thought I’d merge them:

Let me introduce the teapot operator into Cantor’s naive set theory. What this teapot symbol is representing in the equation is twofold: 1) There is an indisputable relation between R and R, 2) That the burden of proof [for a solution] lies with the scientific community. In this case, set theorists, to come up with new axioms and produce a better less fallible system (which they did). The teapot operator basically means “The relationship of the left and the right transcends this system, get a new one.” For example, another application could be to state that P tea NP:

Although I may be getting ahead of myself thinking that the P vs NP argument transcends todays computational theory, hopefully you see where I’m comming from. All in all it was a bit of fun to portray the ideology of the scientific method by continuously putting the onus upon themselves to come up with solutions or corrections.

Fibonacci? Too mainstream..

So I really enjoy infinite sequences that correspond to a constant value (most cool people do) and I wanted to post about a couple including a recent favourite the Look-and-say Sequence. The Fibonacci Sequence is probably the most known. Every Nth number (after the first two) is the sum of the previous two:

F_n = F_{n-1} + F_{n-2} where F_0 = 0 and F_1 = 1
\boldsymbol{F} = ( 0,1,2,3,5,7,13,21,34,55,89,\dots )

The constant value that corresponds to this sequence is called the Golden Ratio, which I use in this previous post. The sequence converges to the ratio:

\displaystyle \lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\varphi \approx 1.61803

The Lucas Numbers:
Changing the starting value of the Fibonacci sequence from 0 to 2 gives:

L_n = L_{n-1} + L_{n-2} where L_0 = 2 and L_1 = 1
\boldsymbol{L} = ( 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123,\dots )

The Lucas Numbers relate directly to the Fibonaci Numbers, for example: L_n = F_{n-1} + F_{n+1} and even though almost all the numbers are different, they both converge to the same ratio:

\displaystyle \lim_{n\to\infty}\frac{L_{n+1}}{L_n}=\varphi \approx 1.61803

The Pell Numbers:
With a slight alteration (multiplying one term by 2) the Fibonacci equation produces:

P_n = 2 P_{n-1} + P_{n-2} where P_0 = 0 and P_1 = 1
\boldsymbol{P} = ( 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378,\dots )

The Pell numbers related to \sqrt{2} which can be seen when the series converges to the Silver Ratio (1+\sqrt{2}):

\displaystyle \lim_{n\to\infty}\frac{P_{n+1}}{P_n}=\delta_s \approx 2.1412

The Look-and-say Sequence:
This is a very interesting sequence because of it’s strange method. Take a ‘seed’ number (the first in the sequence), write it down then say what you see, repeat. For example if you start with 1 then that is “One 1”, written down that is “Two 1s”, written down that is “One 2, One 1”, etc. In the sequence no number greater than 3 is needed unless that number is used as the seed:

\boldsymbol{Ls} = ( 1,11,21,1211,111221,312211, 13112221, 1113213211, \dots )

If the seed number is 22 then nothing ever changes, otherwise the numbers eventually grow at a rate of 30% in size each time. Just as before a ratio emerges from the sequence and that gives us Conway’s Constant:

\displaystyle \lim_{n \to \infty}\frac{{Ls}_{n+1}}{{Ls}_{n}} = \lambda \approx 1.30357

This ratio is found with any seed number that isn’t 22. On a side note, the Golden Ratio is the root to the 2nd degree polynomial x^2 - x - 1 but Conway’s Constant isn’t as simple an equation, it is the root to a 71st degree polynomial! It also has links to chemistry and was first published about in a paper called “The Weird and Wonderful Chemistry of Audioactive Decay”.

The Super Fibonacci Numbers:
I can’t find anything on this so i’m not sure if they have a name or if I’ll be the first to write about them. I started with a number: The Super Golden Ratio (Sg) which is the solution to x^3 - x^2 - 1 = 0 and has the value of 1.4655\dots
Using the closed-form expression called Binet’s Formula I could easily get the set of numbers, all I had to do was swap the golden ratio and golden conjugate with the super versions: Sf_n = \frac{Sg^n-Sc^n}{Sg-Sc} . After looking at the pattern I arrived at my answer:

Sf_n = Sf_{n-1} + Sf_{n-3} where Sf_0 = 0 , Sf_1 = 1 and Sf_2 = 1
\boldsymbol{Sf} = (0,1,1,1,2,3,4,6,9,13,22,31,44,66,97,\dots)

\displaystyle \lim_{n \to \infty}\frac{{Sf}_{n+1}}{{Sf}_{n}} = Sg \approx 1.4655

The Extended Fibonacci Numbers:
Before I derived the Super numbers from the Super ratio I thought maybe they would be the same as the Fibonacci but with an extra term: Sf_n = Sf_{n-1} + Sf_{n-2} + Sf_{n-3} …it wasn’t. The sequence, which I am calling the Extended Fibonacci numbers (Ef), looked like this:

Ef = (0,1,1,2,4,7,13,68,81,149,274,504,927,1705,3136, \dots)

\displaystyle \lim_{n \to \infty}\frac{{Ef}_{n+1}}{{Ef}_{n}} = Eg \approx 1.8392

I looked back at my scribblings and noticed that the polynomial didn’t just have an extra term on it like I thought, an x was also removed. I whipped up WolfRamAlpha and punched in x^3 - x^2 - x - 1 = 0 (putting the -x back in) and needless to say x \approx 1.83929

Related reading: