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Other Approaches to 102 Linear algebra, Groups and polynomials What We’ve Done So Far Discovered some cool properties of Rule 102. Used straight-forward brute-force proofs. But we can take advantage of techniques from other branches of mathematics. Linear Algebra Group theory. Polynomials. Other... (you discover it!) Optional Sections The sections on linear algebra and group theory may be skipped. Depends on time constraints. Depends on student background and interest. Neither section is critical to later developments. The section on polynomials and derivatives is important to later developments. Linear Algebra Basic idea is to use matrix multiplication to get the derivative. Consider 1 0 0 D 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 1 First Derivative Matrix What happens if multiply D by x? Get first derivative! Try it! These give the sum of the second two elements. These give the sum of the first two elements. D 1 0 0 Dx 0 0 1 1 1 0 0 0 0 x 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 d(x) 0 1 0 0 1 1 1 0 1 0 0 1 1 1 1 0 Kth Derivative Matrix So call D the first derivative matrix. What if I want the kth derivative? Apply first derivative k times. D(Dx) = 2nd derivative. D(D(Dx) = 3rd derivative. That’s just matrix multiplication. The kth derivative is just 1 0 0 k D 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 1 k 2nd Derivative Matrix What’s the second derivative matrix look like? 1 0 0 2 D 0 1 1 0 0 0 0 2 1 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 Can you guess the pattern? 0 0 0 1 0 0 0 0 0 1 Pascal Triangle Gives kth Derivative The rows of the kth derivative matrix are just the kth derivative of Pascal’s triangle. Example, 4th row of Pascal is Modified for finite length. So 4th 0001 0011 0101 1111 1 0 0 4 D 0 1 derivative matrix is 1 0 0 0 0 4 1 1 1 1 0 0 0 0 1 0 0 0 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 Eigenvector Solutions So now can use all the power of linear algebra. Example: Recall eigenvectors? Eigenvectors are vectors x such that Dx = l x for some scalar l and l is the eigenvalue). Eigenvectors Give Cycle Lengths If we want the cycle length Just want the derivative that gives back x. i.e., dkx = x. Theorem (eigenvector cycle lengths): The eigenvectors x associated with eigenvalue 1 of the kth derivative matrix are those vectors with cycle length k (i.e., vectors such that dkx = x). Eigenvector Implication Implication: If we want to find the vectors that have cycle length k, we just have to find the eigenvectors of Dk. So use linear algebra to find those eigenvectors! I leave that to you. The Identity Matrix If Dkx = x, then that means Dk is some kind of identity matrix. Is it 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 ? Not the Real Identity No! The real identity matrix would imply that any and every vector cycles. For example, consider 000001. Does not cycle. Why not? The “odd order corollary” says its cycle length is zero. But if Dk is the standard identity then that implies 00001 will multiply times Dk to give itself. i.e., has cycle length k. Oops! Even Identity Is there a matrix that lets 000011 cycle but not 000001? 0 1 I 1 1 Yes! Let That’s our identity! 1 1 1 0 1 1 1 0 1 1 1 0 Group Theory And now with an appropriate identity we can take a small diversion. Some of you may not have taken group theory. Don’t panic! I will explain what you need. We will barely use it, and I will not test you on it. Just want you to consider something delicious and cool. Sit back and enjoy the ride. What’s a Group? Any collection of objects G with an operation “.” that satisfies the following properties. 1. 2. 3. 4. Closure: if a, b are in G, then a.b is also in G. Associative: a.(b.c) = (a.b).c. Identity: There exists and identity e such that a.e = e.a = a for all a in G. Inverse: For every a in G there exists a-1 such that a-1.a = e. Example of a Group The set of all integers ..., -2, -1, 0, 1, 2, ... with operation “+”. i.e., a.b = a+b. Closure obvious. Associative obvious. 0 is the identity. -a is the inverse of a. This is an infinite sized group. Example of Finite Sized Group The numbers 1 and -1 with the operation multiplication. Identity is 1 Inverse of 1 is 1. Inverse of -1 is -1. Has size or order of 2. Example of Geometric Group Rotations. Consider a triangle. Can rotate 120 degrees three times before back at beginning position. 0th rotation is the identity e. (No rotation.) 1st rotation is a. 2nd rotation is a2 3rd rotation is a3 = e So we have a group {e, a, a2}. What’s the inverse of each element? Consider a pentagon. Rotations create group {e, a, a2 , a3 , a4}. Rotations and Reflections Group Consider triangle with rotation “a” AND reflection “b” about vertical axis. Have e, a, a2 for rotations. And e, b for reflections (two reflections give back e). Also have ab (rotation followed by reflection). And ba, a2b, ba2. Note ab does not equal ba! But ba = a2b and ab = ba2. Try it! Draw a triangle. Label vertices 1, 2, and 3. Now try combinations of rotations and reflections. Whole group is {e, a, a2, b, ab, ba}. Other combinations give one of the elements listed above. Cyclic Groups Consider an integer n. Construct a group of order n by considering any symbol “a” and the symbols a0, a1, a2, a3, ...an-1 where we insist that a0 = an = e, ai . aj = ai+j mod n. Called cyclic groups. “a” is the generator. Example: triangle rotation group! Example: integers modulo n with addition. Consider modulo 5. Have 0, 1, 2, 3, 4. Generator is 1. Identity is 0. Inverse is the number plus 5 (ai+5 . aj = 0). Rule 102 Derivative Matrix Group Now coolness! The kth derivative matrices form a cyclic group. Theorem (derivative matrix group): Suppose Dkx = x. Then D1, D2, ...Dk are a cyclic group of order k. Proof: Identity is I = Dk (the “even” identity matrix). Inverse of Di is Di+(k-1). Associative and closure are obvious. Who Cares if it is a Group? Can now use group theory to prove results about rule 102! Example: Subgroups are subsets of a group that also form their own group. 0, 1, 2, 3 with addition mod 4. Has subgroup 0, 2. Subgroups always have a size that divides the size of their parent group. Whoa, is that cool or what! Cycle Length Divisors So consider subgroups of the “derivative matrix group”. Their size will divide k (the cycle length). Can prove from this that any vector x will have a cycle length that divides the cycle length of 000...011. That’s the divisor theorem again! But can now see how to expand the divisor theorem to find out exactly what vectors will be in each cycle. An Aside on Other CA and Groups I find group theory to be beautiful. If you do too, you may want to explore the following. Many CA correspond to a group! How? Treat each state of the CA as a group element. E.g., 0 and 1 But call them a and b. Or 0, 1, and 2 But call them a, b, and c. Group Z2 Now consider rules for their combination. E.g., for rule 102 aa = a ab = b ba = b bb = a (0+0 = 0) (0+1 = 1) (1+0 = 1) (1+1 = 0) Note the identity, associativity, inverse, closure. This group is called Z2. The integers 0, 1, ..., n with sum mod n form the “cyclic groups” Zn. Evolution of Group Elements Now consider the time evolution as a CA. a a aa b ab aaab aaababba a ba abba ... ... ... ... The patterns that we see in the CA (e.g., fractal) reflect underlying structures in the groups! CA with “Reflection” So now treat a and b as rotation and reflection of a triangle! Then have other states as well (besides a and b). Remember {e, a, a2, b, ab , ba}? So have 6 states. Each one of these states should get a different color. Now can look at the CA picture. a a aa = a2 b ab ba aaab = b abba = a2 aaababba = ab a ... ... ... ... Explore CA With Groups Fascinated? Explore what CA correspond to what groups. For a given CA, explore what group elements form different visual patterns. Find subgroups, cyclic subgroups, normal subgroups, cosets, etc. in the visual patterns. Group theory is deep and powerful. There are many other possibilities for exploration. Perhaps good idea for class final project... One More Approach: Polynomials Just briefly, another avenue is to treat each vector as a polynomial. Consider a = (a0, a1, a2, a3, ..., an). Write this as a polynomial i n i i i 0 p( x) a x Derivative as Polynomial Multiplication The derivative is found by multiplying by 1 x 1 Why? Multiplying a polynomial by xj shifts each coefficient j places to the right. Multiplying a polynomial by x-j shifts each coefficient j places to the left. So x-1+1 adds the (i+1)st polynomial term to the ith term. The term to the right has been shifted to the left. Polynomial Example p( x) 1011 1x 0 0 x1 1x 2 1x 3 d ( x) (1x 0 0 x1 1x 2 1x 3 )( x 1 1) 1 (1x 0 x 1x 1x ) 0 1 2 (1x 0 0 x1 1x 2 1x 3 ) 1 1x 1x 1x 0 x 1x 0 1 2 3 Term due to x-1. Term due to 1. But wait! Now, my polynomial is too long. See next slide. But basic idea is to wrap around that term and add to the end. Boundary Conditions How do you get wrap around? Take the polynomial modulo (xn+1). In other words, the remainder after dividing by xn+1. Yeah, slight pain in the rear. Advantage of Polynomial Approach? Easily generalized. Suppose we are using rule 90. That’s the same as adding the left and right neighbors mod 2. Try it! Try writing out rule 90. 000 goes to what? 001 maps to what? Etc. Now the derivative is found by multiplying by x 1 0 x Other Rules Can look at any other additive rules by changing the derivative polynomial. For example, what rule number has a derivative that is found by multiplying by 1 x 1 Reference The polynomial approach was used by Stephen Wolfram. See his 1984 paper, Algebraic properties of cellular automata, in Communications in Mathematical Physics, vol. 93, pp. 219-258. Good stuff. Try Exploring! I leave it to you to explore with polynomials. And groups And linear algebra And ...!