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Exploring Cellular Automata and Computational Theory

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  1. Governor’s School for the Sciences Mathematics Day 13

  2. MOTD: John von Neumann • 1903 to 1957 • Built mathematical framework for quantum mechanics • Worked in game theory • Developed first modern day computer • Developed cellular automata • Pioneer of computer science

  3. How GSS is like college • More freedom and responsibility • Some classes of interest; some required • Classes tough; pace is fast • Work is done outside of class • Students better than average HS students

  4. How college is not like GSS • Freedom to get in big trouble and be completely irresponsible • Instructor not always good or caring • Classes not designed to be fun or neccesarily interesting • Workload can be overwhelming • Being smart does still not make you popular

  5. Cellular Automata • Cells – d-dim’l set of squares (cells): • Each cell is in one of k states (or colors) and is a simple computer • Each cell runs the same simple ‘program’ called a transition or update rule • Rule gives next state depending on state of neighbors

  6. Executing a CA • Assign initial states to the cell array • For each time cycle, execute the rule at each cell (all updates occur at the same time) • The results can model physical patterns, generate random numbers and have other applications Group Exercise

  7. Examples (d=1)

  8. Details for 1D • Associate the cells with the integers: …,-4,-3,-2,-1,0,1,2,3,4,… (infinite) • State is designated: x(i) or xi • Rule based on states of cells in neighborhood: Ni = {j : |i-j|r }, r>0 is range • Example r=1, Ni = {i-1, i, i+1} • k states means k 2r+1 different possible neighborhood state combinations • Rule specifies future state for each possible combination

  9. Simplest Case: d=1, r=1, k=1 • Neighborhood of i has states x(i-1), x(i), x(i+1) • Call them p, q, r • Rules are arithmetic/logical combinations of p, q and r, like (p+r) mod 2 1 – (1-p)(1-q)(1-r) r and q p • Or just an explicit set of outputs for each of the 8 possible neighbor state combos

  10. Worksheet • Start with the pattern and apply these rules 5 timesEx: p + r (mod 2) You Try A: r B: (p or q) or r C: 1-p*q

  11. Results

  12. (aside) Binary Arithmetic • 2 states: denote them 0 and 1 • The states of a collection of cells is just a list of 0s and 1s • A list of 0s and 1s can be interpreted as the base-2 representation of a positive integer (and vice-versa!) 101101002 = 27 + 25 + 24 + 22 = 18010 • Use for numbering neighborhoods and rules

  13. … and Boolean Algebra • 1=on and 0=off or 1=true and 0=false • If p and q are two values then not p = 1-p p and q = pq p or q = 1 – (1-p)(1-q) p xor q = p+q (mod 2) • Any logical rule using two states can be written as an binary arithmetic expression

  14. Naming Rules 8 possible nbhd patterns 2 possible outputs for each pattern 28 = 256 possible rules

  15. Naming Rules (2) • Given any rule, it produces 8 output values, one for each neighborhood • Write outputs in order from 7 to 0 and interpret as a binary number • Decimal value is the Rule Number • Ex: Rule 30: 30 = 000111102Rule 110: 110 = 011110002 • Remember to read from right to left for states 0-7

  16. Classification • 4 basic cases for CAsClass 1: tends toward a limit point (like all 0s)Class 2: tends toward a cycle Class 3: chaotic behaviorClass 4: more complex; capable of universal computation

  17. Other Classification • Totalistic: rule only depends on sum of neighboring values (24 = 16) • Legal: totalistic, plus if sum=0, state must be 0 (23 = 8) • Reversible: can go forward or back in time (without losing information)

  18. #254 (C1) #170 (C2) #30 (C3) #110 (C4)

  19. Finite State Machine • S = finite set of states • I = finite set of inputs • O = finite set of outputs • f s= a function that takes a state and an input and produces a state • f o= a function that takes a state and produces an output • Given a starting state and a string of inputs, apply f o and f s repeatedly to produce a string of outputs

  20. Turing Machines • A Turing machine is like a finite state machine, except: It can erase/write on the input It can read input forward or backward • A Turing machine is an abstract algorithm and its existence and properties are used to say clever things about the algorithm, e.g. decidability and complexity

  21. Universal Turing Machines • There is a Turing machine which can simulate the action of any other Turing machine on any input • It is called the Universal Turing Machine • Basic idea: write a ‘program’ for a Turing machine and encode it on the input tape, the UTM reads the ‘program’ and executes it on the rest of the input; complicated due to lack of memory

  22. CA and Turing Machines • Rule 110 is a Universal Turing Machine • There are CAs which are Universal CAs! (much more complicated than a UTM; serial versus infinitely parallel)

  23. Lab Time • Classification of all 256 rules for the simple case • Each person will do 16 different rules; assignment in Lab • Everyone must do their part! • When done: work on your project

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