Cellular automata

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Cellular automata

  1. 1. From :- Abhisek Kundu (11081026) Nur Islam (11081017) Pabitra Paramanik (11081005)
  2. 2. TIME FRAME MAJOR PLAYERS CONTRIBUTION Early 50’s J. Von Neuman , E.F. Codd , Henrie & Moore , H Yamada & S. Amoroso Modeling biological systems - cellular models ‘60s & ‘70s A. R. Smith , Hillis, Toffoli Language recognizer, Image Processing ‘80 s S. Wolfram ,Crisp,Vichniac Discrete Lattice,statistical systems, Physical systems ‘87 - ‘96 IIT KGP, Group Additive CA, characterization,applications ‘97 - ‘99 B.E.C Group GF (2p) CA
  3. 3. 1 2 3 • ANALYSIS AND SYNTHESIS OF NONLINEAR REVERSIBLE CELLULAR AUTOMATA • GUI IMPLEMENTATION OF RECHABILITY TREE • VLSI DESIGN AND TESING BASED ON CELLULAR AUTOMATA
  4. 4. NEXT CELLULAR AUTOMATA(CA) BASICS NEXT CA RULES REVERSIBLE CA NEXT REACHABILITY TREE
  5. 5. A Cellular Automata (CA) is a discreet model studied in computability theory , mathematics , physics , complexity science , theoretical biology and microstructure modeling.  A cellular automaton consists of a regular grid/lattice of cells.  It evolves in discrete space and time , and can be viewed as an autonomous Finite State Machine(FSM).  Each cell follows a simple rule for updating its state.  The cell's state s at time t+1 depends on its own state and the states of its neighbouring cells at t. Cell State = empty/off/alive/0 Grid/Lattice State = filled/on/dead/1
  6. 6. CAs have been (or could be) used to solve a wide range of computing problems including: Image Processing: Each cell correspond to an image pixel and the transition rule describe the nature of the processing task. Random Number Generation: CAs can generate large sequences of random numbers. NP-Complete Problems: CAs can address some of the more difficult problems in computer Science. OTHERS: VLSI Testing, Data Encryption, Error Correcting Code Correction, Testable Synthesis, Generation of hashing Function.
  7. 7. Cellular Automata offer many advantages over standard computing architecture including: Implementation: CAs require very few wires. Scalability: It is easy to upgrade a CA by adding additional cells. Robustness: CAs continue to perform even when a cell is faulty because the local connectivity property helps to contain the error.
  8. 8. The three main components of a Cellular Automata are : The array dimension The neighborhood structure The transition rule Neighborhood:Von Neumann Moore Extended Moore Periodic Boundary CA :- Left neighbor of the left most cell is the right most cell and vice versa. Null Boundary CA :- State of left neighbor of the left most cell and the right neighbor of the right most cell is Zero/Null.
  9. 9.  Next State Function:- In a CA next state Si t+1 of the ith cell is specified by the Next State function fi as Si t+1 = fi (S i-1t , S it , S i+1 t ) Each cell has a next state function . If the next state function of the ith cell is expressed in the form of a truth table then the decimal equivalent of the output is conventionally referred to as the ‘Rule’ Ri.
  10. 10. We can form the next state combinational logic corresponding to a cell’s rule that determines next state of the cell. Linear/Additive Rule :- The rule that employ only XOR logic or XNOR logic in its next state combinational logic is called linear rule otherwise it is a non-linear rule . Out of 256 rules there are only 14 rules (Rule-15,51,60,85,90,102,105,150,153,165,170,195,204,240) are linear / additive rule.
  11. 11. D Cell 0 Q Cell 1 Cell 2 Cell 3 0 0 4-Cell CA Structure
  12. 12.  A small number of sensible rules, for any given suitable application.  Every CA rule says: A cell in state X changes to a cell of state Y if certain neighbourhood conditions are satisfied  For 1d,2 state, 3 neighbour CA have total number of 2^2^3 = 2^8 = 256 rules.
  13. 13.  A combination of the present states can be viewed as the Min Term of a 3varible (S i-1t , S it , S i+1 t ) switching function . Therefore each column of the first row of table2 is referred to as Rule Min Term (RMT).  RMT 7 of rule 105 of cell1= d(don’t care) RMT 4 of rule 129 for cell2 = 0  RMT 3 of rule 171 of cell3= 1 RMT 1 of rule 65 of cell4= d(don’t care)
  14. 14. Relationship among RMTs of cell i and cell (i+1) for next state computation  CA in n-neighborhood , an RMT can be considered as n-bit window(i-1 , i ,i+1).  The n-bit window for the (i+1)th cell can be found from the window of ith cell with one bit right shift. 
  15. 15. The RMT window for ith cell is (bi-1 bi bi+1), bi =0/1, then the RMT window for (i+1)th cell is either (bibi+1 0) or (bibi+1 1).  Therefore if ith CA cell changes it state following the RMT k of the rule Ri, then (i+1)th cell will change state following the RMT 2kmod8 or 2kmod8+1. 
  16. 16.  2D cellular automata system.  Each cell has 8 neighbors - 4 adjacent orthogonally, 4 adjacent diagonally. This is called the Moore Neighborhood.  Simple rules, executed at each time step:  A live cell with 2 or 3 live neighbors survives to the next round.  A live cell with 4 or more neighbors dies of overpopulation.  A live cell with 1 or 0 neighbors dies of isolation.  An empty cell with exactly 3 neighbors becomes a live cell in the next round.
  17. 17. Definition 2 :- A rule is balanced if it contains equal number of 1s and 0s in its 8-bit binary representation ; otherwise it is an unbalanced rule.  Definition 3 :- A rule Ri’ is the complement rule of R if each RMT(Rule Min Term) of Ri’ is the complement of the corresponding RMT of Ri , Therefore , Ri + Ri’ = 11111111 (255).  Definition 4 :- Two RMTs are equivalent if both result in the same set of RMTs effective for the next level of Reachability tree.  Definition 5 :- Two RMTs are sibling at level i+1 if these are resulted in from the same RMTs at the level i of the Reachability tree. 
  18. 18. In case of reversibility there are two types of CA : Reversible CA :- The initial CA state repeats after certain no of time steps . Therefore all the states of a reversible CA are reachable from other states. A state must have only one predecessor. It contains only cyclic states in it state transition diagram.  Irreversible CA :- There are some states which are not reachable(non-reachable states) from other state and a state may have more than one predecessor.
  19. 19.  The reachability tree is defined to characterize the CA states. It is a binary tree and represents the reachable states of a CA.  Each node of the tree is constructed with RMT(s) of a rule .  Left Edge : - 0-edge Right Edge :- 1-edge.  The no of levels of the reachability tree for an n-cell CA is (n+1).Root node is at level 0 and leaf nodes are at level n.  The node at level I are constructed with the selected RMTs of Ri+1 for the next state computation of cell (i+1).
  20. 20. WE ARE IN PROGRESS…...….

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