cellular automata

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DEVELOPMENT OF SOFTWARE PACKAGE FOR CELLULAR AUTOMATA BASED SOLUTIONS OF VARIOUS PROBLEMS From :- Abhisek Kundu (11081026) Nur Islam (11081017) Pabitra Paramanik (11081005)

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Page 1: Cellular automata

DEVELOPMENT OF SOFTWARE PACKAGE FOR CELLULAR AUTOMATA BASED SOLUTIONS OF VARIOUS PROBLEMS

From :- Abhisek Kundu (11081026) Nur Islam (11081017)

Pabitra Paramanik (11081005)

Page 2: Cellular automata

BACKGROND OF CA

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

Page 3: Cellular automata

OBJECTIVE

1• ANALYSIS AND SYNTHESIS OF NONLINEAR

REVERSIBLE CELLULAR AUTOMATA

2• GUI IMPLEMENTATION OF RECHABILITY

TREE

3• VLSI DESIGN AND TESING BASED ON

CELLULAR AUTOMATA

Page 4: Cellular automata

INDEX

CELLULAR AUTOMATA(CA)

BASICS

CA RULES REVERSIBLE CA

REACHABILITY TREE

NEXT

NEXT

NEXT

Page 5: Cellular automata

CELLULAR AUTOMATA

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

State = filled/on/dead/1

Grid/Lattice

Page 6: Cellular automata

APPLICATION OF CA’s

CAs have been (or could be) used to solve awide range of computing problems including:

Image Processing: Each cell correspond to an image pixel and the transition rule describe thenature of the processing task.

Random Number Generation: CAs cangenerate large sequences of random numbers.

NP-Complete Problems: CAs can address someof the more difficult problems in computer Science.

OTHERS: VLSI Testing,Data Encryption, Error Correcting Code Correction,Testable Synthesis, Generation of hashing Function.

Page 7: Cellular automata

ADVANTAGES OF CA’s

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 localconnectivity property helps to contain the error.

Page 8: Cellular automata

CELLULAR AUTOMATA BASICS

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.

Page 9: Cellular automata

CELLULAR AUTOMATA BASICS

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 i

t , 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.

Page 10: Cellular automata

CELLULAR AUTOMATA BASICS

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.

Page 11: Cellular automata

EXAMPLE :1D,2 STATE,4 CELL,3 NEIGHBOUR NULL BOUNDARY CA

4-Cell CA Structure

0

0

D

QCell 0 Cell 1 Cell 2 Cell 3

Page 12: Cellular automata

CA RULES 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.

Page 13: Cellular automata

RULE MIN TERM (RMT) A combination of the present states can be viewed as the Min Term of a 3-

varible (S i-1t , S i

t , 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)

Page 14: Cellular automata

RULE MIN TERM (RMT) 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.

Page 15: Cellular automata

RULE MIN TERM (RMT) 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.

Page 16: Cellular automata

EXAMPLE:JOHN CONWAY’S GAME OF LIFE

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.

Page 17: Cellular automata

CA RULES TYPES 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.

Page 18: Cellular automata

CLASSIFICATION OF CAIn 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.

Page 19: Cellular automata

REACHABILITY TREE 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).

Page 20: Cellular automata

REACHABILITY TREE

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REACHABILITY TREE

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REACHABILITY TREE

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THANK YOU ALL !!

WE ARE IN PROGRESS…...….