cellular automata

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

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

Pabitra Paramanik (11081005)

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

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

INDEX

CELLULAR AUTOMATA(CA)

BASICS

CA RULES REVERSIBLE CA

REACHABILITY TREE

NEXT

NEXT

NEXT

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

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.

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.

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.

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.

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.

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

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.

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)

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.

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.

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.

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.

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.

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).

REACHABILITY TREE

REACHABILITY TREE

REACHABILITY TREE

THANK YOU ALL !!

WE ARE IN PROGRESS…...….