From :- Abhisek Kundu (11081026)
Nur Islam (11081017)
Pabitra Paramanik (11081005)
MAJOR PLAYERS CONTRIBUTION
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
S. Wolfram ,Crisp,Vichniac
systems, Physical systems
‘87 - ‘96
IIT KGP, Group
‘97 - ‘99
GF (2p) CA
• ANALYSIS AND SYNTHESIS OF NONLINEAR
REVERSIBLE CELLULAR AUTOMATA
• GUI IMPLEMENTATION OF RECHABILITY
• VLSI DESIGN AND TESING BASED ON
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
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.
State = empty/off/alive/0
State = filled/on/dead/1
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.
Error Correcting Code Correction,
Generation of hashing Function.
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
Robustness: CAs continue to perform even
when a cell is faulty because the local
connectivity property helps to contain the error.
The three main components of a Cellular Automata are :
The array dimension
The neighborhood structure
The transition rule
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.
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
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.
4-Cell CA Structure
A small number of sensible rules, for any given
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.
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)
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.
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
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.
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
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’ =
Definition 4 :- Two RMTs are equivalent if both result
in the same set of RMTs effective for the next level of
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.
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
Irreversible CA :- There are some states which are not
reachable(non-reachable states) from other state and a state may have more than one
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).