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Cellular Automata (CA) - Theory & Application SUSMITA SUR KOLAY P. PAL CHAUDHURI.

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Presentation on theme: "Cellular Automata (CA) - Theory & Application SUSMITA SUR KOLAY P. PAL CHAUDHURI."— Presentation transcript:

1 Cellular Automata (CA) - Theory & Application SUSMITA SUR KOLAY P. PAL CHAUDHURI

2 I BACKGROUND Time FrameMajor Players Contribution Early 50’s J. Von Neuman, E.F. Codd, Henrie & Moore, H Yamada & S. Amoroso Modeling biological systems - cellular models ‘60s & ‘70sA. R. Smith, Hillis, ToffoliLanguage recognizer, Image Processing ‘80 sS. Wolfram,Crisp,VichniacDiscrete Lattice,statistical systems, Physical systems ‘87 - ‘96 IIT KGP, GroupAdditive CA, characterization,applications ‘97 - ‘99B.E.C GroupGF (2 p ) CA

3 II CA PRELIMINARIES - GF(2) CA 2.1 CA Structure and Rule (a) 4-Cell Group CA Structure Non-Group CA structure D Q Cell 0Cell 1Cell 2Cell (b) CA RULE : q i t+1 = f (q i-1 t, q i t, q i+1 t ) Neighborhood State (90) (150)

4 ( c) XOR/ XNOR Rule List With XORWith XNOR Rule 60: q i t+1 = q i-1 t (+) q i t Rule 90: q i t+1 = q i-1 t (+) q i+1 t Rule 102: q i t+1 = q it (+) q i+1 t Rule 150: q i t+1 = q i-1 t (+) q i t (+) q i+1 t Rule 170: q i t+1 = q i+1 t Rule 204: q i t+1 = q i t Rule 240: q i t+1 = q i-1 t 195: q i t+1 = q i-1 t (+) q i t 165: q i t+1 = q i-1 t (+) q i+1 t 153: q i t+1 = q i t (+) q i+1 t 105: q i t+1 = q i-1 t (+) q i t (+) q i+1 t 85: q i t+1 = q i+1 t 51: q i t+1 = q i t 15: q i t+1 = q i-1 t

5 2.2 State Transition Behavior (a) Group CA structure state transition behavior Non-Group CA structure & behavior T = Ch. Poly : x 4 + x Min Poly : (x 2 +x+1) 2 No. Of Pred = 2 Cycle Structure = 1, 1(3), 2(6) T = Ch. Poly :x 4 + x 3 + x 2 Min Poly : x 2 (x 2 +x+1) No. Of Pred = 2, Height = 2. Cycle Structure = 1(3), 1(1)

6 Complete Characterizations Th1 : A CA is a group CA iff the determinant det T = 1, where T is the characteristic matrix for the CA. Th2 : A group CA has cycle lengths of p or factors of p with a non-zero starting state iff det[T p (+) I] = 0 Th3 : If d is the dimension of the null space of the characteristic matrix of a non-group CA, then the total numbers of predecessors of the all-zero state(state 0) is 2 d Th4 : If the number of predecessors of a reachable state and that of the state 0 in a non-complemented CA, are equal. Th5 : An exhaustive CA and exhaustive LFSR are isomorphic to each other. A few important theorems for characterization on height,cycle length & no. of components.

7 2.4 List Of Applications VLSI Testing Data Encryption Error Correcting Code Correction Testable Synthesis Generation of hashing Function

8 CA for generation of exhaustive Patterns A 4-cell CA : T = Ch. Poly : x 4 + x + 1 Factors : ( ) Minimal Polynomial : ( ) Rank of matrix = 4 Depth Of CA = 0 Null Space = ( ) Cycle Structure : 1(1), 1(15) Determinant =

9 CA for Generation of exhaustive Two Patterns T = STG ………... State Pattern (tapping 0,2,4) …….. ….. S S 0 S 2 = S 1 S S 2 S 3 S 4 S S S 1 = 010 S S S S 5 S v = T v X S v + T 0 X S 0

10 Th. For the given characteristic matrix of a 2n-cell CA and a set of n visible bits, an exhaustive 2-pattern generation is ensured if the rank of the corresponding obscurity matrix is n. Proof : An arbitrary two-pattern S v ->S v is obtained iff the following equation is satisfied. S v = T v X S v + T 0 X S 0 => T 0 X S 0 = S v - T v X S v = X where X is the n-dimensional vector (S v - T v X S v ). A solution for S 0 exists iff rank [T 0 ] = rank [T 0 X] and is ensured if rank [T 0 ] = n. Hence the theorem.

11 2.7 Generalization Lemma : A 2n-cell null boundary CA with any combination of Rule 90 & 150 over the cell is capable of generating exhaustive two-patterns at the cells 0,2,4,…,(2n-1) Proof : The characteristic matrix of a 2n-cell CA with an arbitrary combination of Rule 90 (I.e. g i = 0) & Rule 150 (g i = 1) is given by : T = g0 1 0 …. 0 1 g1 1 … g2 ….0 …………… g2n-1 Here, T 0 = 1 0 … … ……… …. 1 Obviously, Rank(T 0 ) = n. Hence, the lemma is proved by the prev. th.

12 2.8 ILLUSTRATION Consider a six-cell null boundary CA with rule vector (90, 150, 150, 150, 90, 150). The next state function of this is represented by S0S1S2S3S4S5S0S1S2S3S4S5 = S0S1S2S3S4S5S0S1S2S3S4S5 Or, in general, S = T X S Since, T 0 has rank n=3, the CA generates exhaustive two patterns on 0,2,4 bit positions. VISIBLE BITS : T v = T 0 =

13 CA Based Test Pattern generator Exhaustive two patterns - All possible transitions on Primary inputs (PI’s). For circuits with large number of PI’s, tune CATPG for the CUT – For a CUT with n PI’s, use k-cell GF(2 p ) CA where n <= kp – Fix the value of p – Define the CA structure that suits best for testing the CUT. C U T ….. q 1 2 i k.. 12 p K-cell GF(2 p )

14 GF(2 p ) CA based CATPG and Experimental results on synchronous circuit. TEST RESULTS WITH UNIFORM CA :

15 Table 2 : Test Results with Hybrid CA

16 CA based Response Evaluator C U T C A R E CUT OUTPUT : 7-value logic,,,,,, For n PO CUT, CARE : n cell GF(2 3 ) CA - Maximum Length GF(2 3 ) CA - minimum aliasing. - Compare with Golden signature - Diagnosis ?

17 DIAGNOSIS CUT is divided into k number of blocks B1 B2 B3 B4 ….. …... C A R E CA Classifier Faulty Block

18 DIAGNOSIS (Contd.) Introduce a fault on j th component /gate of the i th block B i and generate the signature B ij. Design the CA-based classifier based on the given classes. {{B 1 },{B 2 },……….{B i },…….} With B ij as the input, the classifier identifies the faulty block B i.

19 Illustration G6 G4 G8 G11 G10 G1 G2 G3 G7 G4 G9 G5 G1 G2 Fig : t4.v Test Vectors : G1 G2 G3 ; Wireinstanceclass G3,G7AND2_0 1 G2NAND2_2 G9NAND2_2 G5NAND2_3 2 G1NAND2_3 G1base & NAND2_0 G4base & NAND2_4 G2base & NAND2_0, 3 INV1_0 G5base & NAND2_5 G4NAND2_5 G1NAND2_5 4 G6BUF1_ AND2_0 NAND2_0 NAND2_2 NAND2_4 BUF1_0 INV1_0 NAND2_3 NAND2_5

20 Faults - detected and diagnosed GOLDEN SIGNATURE : Faulty Signature Faults BLOCK 3 [G1(base),G2(base)] BLOCK 3 [G1(NAND2_0)] BLOCK 3 [G2(INV1_0),G5(base)] BLOCK 3 [G5(NAND2_4)] CA CLASSIFIER

21 Why Cellular Automata ? Regular, Modular, Cascadable structure. Use of GF(2 p ) CATPG with p and CA structure tuned to match the test requirement of the circuit. (Implemented for synchronous circuits with promising results) CA Toolkit has been developed based on the Theory of Extension Field and analytical study of GF (2 p ) CA state transition behavior. The Toolkit enables – identification of cycle structure of the CA. – Design a CA, to realize a given state transition behavior Diagnosis tool based on Multiple Attractor CA to diagnose the faulty block within the CUT.

22 THANK YOU ALL !!


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