Sudheer Sahu John H. Reif Duke University X Y Z XY YZ Capability and Limitations of Redundancy Based Compact Error Resilient Methods Sudheer Sahu John H. Reif Duke University

1 1 Tile [Yan03] Pad Output 1 Output 2 Input 2 Input 1 [Rothemund04] Input 1 Input 2 Output 1 Output 2 1 1 Output 1 = Input 1 XOR Input 2 Output 2 = Input 1 XOR Input 2

Errors…… 1 1 1 1 1 1 1 1

Work in Error correction Winfree’s seminal work Goel’s snake tiling Reif’s compact error resiliency Original tiles: Error resilient tiles: (Excerpted from Winfree 03) Error rate   2 Assembly size increased by 4

Work in Error correction Winfree’s seminal work Goel’s snake tiling Reif’s compact error resilient scheme Original tiles: X Y Z XY YZ Error resilient tiles: Error checking pads

Work in Error correction Winfree’s seminal work Goel’s snake tiling Reif’s compact error resilient scheme [Schulman05] [Soloveichik05]

Two Dimensional Assembly Tile T(i,j) Input: U(i,j), V(i,j) Output : U(i+1,j), V(i,j+1) U(i+1,j)=U(i,j)OP1V(i,j) V(i,j+1)=U(i,j)OP2V(i,j) V(I,j+1) T(i,j) U(i+1,j) U(I,j) V(I,j)

Error Model The assembly takes place in a kinetic manner, but the error analysis is done at equilibrium. When the system is in equilibrium, the probability that a mismatch occurs is ε. Independent errors. Emphasis on the correctness of complete assembly Redundancy based compact error resilience scheme Immediate neighborhood 8 surrounding tiles a-dependent, a-independent T(i’,j’) a-dependent on T(i,j), if i’≥i and j’≥j a-independent otherwise 4 6 3 5 2 1

Error reduction from ε to ε2 Theorem: There exists a redundancy based compact error resilient scheme to reduce error from ε to ε2 for arbitrary boolean functions OP1 and OP2. T(i,j) U(i,j) V(i,j) U(i,j+1) U(i+1,j) V(i+1,j) U(i+1,j+1) V(i,j+1) V(i+1,j+1)

T(i,j+1) T(i-1,j+1) T(i+1,j) T(i,j) T(i-1,j) T(i+1,j-1) T(i,j-1) U(i,j+2) U(i,j+1) V(i,j+1) U(i,j+2) U(i,j+1) V(i,j+1) V(i+1,j+1) V(i,j+1) U(i,j+1) V(i+1,j+1) V(i,j+1) U(i,j+1) T(i+1,j) T(i,j) T(i-1,j) U(i+1,j+1) U(i+1,j) V(i+1,j) U(i+1,j+1) U(i+1,j) V(i+1,j) U(i,j+1) U(i,j) V(i,j) U(i,j+1) U(i,j) V(i,j) V(i+2,j) V(i+1,j) U(i+1,j) V(i+1,j) V(i,j) U(i,j) V(i+2,j) V(i+1,j) U(i+1,j) V(i+1,j) V(i,j) U(i,j) T(i+1,j-1) T(i,j-1)

Error Reduction from ε to ε3 Theorem: For arbitrary Boolean functions OP1 and OP2 error reduction from ε to ε3 is not possible using redundancy based compact error resilient schemes. g’(f(V(i,j-1))) h’(f(V(i-1,j))) g(V(i,j-1)) h(V(i-1,j)) f(V(i,j)) U(i,j) U(i,j) f(V(i-1,j)) V(i,j) V(i,j-1) V(i,j) V(i-1,j) f(V(i,j)) f(V(i,j-1))

Restricted class of functions for error reduction from ε to ε3 For constant V(i,j), OP1 is input-sensitive to U(i,j) For constant U(i,j), OP2 is input-sensitive to V(i,j) If both U(i,j) and V(i,j) change at least one of the U(i,j) OP1V(i,j) or U(i,j)OP2V(i,j) should change. Example: Number of Boolean functions in this class= 8 U V U OP1V U OP2V 1

Error reduction from ε to ε3 Theorem: For restricted class of Boolean functions OP1 and OP2 such that at least one of the U(i+1,j) or V(i,j+1) changes for any change in U(i,j) or V(i,j), there exists a redundancy based compact error resilience scheme that can reduce the error from ε to ε3. Theorem: For any combination of Boolean functions OP1 and OP2 outside the restricted class, there exists no redundancy based compact error correction schemes that can reduce the error from ε to ε3 in two-dimensional self-assembly.

Error reduction from ε to ε4 Theorem: For any Boolean functions OP1 and OP2, there exists no redundancy based compact error correction scheme that can reduce error from ε to ε4

Self-assembly in three dimensions Microelectronics assembly Heterogeneous 3D integration of next-generation microsystems [Xiong03,Martin01,Clark02,Whitesides02] Building complex systems from microscaled templates Three-dimensional structures for computing [Jonoska99] generalization of Pascal triangle to 3D [Bondarenko93] 3D multiplexers simulation of a 2D cellular automata matrix multiplication integer multipliers context free language recognition Crystal structure of 3D DNA lattices [Seeman04] Big question : Fault-tolerance ???? extension of error-correction techniques in 2D assemblies to 3D.

Assembly in Three Dimensions W(i,j,k+1) V(i,j+1,k) Tile: T(i,j,k) Inputs: U(i,j,k),V(i,j,k),W(i,j,k) Outputs: U(i+1,j,k) = f1(U(i,j,k),V(i,j,k),W(i,j,k)) V(i,j+1,k) = f2(U(i,j,k),V(i,j,k),W(i,j,k)) W(i,j,k+1)= f3(U(i,j,k),V(i,j,k),W(i,j,k)) U(i,j,k) U(i+1,j,k) W(i,j,k) V(i,j,k)

The error model Obvious extension of the 2D model Independent error model Correctness of complete pattern not just the output Redundancy based compact error resilience scheme

Error reduction to ε2 Theorem: There exists a redundancy based compact error resilient scheme for error reduction from ε to ε2 for any arbitrary boolean functions f1, f2, f3.

Error reduction to ε3 in restricted class of functions Theorem: If Boolean functions f1, f2, and f3 satisfy the following conditions: for fixed V(i,j,k) and W(i,j,k), f1(U,V,W) is input-sensitive to U(i,j,k). for fixed U(i,j,k) and W(I,j,k), f2(U,V,W) is input-sensitive to V(I,j,k). for fixed U(i,j,k) and V(i,j,k), f3(U,V,W) is input-sensitive to W(i,j,k). Then there exists a compact error resilient scheme to reduce error from ε to ε3 for three-dimensional self-assembly.

Error reduction to ε4 Theorem: For arbitrary Boolean functions f1,f2 and f3 there exists no redundancy based compact error correction scheme that can reduce error from ε to ε4 in three dimensional self-assembly.

Conjectures For three dimensional self-assembly and redundancy based compact error correction schemes: Conjecture: For arbitrary Boolean functions f1,f2 and f3 there exists no error reduction scheme that can reduce error from ε to ε3. Conjecture: For any Boolean functions f1, f2 and f3 outside restricted class of functions there exists no error reduction scheme to reduce error from ε to ε3. Conjecture: For any Boolean functions f1, f2 and f3 there exists no error reduction scheme to reduce error from ε to ε4.

Capabilities and Limits ε2 Arbitrary Yes ε3 No conjecture, No Restricted Outside restricted ε4 Any

Acknowledgement This work is supported by NSF EMT Grants CCF-0523555 and CCF-0432038 Prof. Winfree for useful discussions.