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71 Pages

### D-algorithm

Course: ECE 545, Spring 2012
School: Alliant
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Word Count: 2118

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ATPG Overview Combinational Major ATPG algorithms Definitions D-Algorithm (Roth) -- 1966 D-cubes Bridging faults Logic gate function change faults PODEM (Goel) -- 1981 X-Path-Check Backtracing Summary 2/10/2012 2 Forward Implication Results in logic gate inputs that are significantly labeled so that output is uniquely determined AND gate forward implication table: 2/10/2012 3 Backward...

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ATPG Overview Combinational Major ATPG algorithms Definitions D-Algorithm (Roth) -- 1966 D-cubes Bridging faults Logic gate function change faults PODEM (Goel) -- 1981 X-Path-Check Backtracing Summary 2/10/2012 2 Forward Implication Results in logic gate inputs that are significantly labeled so that output is uniquely determined AND gate forward implication table: 2/10/2012 3 Backward Implication Unique determination of all gate inputs when the gate output and some of the inputs are given 2/10/2012 4 Implication Stack, Decision Tree, and Backtrack Stack Tree Unexplored Present Assignment Searched and I f S h d d Infeasible ibl 0 E 1 1 0 0 0 2/10/2012 B B 1 0 1 F 1 0 F F 1 5 Objectives and Backtracing in ATPG Objective desired signal value goal for ATPG Guides it away from infeasible/hard solutions Uses heuristics Backtrace Determines which primary input and value to set to achieve objective U h i ti such as nearest PI Use heuristics h t Forward trace Determines gate through which the fault effect should be sensitized Use heuristics such as output that is closest to the present fault effect 2/10/2012 6 Branch and Bound Branch-and-Bound Search Efficiently searches binary search tree Branching At each tree level selects which input level, variable to set to what value Bounding Avoids exploring large tree portions by artificially restricting search decision choices ifi i ll i i hd i i h i Complete exploration is impractical Uses heuristics Backtracking Search fails, therefore undo some of the work completed and start searching from a location where search options still exist p 2/10/2012 7 D Algorithm D-Algorithm Roth (1966) Fundamental concepts invented: First complete ATPG algorithm D-Cube DC b D-Calculus Implications forward and backward Implication stack Backtrack Test Search Space 2/10/2012 8 Singular Cover Example g p Minimal set of logic signal assignments to represent a function show prime implicants and prime implicates of Karnaugh p p p p g map (with explicitly showing the outputs too) 2/10/2012 Gate AND 1 2 3 Inputs Output Gate B A d NOR X 0 0 1 0 X 0 2 1 1 1 3 d Inputs Output X 1 0 1 X 0 e F 0 0 1 9 Primitive D-Cube of Failure D Cube Models circuit faults: Stuck-at-0 Stuck-at-1 Other f lt Oth faults, such as B id i fault ( h t circuit) h Bridging f lt (short i it) Arbitrary change in logic function AND Output sa0: "1 1 D" 1 D AND Output sa1: "0 X D" "X 0 D" Wire sa0: "D" p g Propagation D-cube models conditions under which fault effect propagates through gate 2/10/2012 10 Construction of Primitive D-Cubes of Failure 1. 1 Make cube set 1 when good machine output is 1 and set 0 when good machine output is 0 2. Make cube set 1 when failing machine output is 1 and 0 when it is 0 3. Change 1 outputs to 0 and D-intersect each cube with every 0. If intersection works, change output of cube to D 4. 4 Change 0 outputs to 1 and D intersect each D-intersect cube with every 1. If intersection works, change output of cube to D g p 2/10/2012 11 Gate Function Change D-Cube of Failure Fail re Cube-set a b c 0 0 1 0 1 2/10/2012 Cube-set a b 0 1 1 0 c D D 0 X 1 0 1 X X 0 1 0 X 1 0 0 PDFs for 1 AND changing g g 0 to OR 1 1 12 Propagation D Cube D-Cube Collapsed truth table entry to characterize logic Use Roth's 5-valued algebra AND gate: use the rules given earlier using and but in thi b t i this case work with good circuit only k ith d i it l Write all primitive Cubes of AND gate and then create propagation cubes A D 1 D D 1 D B 1 D D D D 1 d D D D D D D 13 2/10/2012 D-Cube Operation of D-Intersection DC b O i f DI i undefined (same as ) or requires inversion of D and D D-intersection: 0 0 = 0 X = X 0 = 0 1 1 = 1 X = X 1 = 1 X X = X D-containment 0 1 X D D Cube a contains Cube b if b is a subset of a 2/10/2012 0 1 X D D 0 0 1 1 0 1 X D D D D 14 Implication Procedure 1. 1 Model fault with appropriate primitive D D2. 2 3. cube of failure (PDF) Select propagation D-cubes to propagate fault effect to a circuit output (D-drive procedure) Select singular cover cubes to justify internal circuit signals (Consistency procedure) Put signal assignments in test cube g g Regrettably, cubes are selected very arbitrarily by D-ALG y y 15 2/10/2012 D Algorithm D-Algorithm Top Level 1. Number all circuit lines in increasing level order from PIs to POs; 2. Select a primitive D-cube of the fault to be the test cube; Put logic outputs with inputs labeled as D (D) onto the D-frontier; 3. D-drive (); 4. Consistency (); 5. return (); 2/10/2012 16 D-Algorithm D-drive while (untried fault effects on D-frontier) select next untried D-frontier gate for propagation; while (untried fault effect fanouts exist) select next untried fault effect fanout; generate next untried propagation D-cube; D-intersect selected cube with test cube; if (intersection fails or is undefined) continue; if (all propagation D-cubes t i d & f il d) break; ( ll ti D b tried failed) b k if (intersection succeeded) add propagation D-cube to test cube -- recreate D-frontier; Find all forward & backward implications of assignment; save D-frontier, algorithm state, test cube, fanouts, fault; break; else if (intersection fails & D and D in test cube) Backtrack (); 2/10/2012 if (intersection fails) break; else 17 if (all fault effects unpropagatable) Backtrack (); D-Algorithm -- Consistency g = coordinates of test cube with 1' & 0' di t f t t b ith 1's 0's; if (g is only PIs) fault testable & stop; for (each unjustified signal in g) Select highest # unjustified signal z in g, not a PI; if (inputs to gate z are both D and D) break; while (untried singular covers of gate z) select next untried singular cover; if (no more singular covers) If (no more stack choices) fault untestable & stop; ( ) p; else if (untried alternatives in Consistency) pop implication stack -- try alternate assignment; else Backtrack (); D-drive (); If (singular cover D-intersects with z) delete z from g, add inputs to singular cover to g, fi d all forward and backward implications of i l t find ll f d db k d i li ti f new assignment, and break; 2/10/2012 18 If (intersection fails) mark singular cover as failed; Backtrack if (PO exists with fault effect) Consistency (); else pop prior implication stack setting to try alternate assignment; if (no untried choices in implication stack) fault untestable & stop; f else return; 2/10/2012 19 Circuit Example 7.1 and Truth Table a 0 0 0 0 1 1 1 1 2/10/2012 Inputs b 0 0 1 1 0 0 1 1 c 0 1 0 1 0 1 0 1 Output F 0 0 0 1 0 0 0 0 20 Singular Cover & Propagation D-Cubes A 1 0 B 1 0 1 0 C 1 0 d 1 0 0 e 0 1 1 1 0 D D D 0 D D F Singular cover Used for justifying lines D 1 D 1 D D D 1 D 1 D D 1 0 D D D D 0 D 0 0 1 Propagation D-cubes Conditions under which difference between good/failing b t d/f ili machines propagates D D D 2/10/2012 21 Steps for Fault d sa0 Fa lt Step A B C d e F 1 1 1 D 2 3 Cube type PDF of AND gate D 0 D Prop. D-cube for NOR D cube 1 1 0 Sing. Cover NAND 2/10/2012 22 Example of 7.3 Fault u sa1 Primitive D-cube of Failure 1 0 sa1 D 2/10/2012 23 Example 7.3 Step 2 u sa1 Propagation D-cube for v 1 0 0 sa1 D D 2/10/2012 24 Example 7.3 Step 2 u sa1 Forward and backward implications 1 0 0 1 0 D sa1 D 1 0 0 2/10/2012 25 Inconsistent d = 0 and m = 1 cannot justify r = 1 (equivalence) Backtrack Remove B = 0 assignment 2/10/2012 26 Example 7.3 Backtrack Need alternate propagation D-cube for v 1 0 sa1 D 2/10/2012 27 Example 7.3 Step 3 u sa1 Propagation D-cube for v 1 1 0 sa1 D D 2/10/2012 28 Example 7.3 Step 4 u sa1 Propagation D-cube for Z 1 1 1 0 sa1 D D D 1 2/10/2012 29 Example 7.3 Step 4 u sa1 Propagation D-cube for Z and implications 1 1 1 1 1 0 D 1 2/10/2012 0 0 sa1 0 D D 30 PODEM -- G l Goel (1981) New concepts introduced: Expand binary decision tree only around primary inputs Use X-PATH-CHECK to test whether DA C C frontier still there Objectives -- bring ATPG closer to propagating D (D) to PO Backtracing 2/10/2012 31 Motivation IBM i t d d semiconductor DRAM introduced i d t memory into its mainframes late 1970's Memory had error correction and translation circuits improved reliability D-ALG unable to test these circuits D ALG Search too undirected Large XOR-gate trees g g Must set all external inputs to define output Needed a better ATPG tool 2/10/2012 32 PODEM High Level Flow High-Level 1. 2. 3. 4. Assign binary value to unassigned PI Determine implications of all PIs Test Generated? If so, done. Test possible with more assigned PIs? If maybe, go t St 1 b to Step 5. Is there untried combination of values on assigned PIs? If not, exit: untestable fault not 6. Set untried combination of values on assigned PIs using objectives and backtrace. Then, g g j , go to Step 2 2/10/2012 33 Example 7.3 Again Select path s Y for fault propagation sa1 2/10/2012 34 Example 7.3 -- Step 2 s sa1 Initial objective: Set r to 1 to excite fault 1 sa1 2/10/2012 35 Example 7.3 -- Step 3 s sa1 Backtrace from r 1 sa1 2/10/2012 36 Example 7.3 -- Step 4 s sa1 Set A = 0 in implication stack 1 0 sa1 2/10/2012 37 Example 7.3 -- Step 5 s sa1 Forward implications: d = 0, X = 1 1 0 0 sa1 1 2/10/2012 38 Example 7.3 -- Step 6 s sa1 Initial objective: set r to 1 1 0 0 sa1 1 2/10/2012 39 Example 7.3 -- Step 7 s sa1 Backtrace from r again 1 0 0 sa1 1 2/10/2012 40 Example 7.3 -- Step 8 s sa1 Set B to 1. Implications in stack: A = 0, B = 1 1 0 1 0 sa1 1 2/10/2012 41 Example 7.3 -- Step 9 s sa1 Forward implications: k = 1 m = 0 r = 1 q = 1 Y = 1, 0, 1, 1, 1, s = D, u = D, v = D, Z = 1 0 1 1 1 0 0 sa1 1 D D D 1 1 1 2/10/2012 42 Backtrack -- Step 10 s sa1 X PATH CHECK shows paths s Y and X-PATH-CHECK u v Z blocked (D-frontier disappeared) 1 0 0 sa1 s 1 2/10/2012 43 Step 11 -- s sa1 Set B = 0 (alternate assignment) 1 0 0 sa1 2/10/2012 44 Backtrack -- s sa1 Forward implications: d = 0 X = 1 m = 1 r = 0 0, 1, 1, 0, s = 1, q = 0, Y = 1, v = 0, Z = 1. Fault not sensitized. 0 0 0 0 1 sa1 1 0 1 1 0 1 1 2/10/2012 45 Step 13 -- s sa1 Set A = 1 (alternate assignment) 1 1 sa1 2/10/2012 46 Step 14 -- s sa1 Backtrace from r again 1 1 sa1 2/10/2012 47 Step 15 -- s sa1 Set B = 0. Implications in stack: A = 1, B = 0 1 1 0 sa1 2/10/2012 48 Backtrack -- s sa1 F Forward i li ti d implications: d = 0 X = 1 m = 1 r = 0 0, 1, 1, 0. Conflict: fault not sensitized. Backtrack 0 1 0 0 1 sa1 1 0 1 1 0 1 1 2/10/2012 49 Step 17 -- s sa1 Set B = 1 (alternate assignment) 1 1 1 sa1 2/10/2012 50 Fault Tested -- Step 18 s sa1 Forward implications: d = 1 m = 1 r = 1 q = 0 s = 1, 1, 1, 0, D, v = D, X = 0, Y = D 1 1 1 1 1 sa1 0 D X 2/10/2012 51 0 D D D Backtrace (s, vs) Pseudo-Code v = vs; while (s is a gate output) if (s is NAND or INVERTER or NOR) v = v; if (objective requires setting all inputs) select unassigned input a of s with hardest controllability to value v; else select unassigned input a of s with easiest controllability to value v; s = a; return (s, v) /* Gate and value to be assigned */; 2/10/2012 52 Objective Selection Code if (gate g is unassigned) return (g, v); select a gate P from the D-frontier; select an unassigned input l of P; if (gate g has controlling value) c = controlling input value of g; else if (0 value easier to get at input of XOR/EQUIV gate) c = 1; else c = 0; ; return (l, c ); 2/10/2012 53 PODEM Algorithm while (no fault effect at POs) if (xpathcheck (D-frontier) (l, vl) = Objective (fault, vf lt); fault (pi, vpi) = Backtrace (l, vl); Imply (pi, vpi); if (PODEM (fault vfault) == SUCCESS) return (SUCCESS); (fault, (pi, vpi) = Backtrack (); Imply (pi, vpi); p if (PODEM (fault, vfault) == SUCCESS) return (SUCCESS); Imply (pi, "X"); return (FAILURE); else if (implication stack exhausted) return (FAILURE); else Backtrack (); return (SUCCESS); 2/10/2012 54 Summary y D-ALG First complete ATPG algorithm D Cube D-Cube D-Calculus Implications forward and backward Implication stack Backup PODEM Expand decision tree only around PIs Use X-PATH-CHECK to see if D-frontier exists f Objectives -- bring ATPG closer to getting D (D) to PO Backtracing 2/10/2012 55 Appendices A di 2/10/2012 56 Push-down stack. Records: Implication Stack Each signal set in circuit by ATPG Whether alternate signal value already tried Portion of binary search tree already searched 2/10/2012 57 Objectives and Backtracing in ATPG Objective desired signal value goal for ATPG Guides it away from infeasible/hard solutions y Uses heuristics Backtrace Determines which primary input and value to set to achieve objective l t tt hi bj ti Use testability measures 2/10/2012 58 Bridging Fault Circuit B id i F lt Ci it 2/10/2012 59 Bridging Fault D-Cubes of Failure Cube-set a 0 0 X 1 1 X 0 0 X 1 1 X 0 X 1 0 1 X b a* b* 0 X 1 X 0 1 1 X 0 1 0 1 D X PDFs for 1 Bridging fault 0 1 D 1 0 1 1 Cube-set a b a* b* 2/10/2012 60 Example 7.2 Fault E ample 7 2 Fa lt A sa0 Step 1 D-Drive Set A = 1 1 D D 2/10/2012 61 Step 2 -- E ample 7 2 Example 7.2 Step 2 D-Drive Set f = 0 0 1 D D D 2/10/2012 62 Step 3 -- E ample 7 2 Example 7.2 Step 3 D-Drive Set k = 1 1 0 1 D D D D 2/10/2012 63 Step 4 -- E ample 7 2 Example 7.2 Step 4 Consistency Set g = 1 1 0 1 D D 1 D D 2/10/2012 64 Step 5 -- E ample 7 2 Example 7.2 Step 5 Consistency f = 0 Already set 1 0 1 D D 1 D D 2/10/2012 65 Step 6 -- E ample 7 2 Example 7.2 St 6 C i t Step Consistency S t c = 0 S t e = 0 Set 0, Set 1 0 D 0 0 D D 1 D 1 2/10/2012 66 D-Chain D Chain Dies -- E ample 7 2 Example 7.2 Step 7 Consistency Set B = 0 p y D-Chain dies X 0 0 1 D 1 0 0 D 1 D D Test cube: A B, C, D, e, f, g h k L A, B C D e f g, h, k, 2/10/2012 67 Example 7.3 Fault s sa1 Primitive D-cube of Failure 1 D sa1 2/10/2012 68 Example 7.3 Step 2 s sa1 Propagation D-cube for v 1 D 1 sa1 0 D D 2/10/2012 69 Example 7.3 Step 2 s sa1 Forward & Backward Implications 1 1 1 1 1 D 0 sa1 0 D D 2/10/2012 70 Example 7.3 Step 3 s sa1 Propagation D-cube for Z test found! 1 1 1 1 1 D 0 sa1 0 D D 1 2/10/2012 D 71
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