1. 1 CSCE 932, Spring 2007 Test Generation for Combinational Logic

2. 2 Outline Fundamental Concepts Structure vs. Function Boolean Difference Test Generation Algorithms Multi-valued Algebras Complexity of test generation PODEM Boolean-Satisfiability Based Summary

3. 3 Origins of Stuck-Faults Eldred (1959) – First use of structural testing for the Honeywell Datamatic 1000 computer Galey, Norby, Roth (1961) – First publication of stuck-at-0 and stuck-at-1 faults Seshu & Freeman (1962) – Use of stuck-faults for parallel fault simulation Poage (1963) – Theoretical analysis of stuck-at faults

4. 4 Functional vs. Structural ATPG

5. 5 Carry Circuit

6. 6 Functional vs. Structural (Contd) Functional ATPG – exhaustive set of tests for 129 inputs, 65 outputs: 2129 = 680,564,733,841,876,926,926,749, 214,863,536,422,912 patterns Using 1 GHz ATE, would take 2.15 x 1022 years Structural test: No redundant adder hardware, 64 bit slices Each with 27 faults (using fault equivalence) At most 64 x 27 = 1728 faults (tests) Takes 0.000001728 s on 1 GHz ATE Designer gives small set of functional tests – augment with structural tests to boost coverage to 98+ %

7. 7 Boolean Difference I will illustrate the concepts in class, along with illustrative examples.

8. 8 Multi-valued Algebras SymbolMeaningGood M/c Failing M/c D1/010 Roth’s Algebra D0/101 00/000 11/111 XX/XXX G00/X0X Muth’s Additions G11/X1X F0X/0X0 F1X/1X1

9. 9 Roth’s and Muth’s Higher- Order Algebras Simultaneously represent two machines: Good circuit machine (1st value) Bad circuit machine (2nd value) Better to represent both in the algebra: Need only 1 pass of ATPG to solve both Good machine values that preclude bad machine values become obvious sooner & vice versa Needed for complete ATPG: Combinational: Multi-path sensitization, Roth Algebra Sequential: Muth Algebra -- good and bad machines may have different initial values due to fault

10. 10 Path Sensitization Method Circuit Example Fault Sensitization Fault Propagation Line Justification

11. 11 Path Sensitization Method Circuit Example Try path f – h – k – L blocked at j, since there is no way to justify the 1 on i 1 0 D D 1 1 1 D D D

12. 12 Path Sensitization Method Circuit Example Try simultaneous paths f – h – k – L and g – i – j – k – L blocked at k because D-frontier (chain of D or D) disappears 1 D D D D D 1 1 1

13. 13 Path Sensitization Method Circuit Example Final try: path g – i – j – k – L – test found! 0 D D D 1 D D 1 0 1

14. 14 Complexity of Test Generation The primary source of complexity is that the number of sensitizable paths grow exponentially with the circuit size and in the worst case the algorithm may have to search all of them for a solution.

15. 15 Algorithmic Complexity of Test Generation Algorithmic complexity refers to the worst- case-time behavior in terms of problem parameter(s) For combinational test generation, circuit size (in number of lines) is used as the parameter. Sahni and Ibarra* showed NP-completeness of test generation by reducing a well NP- complete problem (3-SAT) to the combinational test-generation problem. O.H. Ibarra and S. K. Sahni, “Polynomially Complete Fault Detection Problems,” IEEE Trans. Computers, March 1975, pp.242-249/

16. 16 PODEM Test Generation Algorithm Structure-based and fault oriented Signal values are explicitly assigned at the PIs only; other values are computed by implication Eliminates D-algorithm’s need to justify internal signals Backtracks at PIs only when a contradiction occurs Many extensions exist with more complex heuristics, e.g. the FAN algorithm

17. 17 PODEM Decision Tree   X X  X ~ ~ B=1 D=0 D=1 A=1 A=0 A=1 Success, Halt ~ X  Unused assignments Back up, no test possible No remaining choices C=1

18. 18 PODEM Algorithm Initialize all signal values to unknown Branch Repeat Imply If(fault is detected) then exit loop if(no test possible with current ass’t) then exit Bound else Branch Forever

19. 19 & =1 & & & & & x1 x2 x3 x4 x5 z1 z2 SA0 Fault Assume sequential input selection, applying 1 before 0 PODEM Example

20. 20 PODEM Example & =1 & & & & & x1 x2 x3 x4 x5 z1 z2 SA0 Fault = 0 1 Step 1: Set x1 = 1 and imply D

21. 21 PODEM Example & =1 & & & & & x1 X2 x3 x4 x5 z1 z2 SA0 Fault Step 2: Set x2 = 1 and imply = 0 1 = 1 D

22. 22 PODEM Example & =1 & & & & & x1 X2 x3 x4 x5 z1 z2 SA0 Fault = 0 1 = 1 D = 1 0 1 0 1 Step 3: Set x3 = 1 and imply. Fault cannot be detected at z1 but detection at z2 is still possible.

23. 23 PODEM Example & =1 & & & & & x1 X2 x3 x4 x5 z1 z2 SA0 Fault = 0 1 = 1 D = 1 0 1 0 1 Step 4: Set x4 = 1 and imply. Now the fault cannot be detected at either output hence need to backtrack. = 1 0 1 0 1

24. 24 PODEM Example & =1 & & & & & x1 X2 x3 x4 x5 z1 z2 SA0 Fault = 0 1 = 1 D = 1 0 1 0 1 Step 4: Set x4 = 1 and imply. Now the fault cannot be detected at either output hence need to backtrack. = 1 0 1 0 1

25. 25 PODEM Example & =1 & & & & & x1 X2 x3 x4 x5 z1 z2 SA0 Fault = 0 1 = 1 D = 1 0 1 0 1 Step 5 (backtrack): Set x4 = 0 and imply. Now the fault is detected at output z2. = 0 1 D’D’ D’D’ D D’D’

26. 26 Podem Heuristics Several heuristics are used for speedup Line objectives Selecting alternatives in back-tracing Selecting alternatives for extending D-path to an observable output Many heuristics are based on the controllability and absorbability estimates for lines in the circuit

27. 27 Boolean Satisfiability Given a Boolean formula in CNF (product- of-sums), determine if there is an assignment of variable values that satisfies the formula. Special Cases: 2-SAT: CNF clauses have at most 2 literals 3-SAT: CNF clauses have at most 3 literals

28. 28 Test Generation as a Boolean Satisfiability Problem 1. Gate Representation (AND Gate Example): Express C = AB in conjunctive normal form (CNF). The formula is true iff the values of A, B, and C are consistent with the AND function. 2. Circuit Representation: Logically AND the CNF for each gate. The resulting formula, in CNF, is true all the signal values are consistent. 3. Construct the Boolean difference circuit for the target fault (next slide). 4. Represent the Boolean difference circuit by its CNF formula and set the signal at its output to 1. An input assignment is a test iff it satisfies the CNF.

29. 29 The Boolean Difference Circuit Set F=1 in the circuit CNF. Any satisfying solution to the resulting formula must be a test for the fault. Circuit Under Test (CUT) CUT with Target Fault Inserted F

30. 30 Example For line F sa-1 set F=0, F*=1, and H=1. Then the resulting formula is satisfied by ABC = 0X0 and X01 which are the two tests for the fault. g1 g2 g3 g5 g4 A B C C1 C2 D E F G g1 g3 g5* g4 A B C C1 D E F G* g2 H g6 F*

31. 31 SAT Solvers The DPLL Algorithm: Davis-Putnam [JACM 7(1) 1960], and Davis-Putnam-Longemann-Loveland [CACM May 1962] The line justification step in PODEM [Goel, IEEE TC, March 1981] can also be used as a SAT solver The DPLL algorithm has been enhanced in chaff [Moskewicz et al. DAC 2001] and implemented in mchaff and zchaff, the latter being the most widely known and used. Other SAT solvers to note: GRASP [Marques-Silva and Sakalla, IEEE TC, May 1999], SATO [Zhang, Int. Conf. Automated Deduction, 1997] Generally, the DPLL enhancements try to improve on the original backtrack algorithm through careful engineering of all aspects of the search

32. 32 Basic Davis-Putnam (DP) Backtrack Search for SAT decide() selects an unassigned variable and gives it a value. bcp() identifies any variable assignments required by the current variable state to satisfy f. resolveConflict() undoes implied assignment and backtracks to try the next variable assignment

33. 33 2-SAT Based SAT Solving The boolean satisfiability problem for binary clauses (2-SAT) is polynomially solvable while 3-SAT and the general satisfiability problem is NP-complete. This suggests another heuristic of solving the satisfiability problem: generate and test solutions for the binary clauses against the whole CNF until success.

34. 34 B A’ A B’ AB C DE AB C DE Analyzing 2-SAT Constraints Graphically Step 1 (Construct the implication graph): For each binary clause (A+B) create the subgraph: Step 2: Find the transitive closure graph (TC) of the implication graph.

35. 35 Analyzing 2-SAT Constraints Graphically Step 3 (Analyze the transitive-closure graph): (Contradiction): Both (C’,C) and (C,C’) are in the TC. This implies C must be true and false at the same time. This would occur if the fault was redundant (untestable). (Fixed Signal Values): Only one of (C’,C) and (C,C’) is in the TC this implies C has a fixed binary value. (Identical Signal Values): If both (A’,B) and (A,B’) are in the TC then literals A and B must assume identical signal values. (Excluded Signal Value): If (A’,B) is in the TC and no other edges occur between these four literals, then the combination AB=10 is excluded from the solution.

36. 36 Transitive Closure Test Generation Algorithm (TRAN) * In TRAN test generation is carried out entirely by dynamically updating the TC after each variable assignment and analyzing it. * See, Chakradhar et al. IEEE TCAD, June 1993 and Larrabee, TCAD, January 1992.

37. 37 TRAN Flow Chart Circuit with Fault Transitive Closure Contradiction? Unassigned Variables? Assign value to a variable Transitive Closure Redundant Fault Contradiction? Both values tried? Unassigned Variables? Redundant Fault Assign next value to a variable Y N Test Found N Y NY N Y

38. 38 History of Algorithm Speedups AlgorithmEst. speedupYear D-ALG11966 PODEM71981 FAN231983 TOPS2921987 SOCRATES1574  1988 Waicukauski et al.2189  1990 EST8765  1991 TRAN3005  1993 Recursive Learning4851995 Tafertshofer et al.250571997  Part of Automatic Test Pattern Generation (ATPG) System

39. 39 Combinational Test Generation: Summary Gate level combinational ATPG is a mature technology applicable also to sequential circuits designed with full scan. High fault coverage can be obtained within reasonable time for circuits with hundreds of thousands of gates.

40. 40 Test Generation for Sequential-Circuits

41. 41 Time-Frame Expansion COMB LOGIC C M COMB LOGIC C COMB LOGIC C COMB LOGIC C M M Clock Cycle 0 Clock Cycle 1 Clock Cycle 2 Initial State Final State M Iterative Logic Array (ILA) Representation

42. 42 Time-Frame Expansion Based Test Generation 0 1 0 N -2 -M Time Frames for Fault Excitation and Propagation Time Frames for State Justification The sequential test generation problem is reduced to combinational test generation but for multiple time frames and multiple fault instances. Single Observation: This approach guarantees that by observing the primary outputs at the final time frame, the fault will be detected.

43. 43 Test Generation Example: 5 valued logic =1 & & & & & & Time frame 0Time frame -1 SA0 1 0 1 1 0 0 1 1 Conflict 1

44. 44 Test Generation Example: 9 valued logic =1 & & & & & & Time frame 0Time frame -1 SA0 1/0 1/x 0/1 0/x 1/0 0/x 1/1 1/x 0/x 1/x

45. 45 The “Reset Problem” Synchronizing sequence is a generalized reset for FSMs. It reduces the total uncertainty of the initial state to a known fixed state. However, the reduction in ambiguity for some FSMs may not be fast enough to find the synchronizing sequence by algorithmic means using 3-valued logic. Consider the example machine (from Miczo’s text, p. 273) on the next page

46. 46 Miczo’s Example Machine Input x: 01 S0S1 S3S2 S1S3 S0 It can be verified that (0,1,0,1,0) is the shortest synchronizing sequence Consider, encoding the states using two FFs. What is the state ambiguity after the first 0? What does it say about the complexity of sequential ATPG using 3-value algebra?

47. 47 Multiple Observation Times* Some faults may only be testable by observing outputs at multiple time frames during the application of the test sequence. Consider the following example from the paper: * Pomeranz and Reddy, IEEE TC, May 1992 Input x: 01 AB,0C,0 BC/B,0B,1 CD/A,1A,1 D D,0 Verify that the machine has no reset sequence. If the initial state is unknown the output can be either 0 or 1 in every timeframe. Hence, single observation time does not work. Note: The fault shown is functional affecting state transitions for 0-input in states B and C, as shown.

48. 48 Pair Time Units for Observation 1234 A/A++ A/B++ A/C+++ A/D+++ B/A++ B/B++ B/C+++ B/D+++ C/A++ C/B++ C/C+ C/D+ D/A++ D/B++ D/C+ D/D+ Observation Times for Test Sequence (0000) No single observation time covers all the pairs However, if machine outputs are observed at both time units 2 and 4, all pairs are covered. Hence, independent of the starting state, the fault can be detected only if at least two observations are made.

49. 49 Distinguishing Sequence in Finite State Machine Assume two copies of the same FSM, one starts in state s and the other in state t. A distinguishing sequence for states s and t is any input sequence that produces a different output.

50. 50 Example AB DC 0/0 1/0 0/0 1/1 0/1 1/1 0/1 1/1 What is the shortest distinguishing sequence for states C and D?

51. 51 Product Machine Common inputs Product number of states STG derived from component STGs Represents behavior of concurrent operation of M1 and M2. M1 M2 Product Machine

52. 52 Test as a Distinguishing Sequence of a Product Machine Good Machine Faulty Machine != Assume the good and faulty machines start in state s and t The test is any distinguishing of the product machine If the good and faulty machines do not have a reset state, it may not be possible to find a test with single observation. Product Machine

53. 53 Sequential Circuit ATPG Simulation-Based Methods Contrast with target-fault-based: Main loop starts with an initial sequence of test vectors (could be random or otherwise) The sequence is progressively modified and augmented based on simulation based evaluation of the testability of the current set, until the desired level of coverage is reached. IBM’s SOFTG* is an early example of this approach. * T. J. Snethen, “Simulator-Oriented Fault Test Generator,” Proc. DAC, 1977, pp. 88-93.

54. 54 Other Simulation-Based Methods Contest Directed search Cost functions Genetic Algorithms Spectral Methods Summary

55. 55 Fault-Simulation Based Test Generation* Fault simulator Vector source: Functional (test-bench), Heuristic (walking 1, etc.), Weighted random, random Fault list Test vectors New faults detected? Stopping criteria (fault coverage, CPU time limit, etc.) satisfied? Stop Update fault list Append vectors Restore circuit state Generate new trial vectors Yes No Yes No Trial vectors * See, Agrawal, et al., IEEE-TCAD, 1989

56. 56 Asynchronous Circuit An asynchronous circuit contains unclocked memory often realized by combinational feedback. Almost impossible to build, let alone test, a large asynchronous circuit that is reliable. Clock generators, signal synchronizers, flip-flops are typical asynchronous circuits. Many large synchronous systems contain small portions of localized asynchronous circuitry. Sequential circuit ATPG should be able to generate tests for circuits with limited asynchronous parts, even if it does not detect faults in those parts.

57. 57 Asynchronous Model Clocked Flip-flops Feedback delays Synchronous PIs Synchronous POs System Clock, CK Fast model Clock, FMCK CK Feedback-free Combinational Logic C Combinational Feedback Paths: Feedback set Modeling circuit is Shown in orange. PPOPPI

58. 58 Time-Frame Expansion Time-frame k Time-frame -k+1 Time-frame -k-1 C FMCK C FMCK C FMCK C CK Asynchronous feedback stabilization PI PO Feedback set PPI PPO Feedback set Vector k

59. 59 Asynchronous Example s-a-0 s-a-1 1010 1111 0000 0101 Vectors 1 2 3 4 101101 XX0XX0 101101 101101 Outputs 1 2 3 4 Gentest results: Faults: total 23, detected 15, untestable 8 (shown in red), potentially detectable none Vectors: 4 Sparc 2 CPU time: test generation 33ms, fault simulation 16ms

60. 60 Sequential ATPG Summary Combinational ATPG algorithms are extended: Time-frame expansion unrolls time as combinational array Justification via backward time Nine-valued logic system Unlike combinational ATPG: Completeness not guaranteed using 3-valued logic simulation Some circuits may require 9-valued D-algebra 5-valued is not enough Simulation-based ATPG may be a more attractive alternative Cycle-free circuits (where sequential elements don’t form a cycle through combinational logic) are much easier to test than cyclic circuits Require at most dseq time-frames Always initializable DFT a must for large circuits: Either full or partial scan.