Lecture 11alt Advances in Combinational ATPG AlgorithmsATPG: A Boolean Satisfiability ProblemSAT is NP-CompleteSearch for a SolutionExample: f = 1FAN: Fujiwara and Shimono (1983)PODEM Makes Unwise Signal AssignmentsUnique Sensitization of FAN with No SearchHeadlinesContrasting Decision TreesMultiple BacktraceAND Gate Vote PropagationMultiple Backtrace Fanout Stem VotingPODEM Fails to Determine Unique SignalsFAN -- Early Determination of Unique SignalsTOPS: Dominators Kirkland and Mercer (1987)SOCRATES: Learning (1988)Improved Unique Sensitization ProcedureConstructive DilemmaEST: Search Space Learning (Giraldi and Bushnell)Fault B sa1Fault h sa1SummaryCopyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 1Lecture 11altAdvances in Combinational ATPG AlgorithmsLecture 11altAdvances in Combinational ATPG AlgorithmsBranch and Bound SearchFAN – Multiple Backtrace, head lines (1983)TOPS – Dominators (1987)SOCRATES – Learning (1988)EST – Search space learning (1991)ATPG Performance improvementsSummaryCopyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 2ATPG: A Boolean Satisfiability ProblemATPG: A Boolean Satisfiability ProblemCUTCUTwith fault f(a,b,c) = 1TestVector(a,b,c)Copyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 3SAT is NP-CompleteSAT is NP-Completeaccbb1110000 0cc01f a b cfBinary DecisionDiagram (BDD)Copyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 4Search for a SolutionSearch for a SolutionProblem: Given a value of a Boolean function of binary variables, find values of the variables.Solution: Starting at the root, enumerative traversal of the binary decision diagram (BDD) until a solution is found.BDD is a search tree – search consists ofBranch : Set an untried value for a variable–Backtrack to previous branching point if there is no untried valueStop if solution found, or backtracked to root without untried valuesOr, bound search tree for future traversals if solution is impossible and backtrack to previous branching point (some variable orderings may lead to early bounding)Or, continueCopyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 5Example: f = 1Example: f = 1accbb1110000 0cc01f a b cfBinary DecisionDiagram (BDD)boundCopyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 6FAN: Fujiwara and Shimono(1983)FAN: Fujiwara and Shimono(1983)New concepts:Unique sensitizationStop Backtrace at head linesMultiple BacktraceCopyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 7PODEM Makes Unwise Signal AssignmentsPODEM Makes Unwise Signal AssignmentsBlocks fault propagation due to assignment J = 0Copyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 8Unique Sensitization of FAN with No SearchUnique Sensitization of FAN with No SearchFAN immediately sets necessary signals to propagate faultPath over which fault is uniquely sensitizedCopyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 9HeadlinesHeadlinesHeadlines H and J separate circuit into 3 parts, for which test generation can be done independentlyCopyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 10Contrasting Decision TreesContrasting Decision TreesPODEM decision treeFAN decision treeCopyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 11Multiple BacktraceMultiple BacktraceFAN – breadth-firstpasses – 1 timePODEM –depth-firstpasses – 6 timesCopyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 12AND Gate Vote PropagationAND Gate Vote PropagationAND GateEasiest-to-control Input: # 0’s = OUTPUT # 0’s# 1’s = OUTPUT # 1’sAll other inputs:# 0’s = 0# 1’s = OUTPUT # 1’s[5, 3][5, 3][0, 3][0, 3][0, 3]00000111Copyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 13Multiple Backtrace Fanout Stem VotingMultiple Backtrace Fanout Stem VotingFanout Stem -- # 0’s = Σ Branch # 0’s, # 1’s = Σ Branch # 1’s[5, 1][1, 1][3, 2][4, 1][5, 1][18, 6]Copyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 14PODEM Fails to Determine Unique SignalsPODEM Fails to Determine Unique SignalsBacktracing operation fails to set all 3 inputs of gate L to 1Causes unnecessary searchsa1Copyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 15FAN -- Early Determination of Unique SignalsFAN -- Early Determination of Unique SignalsDetermine all unique signals implied by current decisions immediatelyAvoids unnecessary searchsa1Copyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 16TOPS: DominatorsKirkland and Mercer (1987)TOPS: DominatorsKirkland and Mercer (1987)Dominator of g – all paths from g to PO must pass through the dominatorAbsolute -- k dominates BRelative – dominates only paths to a given POIf dominator of fault becomes 0 or 1, backtrackCopyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 17SOCRATES: Learning (1988)SOCRATES: Learning (1988)Static and dynamic learning:a = 1 f = 1 means that we learn f = 0 a = 0 by applying the Boolean contrapositive theoremSet each signal first to 0, and then to 1Discover implicationsLearning criterion: remember f = vf only if:f = vf requires all inputs of f to be non-controllingA forward implication contributed to f = vf Copyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 18Improved Unique Sensitization ProcedureImproved Unique Sensitization ProcedureWhen a is only D-frontier signal, find dominators of a and set their inputs unreachable from a to 1Find dominators of single D-frontier signal a and make common input signals non-controllingCopyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 19Constructive DilemmaConstructive Dilemma[(a = 0) (i = 0)] [(a = 1) (i = 0)] (i = 0)If both assignments 0 and 1 to a make i = 0, then i = 0 is implied independently of a   Copyright 2001, Agrawal & BushnellVLSI Test: Lecture 11alt 20EST: Search Space Learning (Giraldi and Bushnell)EST: Search Space Learning (Giraldi and Bushnell)E-frontier – partial circuit functional decompositionEquivalent to a node in a BDDCut-set between circuit part with known labels and part with X signal labelsEST learns E-frontiers during ATPG and stores them in a hash tableDynamic programming – when new decomposition generated from implications of a variable assignment, looks it up in the hash tableAvoids repeating a search already conductedTerminates search when