Theorems on Redundancy Identification Vishal J. Mehta Vishwani D. Agrawal Michael L. Bushnell Rutgers University, ECE Dept. Piscataway, New Jersey, USA 4/29/2019 Mehta et al.: Redundancy theorems

Mehta et al.: Redundancy theorems Talk Outline Introduction Problem statement Prior work Primary contribution Completion of previous implementation Fixed-value theorem Stem unobservability theorems Results and conclusion 4/29/2019 Mehta et al.: Redundancy theorems

Mehta et al.: Redundancy theorems Problem Statement For identifying logic redundancy: Implement the implication graph and transitive closure procedures with direct and partial implications. Enhance transitive closure for contrapositive implications and fixed-valued signals. Find new ways to Identify unobservable fanout stems. 4/29/2019 Mehta et al.: Redundancy theorems

Prior Work on Redundancy Automatic Test Pattern Generation (ATPG) Uses exhaustive test pattern generation to determine whether or not a target fault has a test. Can identify all redundancies -- exponential complexity. Boolean satisfiability methods use logic implications: Chakradhar et al., Larrabee, Henftling et al., Zhao et al., etc.. Testability analysis (fault-independent) Mostly approximate, linear complexity Raitu et al., Goldstein, Seth and Agrawal, etc. Fault-independent redundancy identification Implication analysis identifies all or a subset of redundant faults -- polynomial complexity (empirically linear). Agrawal et al., Gaur et al., Iyer and Abramovici, etc. I will talk about fault—independent techniques only due to timing constraints. Other techniques are given in details in thesis. 4/29/2019 Mehta et al.: Redundancy theorems

Fault-Independent Methods Iyer and Abramovici (IEEE-TC, June 1996) use implications to find redundant faults whose tests require contradictory values on a signal. Agrawal et al. (ATS’96) use implication graph, introduce observability variables, and use transitive closure for redundancy identification. Gaur et al. (DELTA’02) include anding nodes to represent higher-order implications among signals and observability variables. 4/29/2019 Mehta et al.: Redundancy theorems

Redundancy Identification by Transitive Closure s-a-0 e s-a-0 b Oc Od d Circuit with two redundant faults TC graph (some nodes and edges not shown) Implication Partial implication Transitive closure edge 4/29/2019 Mehta et al.: Redundancy theorems

Mehta et al.: Redundancy theorems Method Summarized Obtain an implication graph from the circuit topology and compute transitive closure. There are 8 different conditions on the basis of which a fault is identified to be redundant. Examples: If node c implies c then line c is fixed at 0 and s-a-0 fault on it is redundant. If node Oc implies Oc then line c is unobservable and both s-a-0 and s-a-1 faults on it are redundant. These conditions obey the contrapositive rule. 4/29/2019 Mehta et al.: Redundancy theorems

Mehta et al.: Redundancy theorems Talk Outline Introduction Problem statement Prior work Primary contribution Completion of previous implementation Fixed-value theorem Stem unobservability theorems Results and conclusion 4/29/2019 Mehta et al.: Redundancy theorems

Mehta et al.: Redundancy theorems Motivation Incomplete implementation (Gaur et al.) Only few anding nodes implemented Some direct implications missing Not all contrapositive relations determined by transitive closure Effect of fixed-valued nodes not included in transitive closure No observability relation across fanouts Redundancies due to stem unobservability not identified 4/29/2019 Mehta et al.: Redundancy theorems

Completion of Previous Implementation Only one of the possible (n+1) signal anding nodes were implemented by Gaur et al. None of the possible n(n+1) observability anding nodes were implemented. Some direct implications for observability variables were not implemented. 4/29/2019 Mehta et al.: Redundancy theorems

Mehta et al.: Redundancy theorems Example Circuit a1 s-a-0 a d Oa1 a c d b s-a-1 b s-a-0 e b1 c Note: only some nodes and edges are shown. Gaur et al. identified b1 s-a-1 and d s-a-0, but could not identify a1 s-a-0, because of unimplemented anding node for gate d. 4/29/2019 Mehta et al.: Redundancy theorems

Mehta et al.: Redundancy theorems Fixed-Value Theorem If a Boolean variable in the implication graph is fixed to a true (false) value then there exist unconditional edges from all other nodes in the graph to the node representing the true (false) state of the fixed variable. 4/29/2019 Mehta et al.: Redundancy theorems

Mehta et al.: Redundancy theorems Example Circuit e f g s-a-1 e f g s-a-0 Note: Only some edges are shown Initially only 2 out of 7 redundant faults were identified. After the implementation of node fixation concept, g-(s-a-1) was identified. With stem unobservability theorems, rest of the 4 redundant faults were identified. 4/29/2019 Mehta et al.: Redundancy theorems

Stem Unobservability Theorem 1 A fanout stem is unobservable, if each signal in its dominator set assumes a constant value and: either the fanout stem does not hold a constant value or the fanout stem holds a constant value and, in spite of any local change in the stem signal, the dominator set values do not change. Notes: A local change of a signal only affects the portion of the circuit between that signal and POs. Dominator set is the set of signals through which a signal in the circuit should pass in order to reach the primary output. 4/29/2019 Mehta et al.: Redundancy theorems

Mehta et al.: Redundancy theorems Example Circuit 1 unobs. stem fixed dom. a b c d For the fanout stem b, the dominator signal d is fixed to 1. As b is not fixed, Theorem 1 identifies b as unobservable stem. 4/29/2019 Mehta et al.: Redundancy theorems

Mehta et al.: Redundancy theorems A fanout stem is unobservable, if each signal in its dominator set is unobservable and: either the stem does not hold a constant value or the stem holds a constant value and, in spite of any local change in the stem signal, the unobservable status of the dominator set remains unchanged. Note: A lemma by Iyer and Abramovici is a special case of Theorem 2. 4/29/2019 Mehta et al.: Redundancy theorems

Mehta et al.: Redundancy theorems Example Circuit a b c 0(fixed) d e b1 unobs. b2 unobs. Fixed value 0 on line c makes the fanout branches b1 and b2 of stem b unobservable. As b is not fixed, Theorem 2 identifies b as an unobservable stem. Note: Stem a is unobservable by Theorem 1, which does not classify stem c as unobservable. 4/29/2019 Mehta et al.: Redundancy theorems

Mehta et al.: Redundancy theorems Talk Outline Introduction Problem statement Prior work Primary contribution Completion of previous implementation Fixed-value theorem Stem unobservability theorems Results and conclusion 4/29/2019 Mehta et al.: Redundancy theorems

Benchmark Results Identified redundant faults and computation time Circuit C5315 c2670 s9234 s s13207 Total Flts. 5350 2747 6927 9 9815 ATPG Flts. CPU s 59 32.3 115 95.2 452 803.7 TCSTEM Flts. CPU s 58 3.9 82 4.0 233 106.0 TCAND Flts. CPU s 32 3.4 25 1.5 135 11.2 FIRE Flts. CPU s 20 2.8 29 1.5 165 20.6 151 806.5 77 158.8 60 13.6 55 23.2 Coded in C++. Used GNU tools for code compilation, debugging and profiling. Results are obtained on ISCAS’85 and ISCAS’89 benchmark circuits. ATPG: TRAN, Chakradhar et al., IEEE-TCAD’93, Sparc 5 TCSTEM: This work, Sparc 5 TCAND: Gaur et al., DELTA’02, Sparc 5 FIRE: Iyer and Abramovici, IEEE-TVLSI’96, Sparc 2 4/29/2019 Mehta et al.: Redundancy theorems

Mehta et al.: Redundancy theorems Limitations of Method s-a-1 s-a-0 c d e f g h b a a b e d c f unobs. stem Example 1 Example 2 Example 1: None of the stem unobservability theorems can identify stem a as unobservable because the dominator set is neither fixed nor unobservable. Example 2: e s-a-1 is redundant because f=g=1 require b=0, which implies e=1. Because f=1 and g=1 are separately treated in the transitive closure and each has multiple satisfying choices, the essential requirement b=0 is not found. The method fails to find this redundancy. 4/29/2019 Mehta et al.: Redundancy theorems

Mehta et al.: Redundancy theorems Conclusion Partial implications, fixed-value theorem and stem unobservability theorems improve the process of redundant fault identification better than any other known fault-independent technique. Checking for the contrapositive rule to update transitive closure may have benefits. A demonstrated limitation of stem unobservability theorems can be improved upon. Possible ways to find essential signal assignments caused by combinations of multiple signals may provide further improvements. 4/29/2019 Mehta et al.: Redundancy theorems

Mehta et al.: Redundancy theorems Future Work Various applications of the TC technique can be explored: Identifying equivalent faults Checking equivalence of combinational circuits. 2 and 3 valued logic simulators. 4/29/2019 Mehta et al.: Redundancy theorems

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