1. Why consider ND-MV networks? Multi-Values: Multi-valued domains can be used to explore larger optimization spaces. Several interesting direct applications – software, asynchronous, … Non-Determinism: Is useful for compact representations A minimum SOP ND-relation is often much smaller than a minimum deterministic one A minimum SOP ND-relation is often much smaller than a minimum deterministic one Don’t cares are a limited form of non-determinism. when considering multi-valued logic, they generalize to non-determinism when considering multi-valued logic, they generalize to non-determinism ND arises naturally when considering the flexibility of implementing a node in an MV network

2. Definition: ND Multi-Valued Network Directed Acyclic Graph Directed Acyclic Graph Primary inputs (PI) X and outputs (PO) Z Primary inputs (PI) X and outputs (PO) Z External specification provides allowed input and output combinations External specification provides allowed input and output combinations j X PIs - X Z POs - Z yjyj

3. Definition: ND Multi-Valued Network Each node has a single output represented by an MV variable y j with range {0, 1,…, |P j |-1} Each node has a single output represented by an MV variable y j with range {0, 1,…, |P j |-1} Each internal node is represented by an MV non-deterministic relation, R Each internal node is represented by an MV non-deterministic relation, R –single MV output Edge exists from node j to node k if relation R at node k depends syntactically on the variable y j at the output of j Edge exists from node j to node k if relation R at node k depends syntactically on the variable y j at the output of j MV ND X PIs - X Z POs - Z ynyn R

4. Example: Ternary Relations R 1 is completely specified (well-defined and deterministic) R 2 is incompletely specified (well-defined with don’t cares) R 3 is partially specified, or non-deterministic R 1 is contained in R 2 R 2 is not contained in R 3 b/a 01 002 100 210 0100,1,22 100 20,1,20 0100,11,2 10,10 20,1,20,1 R1R1R1R1 R2R2R2R2 R3R3R3R3 R a b 2 All relations are well-defined, i.e. for each input minterm there exists at least one output value 2 3 3

5. ND Network Behavior Given an ND network, what is its behavior? Given an ND network, what is its behavior? –behavior is the set of all PI/PO vector pairs that are related? i.e. is in general a multi-output MV relation R(X,Z ). i.e. is in general a multi-output MV relation R(X,Z ). –for a deterministic, well-defined network, there is exactly one PO vector for each PI vector

6. ND Network Behaviors (PI/PO Pairs) Normal Simulation (NS) Normal Simulation (NS) –most realistic one Normal Simulation made Compatible (NSC) Normal Simulation made Compatible (NSC) –one output at a time Set Simulation (SS) Set Simulation (SS) –like X-valued (or “3-valued”) simulation

7. 0 0 0 Normal Simulation (NS) fanins POs node with a non-deterministic relation 3 1 1 2 2 2 {0,2}2 PI/PO relation contains 3 1 1 / 2 1 0 2 1 0 0 0 1 2 PI/PO relation contains 3 1 1 / 2 1 0 3 1 1 / 0 1 2

8. Normal Simulation made compatible (NSC) fanins 3 1 1 2 2 2 {0,2}2 PO 1 PO 2 2 1 0 3 3 = 2 0 0 1 = 1 PI/PO relation contains 3 1 1 / 2 1 3 1 1 / 2 3 3 1 1 / 4 1 3 1 1 / 4 3 It is the cross product of all PO sets. Thus the MV relation for the network is output-symmetric 0 4 2 3 PO 1 PO 2 0 1

9. Set Simulation (SS) fanins {3} {1} {1} {0,2} PO 1 PO 2 {1,2,4}{0,2}{0,2}{0,2}{1,4}{0,1} {1,3}{1,3} = {1,2,4} = {1,3} PI/PO relation contains 3 1 1 / 1 1 3 1 1 / 1 3 3 1 1 / 2 1 3 1 1 / 2 3 3 1 1 / 4 1 3 1 1 / 4 3 It is the cross product of all PO sets. Thus the MV relation for the network is output- symmetric

10. Comparisons is a general MV Boolean relation is a general MV Boolean relation –relatively hard to compute and store and can be computed for each output and and can be computed for each output and can be obtained by elimination in reverse topological order can be obtained by elimination in reverse topological order can be obtained by elimination in topological order can be obtained by elimination in topological order

11. External Specification Can be specified by Can be specified by –The initial network plus compatible don’t cares e.g. in Boolean networks, we can give compatible external don’t cares, one set for each output. e.g. in Boolean networks, we can give compatible external don’t cares, one set for each output. –A separate specification (ND-network or BDD or other) Requirement: Requirement:

12. Network Manipulations Eliminate a node Eliminate a node Optimize a node Optimize a node Decompose/extract a node Decompose/extract a node Analyzed for SS and NSC

13. Eliminating a node into a fanout If i has been eliminated in all of its fanouts, it can be removed from the network i is eliminated into k k YiYi ykyk i yiyi i k YkYk YiYi yiyi ykyk

14. Elimination SS: Eliminating a node can never increase the SS behavior of a network. NSC: Eliminating a node can increase a network’s NSC behavior if and only if the node is ND and ND and has reconvergent fanout. has reconvergent fanout. ND node

15. Decomposition/Extraction NSC: The NSC-behavior of a network can’t increase due to an decomposition/extraction SS: The SS-behavior of a network is can increase only if the non-disjoint variables of the decomposition have an ND node in their TFIs. Decompose B B A B’ A C

16. Minimizing a Node – Computing the Flexibility at a Node Definition. A flexibility is a node relation such that any well-defined sub-relation used at the node leads to a network that conforms to the external specification. Definition. The complete flexibility (CF) is the maximum flexibility possible at a node.

17. Computing the Global CF for a Behavior of type B There is a slight modification of this for SS-behavior

18. Computing CF - Imaging into the Local Space YiYi

19. Properties of CFs Theorem: If a network conforms, then any well-defined relation contained in is acceptable at node j, for Note: the sub-relation can be non-deterministic.

20. Example mvsis 115> mfs -k m1 Original MV Relation of Node. 0 1 2 3 4 +----------+----------+----------+----------+----------+ 0 | --2----- | ---3---- | ---3---- | -1------ | ---3---- | +----------+----------+----------+----------+----------+ 1 | ------6- | ---3---- | -------7 | -----5-- | -------7 | +----------+----------+----------+----------+----------+ 2 | ------6- | ---3---- | -------7 | -----5-- | -------7 | +----------+----------+----------+----------+----------+ 3 | ----4--- | 0------- | ----4--- | ----4--- | ----4--- | +----------+----------+----------+----------+----------+ Derived CF Relation of Node. 0 1 2 3 4 +----------+----------+----------+----------+----------+ 0 | 012345-- | 012345-- | 01234567 | 012345-- | -1-3-5-- | +----------+----------+----------+----------+----------+ 1 | ------6- | -1-3-5-- | -------7 | -1-3-5-7 | ------67 | +----------+----------+----------+----------+----------+ 2 | ------6- | -1-3-5-- | -1-3-5-7 | -1-3-5-7 | ------67 | +----------+----------+----------+----------+----------+ 3 | 0-2-4--- | 0-2-4--- | 0-2-4-6- | ----4--- | 012345-- | +----------+----------+----------+----------+----------+ Definition: An i-set is the set of minterms which can produce value i

21. Node Simplification Problem: find the smallest well-defined SOP representation contained in CF –Size is measured by the total number of cubes in all i- set SOPs Exact Algorithms: Deterministic – There is no known algorithm for this case. Non-deterministic – A Quine-McCluskey type method exists

22. P0P0 P1P1 P2P2 P3P3 all minterms Quine-McCluskey type exact ND SOP relation minimization For each i -set, generate all its primes, P i For each i -set, generate all its primes, P i Form one covering table with Form one covering table with –one column for each p j in P i for all i –one row for each minterm in the input space Solve minimum covering problem Solve minimum covering problem –Primes chosen from P k is the cover for k th i -set.

23. Comparing Changes in Behaviors OperationNSC-behaviorSS-behavior elimination Can increase Can’t increase node minimization Can’t increase decomposition Can increase node flexibility moreless Nothing’s perfect

24. Experimental Setup These ideas have been implemented in a system, MVSIS 2.0 (source code released May 31, 2003.) These ideas have been implemented in a system, MVSIS 2.0 (source code released May 31, 2003.) The SS behavior has been used primarily throughout. The SS behavior has been used primarily throughout. –SS is the computationally efficient Some experiments with using NSC behavior Some experiments with using NSC behavior

25. Experimental Observations SS Behavior Conformity was rarely lost but it did happen. Conformity was rarely lost but it did happen. –This is because we have not modified decomposition yet. Conformity can be regained by minimizing nodes using CFs. Conformity can be regained by minimizing nodes using CFs. –Theorem: If the CF at a node is well defined, then using any well-defined sub-relation contained in CF, brings the network back to conformity for the outputs in the node’s TFO.

26. Status and Future Work Status: Initial implementation of MVSIS 2.0Initial implementation of MVSIS 2.0 Source release May 31, 2003Source release May 31, 2003 3-5x faster than SIS3-5x faster than SIS Experiments with using NSC behaviorExperiments with using NSC behavior Provides only about 1% more flexibility than SS,Provides only about 1% more flexibility than SS, Is computationally much more complexIs computationally much more complexFuture: Develop efficient Boolean optimization algorithms and explore their common computational coreDevelop efficient Boolean optimization algorithms and explore their common computational core Add sequential synthesisAdd sequential synthesis Add technology mappingAdd technology mapping