Simplification of Non-Deterministic Multi-Valued Networks

1. Simplification of Non-Deterministic Multi-Valued Networks Alan MishchenkoElectrical and Computer EngineeringPortland State UniversityRobert K. BraytonElectrical Engineering and Computer ScienceUniversity of California, Berkeley ICCAD 2002

2. Overview • What is an ND network? • Basic definitions • Motivation • Defining and comparing ND network behaviors • Node Flexibility and Node Minimization • Experimental observations • Conclusions and future work ICCAD 2002

3. What is an ND network? PO Similar to a Boolean network except • Each node has a single output multi-valued variable • Each node has a non-deterministic relation relating its input and output values. F MV ND PI ICCAD 2002

4. Why consider ND networks? • Don’t cares are a form of non-determinism. They generalize to non-determinism when considering multi-valued logic • Multi-valued domains can be used to explore larger optimization spaces. • ND arises naturally when considering the flexibility of implementing a node in a network • Given a ND relation, the minimum well-defined ND sub-relation is always smaller than the minimum deterministic one ICCAD 2002

5. PO F yj j PI Reminder: Boolean Network • Directed Acyclic Graph • Each node represents a Boolean function • Edge from node j to node k if the function at k depends syntactically on the variable yj at the output of j • Primary inputs (PI) X and outputs (PO) Z • All signals are binary • External specification provides allowed input and output combinations (external don’t cares) ICCAD 2002

6. PO F yn MV ND PI Definition: ND Multi-Valued Network • Network of MV-nodes (PI, PO, internal) • Each node is represented by an MV variable ynwith its own range{0, 1,…, |pn|-1} • Internal node is represented by an MV non-deterministic relation ICCAD 2002

7. a R b Examples: Ternary Relations All relations are well-defined, i.e. for each input minterm there exists at least one output value R1 R2 R3 2 R1 is completely specified (deterministic) R2 is incompletely specified R3 is partially specified, or non-deterministic R1is contained inR2 R2is not contained inR3 ICCAD 2002

8. Overview • What is an ND network? • Basic definitions • Motivation • Defining and comparing ND network behaviors • Node Flexibility and Node Minimization • Experimental observations • Conclusions and future work ICCAD 2002

9. ND Network Behavior • Given an ND network, what is its behavior, • i.e. what is the set of all PI/PO pairs that are related? • this question is not straightforward. • For a deterministic, well-defined network, there is exactly one PO vector for each PI vector • however, if there are some external don’t cares, then there may be several PO vectors for a PI vector, • but don’t cares are well understood. ICCAD 2002

10. SS NSC ND NS Det ND Network Behaviors (PI/PO Pairs) • Normal Simulation (NS) • Normal Simulation made Compatible (NSC) – will not be discussed • Set Simulation (SS) • similar to X valued simulation where X ={0,1} Note: all these become the same when the network is deterministic. ICCAD 2002

11. The NS-behavior is the set of all PI/PO vectors that can be obtained this way.is in general a MV Boolean relation Normal Simulation • Network is evaluated in topological order • At each node its fanins have a specific vector of values. • The relation at the node determines a set of possible output values of that node • One of these is chosen randomly and broadcast to all the fanouts ICCAD 2002

12. fanouts 2 2 2 node with a non-deterministic relation 1 3 1 fanins Normal Simulation {0,2} 2 ICCAD 2002

13. The SS-behavior is the set of all PI/PO vectors in the cross product of the PO sets that can be obtained this way. can be expressed using Set Simulation • Done in topological order. • On each signal a set of values is obtained • At each node a vector of fanin sets is known. • The output set of values for a node is the union of the sets obtained for all fanin vectors in the cross product of the fanin sets ICCAD 2002

14. = {1,3} = {1,2,4} {1,3} {1,3} {0,2} {0,2} {0,2} {3} {1} {1} Set Simulation PO2 PO1 {1,2,4} {1,4} {0,1} • 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 {0,2} fanins ICCAD 2002

15. Comparisons • is a general MV Boolean relation • relatively hard to compute and store • . can be computed for each output: . It is outputsymmetric Boolean relation. • . can be obtained by elimination in topological order • SS can be considered as an easy-to-compute over-approximation of normal simulation NS. ICCAD 2002

16. Computing RNS – input determinization • At each ND node introduce one MV parameterpi with the same range as the node output. • Relation at node i is replaced by • pi controls the output value of node i • the operatorm is a special BDD projection operator, defined by Bill Lin, that projects onto the smallest allowed output value. • RNS can be obtained by eliminating all internal nodes and existentially quantifying all parameters { pi }. ICCAD 2002

17. External Specification • Can be specified by • The initial network plus don’t cares • e.g. in Boolean networks, we can give external don’t cares, one set for each output. • A separate specification (network or BDD or other) • Notation: • Requirement: one behavior conforms ICCAD 2002

18. Conformity with External Specification • Can use any one of the behaviors • Just be consistent • For example, we may have but If we use consistently there is no problem. • Ultimately, in most applications we want a final deterministic network. • If any behavior conforms, then it contains only correct deterministic ones ICCAD 2002

19. Overview • What is an ND network? • Basic definitions • Motivation • Defining and comparing ND network behaviors • Node Flexibility and Node Minimization • Experimental observations • Conclusions and future work ICCAD 2002

20. Minimizing a Node – Computing the Flexibility at a Node Definition. A flexibility at node  is a relation such that replacing at  any well-defined deterministic relation contained in implies that the resulting network conforms to the external specification. Definition. The maximum possible flexibility at a node is called its complete flexibility (CF). ICCAD 2002

21. Computing the Global CF ND network Called the global B - CF of the node ICCAD 2002

22. Imaging into the Local Space Yi Called the B - CF of the node ICCAD 2002

23. Properties of Flexibilities • If a current network “B-conforms”, Be{NS,NSC,SS }, then any well-defined deterministic function contained in is acceptable at node j. • For NS or NSC, any ND relation will also be acceptable. • But for SS, it is possible that an ND relation contained in can cause the network to not conform (important point) ICCAD 2002

24. M is the number of the input minterms Vis the size of the output range. ti is the number of output values in the relation for input minterm mi MeasuringFlexibility F is equal to 0% for completely specified functions and 100% for relations that take all values in any minterm. Examples: M=6, V=3 R1 R2 R3 T = 10 T = 12 T = 6 F= 0% F= 33% F= 50% ICCAD 2002

25. Amount of Flexibility (SDC, CODC, SS-CF) ICCAD 2002

26. Node Simplification • Compute and use complete flexibility (CF) to simplify the node. Recall: • CF in global space: • CF in local space: • Use to optimize MV-SOP (heuristic, exact) at node j We will look at how to find the smallest well-defined SOP representation contained in a given ND relation ICCAD 2002

27. Representing an ND relation Definition: The i - set of an ND relation is a binary function that is 1 for each minterm that can output value i Definition:A minimum SOP representation of an ND relation is a well-defined sub-relation where all the i - sets are represented by SOPs and the total number of cubes is minimum. ICCAD 2002

28. Finding minimum deterministic SOP representation • A deterministic SOP is never smaller than the smallest ND representation. • There is no known algorithm for finding the minimum deterministic representation. • we have a few heuristic ones • In contrast, there is a method for finding the smallest ND representation. ICCAD 2002

29. P2 P1 P3 P0 Quine-McCluskey type exactND SOP relation minimization • For each i-set, generate all its primes, Pi • Form covering table with • one column for each pj in Pi for all i • one row for each minterm in the input space • Solve minimum covering problem • Primes chosen from Pkis the cover for kthi-set. all minterms ICCAD 2002

30. Overview • What is an ND network? • Basic definitions • Motivation • Defining and comparing ND network behaviors • Node Flexibility and Node Minimization • Experimental observations • Conclusions and future work ICCAD 2002

31. Experimental Setup • These ideas have been implemented in a system, MVSIS • The SS behavior has been used throughout in the experiments. • it is the easiest to use computationally • behavior can be expressed locally at each node as a BDD of the PI. (SS-behavior is output symmetric) ICCAD 2002

32. Experimental ObservationsSS Behavior • Conformity is rarely lost but it does happen. This usually happens during node minimization. • If we use an ND relation at the minimized node, then conformity is not guaranteed (only deterministic SOP guarantees conformity using SS) • Often conformity is automatically regained by minimizing the next node. • If the CF at the next node is well defined, this means that the network can be brought back to conformity. • If it is not well defined, we leave the node relation alone and move to the next node. • We have never experienced a final network that does not conform to the external specification. ICCAD 2002

33. Future Work We believe NSC behavior will be superior. • need to solve computation efficiency problems • equivalent to elimination in reverse topological order • means that intermediate variables have to be used (rather than only PI) • means that it is easier to maintain conformity. • implies that NSC-CF contains more flexibility than SS-CF • however, elimination can cause non-conformity ICCAD 2002

34. The End ICCAD 2002