Process Variation

1. Prime Indicants: A Synthesis Method for Indicating Combinational Logic BlocksWill TomsSchool of Computer ScienceUniversity of Manchester

2. Process Variation • Process Variation at 45nm: • Transistor Delay: 20-23% • Interconnect Delay: 10% • Self-Timed circuits more robust to variations in propagation delay • No external timing assumptions • Indication - status of internal signals can be determined by the outputs of the gates

3. Indicating Combinational Logic • Validity must be encoded into the datapath • DI (unordered) encoding (dual-rail/m-of-n) • Each data word transmitted explicitly • Need to return to known state (Four-phase/RTZ) • Functions difficult to specify: • Functions completely specified • All minterms must be enumerated • Impractical for large datapath operations

4. Desynchronisation • Can construct large indicating datapaths from synchronous circuits by expanding gates into dual-rail equivalents

5. Desynchronisation • Desynchronised networks can be optimised by reducing indication: • Relaxation (Jeong, Zhou): • Removing redundant indication (fanout of gates) • Relative-Timing (Chelcea): • Applying timing constraints to avoid indication • Validity (NCL-X): • Adding additional signals for indication

6. Desynchronisation • Properties of self-timed datapaths differ from conventional datapaths: • Invertions are free (in 1-of-N codes) • N-input complex gates (approximately) same cost as N-input simple gate • Mapping desynchronised networks to DI-codes other than dual-rail (possibly 1-of-4) expensive

7. Block-Level Approach • Self-timed datapaths constructed from complex function-blocks: • Small blocks consisting of up to 10 binary inputs and outputs

8. Block-Level Approach • Existing optimisations applied between function blocks • Distribute indication between the outputs of function blocks • Can use any encoding between function blocks

9. Block-Level Approach • How to synthesise arbitrary indicating function blocks?

10. Indication • In four-phase indicating logic, transitions between spacer and data-values form allowed-transition sets (ATS): • Each ATS describes all the possible states of the inputs (or outputs) within the transition 0000  0101 = {0000,0100,0001,0101}

11. Indication • The adjacent transitions of an ATS are the states distance 1 from final state • Each adjacent transition has an associated variable ε that transitions 0000  0101 = {0100(x4),0001(x2)} • Function f indicates the input transitions of an ATS if for each adjacent transition f is dependent on variable ε f(ε=0) ≠ f(ε=1)

12. Indication • In order for a function block to be indicating for each adjacent transition in each ATS, at least one output function must be dependent on variable ε • Checking Indication is expensive: • 8 input/8 output-function block has 256 ATS each with 8 adjacent transitions = 16,384 (256*8*8) dependency checks to verify an implementation

13. Indication Architecture • Need an architecture that is correct by construction • Implement each output function as a sum-of-products where each product is given by ε(a,b) (the inputs that transition in ATS (a-b)) • Each product implemented by a C-element to cover return-to-zero transitions

14. Indication Architecture • Each term ε(a,b) is mutually-exclusive and so function indicates all of its input transitions. a0 C C b0 ci0 f a0 C b1 ci1

15. Indication Architecture • Large • Slow • As function block has several outputs, the number of outputs that indicate each transition is often >1 • Can optimise architecture by reducing the number of outputs that indicate each input transition

16. Prime Implicants • Implicant of a function, f, is a function p where: p f • A Prime Implicant is an implicant not contained in any other • Minimum cost Sum-of-Products implementation must always consist of a sum of prime implicants

17. 1 0 1 0 100 110 0 010 0 0 0 000 0 0 101 1 1 111 001 1 1 0 011 0 _ _ f = x1x2x3 + x1x2x3 + x1x2x3 Prime Implicants • Prime Implicants generated from minterms by consensus: f = x1x2 + x2x3

18. Prime Implicants • Function constructed by minimum cost covering of prime implicants • Unate Covering Problem (UCP)

19. Optimisation of Indicating Logic • Removing literals from function terms reduces indication • Optimising one function can prevent further optimisations of other functions • Need to consider all functions together • May not be possible to construct indicating solution from prime implicants

20. Optimisation of Indicating Logic • Two phase approach: • Construct minimum cost indicating cover for each function • Determine un-indicated input transitions and reduce existing function terms to cover them

21. Indicant • In indicating function terms have two roles: • Implicate AND Indicate • An indicant is an implicant that indicates the transitions on its literals • The indicants of a function must be mutually-exclusive • A function block constructed from indicant covers of its functions will indicate all internal transitions and some input transitions • Minimum cost implementation of Function Block

22. Indicant Cover • Construct the lowest cost mutually-exclusive cover for each function • Unate Covering Problem: • Determine prime implicants of each function • Enumerate all the expanded implicants • Determine non-overlapping cover

23. Unindicated Inputs • Can easily determine unindicated input transitions • Literals not present in the indicants of any function • Reduce Indicants by re-inserting the literals • Partitions Indicant into multiple indicants • All functions must remain unate • Need to ensure that new indicants: • Cover exactly the same minterms • Are mutually-exclusive

24. Unindicated Literals • Indicant a0 covers 6 minterms: a0b0c0, a0b0c1, a0b0c2, a0b1c0, a0b1c0, a0b1c1, a0b1c2 • Reducing by literal b0 results in two indicants a0b0: {a0b0c0, a0b0c1, a0b0c2} a0b1: {a0b1c0, a0b1c1, a0b1c2} • Reducing by literal c0 results in three indicants a0c0: {a0b0c0, a0b1c0}, a0c1: {a0b0c1, a0b1c1} a0c2: {a0b0c2, a0b1c2}

25. Unindicated Literals • Reducing the indicants by more than one literal multiplies the number of indicants • Reducing Indicant a0 by b0and c0 gives 6 indicants: a0b0c0, a0b0c1, a0b0c2, a0b1c0, a0b1c0, a0b1c1, a0b1c2

26. Unindicated Literals • Use UCP to determine the lowest cost reductions that will cover all of the unindicated inputs • Cost of a reduction can change depending on selection of other reductions • Eliminates several reduction strategies • But distributes the reductions between functions

27. Offset Optimisation • Initial function implementations use C-elements • Can distribute the indication of the RTZ transitions throughout the indicants • Can use conventional UCP • A range of strategies introduced that target generalised C-elements or AND-gates

28. Results • Use clustering algorithm to create suitable-sized function blocks from ISCAS Benchmarks • Results show the effectiveness of optimisation • Literal Count • Not a direct comparison with desynchronisation techniques • Techniques fit into desynchronisation framework

29. 99.1% Results Approx Equal Literal Count Significantly Reduced

30. Summary • Developed new technique to synthesis arbitrarily encoded indicating function blocks • Large reduction in literal count over initial specification • Can be employed within desynchronisation framework

31. VERDAD • New project to look at incorporating verification techniques in self-timed synthesis operations • Inconjunction with Newcastle University. • Funded by EPSRC • PhD Studentship available