The family of Setting: notation in , out STG: Furber and Day, sect - PowerPoint PPT Presentation

The family of Setting: notation in , out STG: Furber and Day, sect 6 4-phase latch controllers Graham Birtwistle Ken Stevens DCS, Sheffield ECE, Utah ✲ ✲ ✲ lr ↑ lr ↓ la ↑ la ↓ ✶ ✻ LOGIC FF/LATCH dIN dOUT Async 2008 (April) Newcastle ❄ ❘ ❄ ✲ ✲ ✲ ✲ ✲ ✻ ✻ a ↑ lt ↑ b ↑ a ↓ lt ↓ b ↓ ② ✯ lt open/closed A systematic way of studying the design- ❄ ❄ ✲ ✲ lr rr space for untimed, bundled, 4-phase latch ✲ ✲ ✲ rr ↑ ra ↑ rr ↓ ra ↓ LC controllers. ✛ ✛ ra la • In our abstractions, internal states a and 1. Shape of the most concurrent protocol. b are hidden; likewise lt . Suitable abstraction for LC behaviours? 2. Its family of less-state rich shapes. • The constraints they impose will remain. We argue that: 3. Categorising/relating the family. • internal states: implicit part of a spec • Each abstraction will have several cir- cuit implementations, each of which have • logic: just delays lr (or la ) 4. Tabulation of pipeline behaviours. the same pipeline characteristics. • enable: another lateral delay 1 2 3

Pipeline models STGs/Abstract Shape/Family Max latch controller protocol L = lr ↑ . la ↑ . lr ↓ . la ↓ . L LC : STG 1 STG 2 MAX SHAPE ✲ ✲ lr rr R = rr ↑ . ra ↑ . rr ↓ . ra ↓ . R LC ✛ ✛ ra la ❄ ❄ ❄ LC = ( L | R ) COMMON SHAPE CUTAWAY RULES SP d : In CCS lr ↑ . la ↑ is read as ✲ ✲ ✲ ✲ ✲ ✲ lr rr LC 1 LC 2 LC 3 ........... LC d lr ↑ then some time later la ↑ ✛ ✛ ✛ ✛ ✛ ✛ ra la ❄ ❄ All possible interleavings are traced FAMILY SP d ; PP w,d DESIGN L = lr ↑ . . • . la ↑ . lr ↓ . la ↓ . L ✲ ✲ ✲ ✲ SPACE lr rr R = • . rr ↑ . ra ↑ . . rr ↓ . ra ↓ . R LP d ✛ ✛ PP 2 ,d : F 2 J 2 1. Several STG’s → same abstract shape. LC max = ( L | R | • | ) \ { • , } ✲ ✲ LP d ✛ ✛ ✛ ✛ ra la 2. We can compare shapes. ✲ ✲ ✲ lr ↑ lr ↓ la ↑ la ↓ ✻ ❄ • 3. We can define the maximal shape ✻ ❄ Outputs rr / la and Inputs lr / ra ... and its derivative family. ✲ ✲ ✲ ✲ rr ↑ ra ↑ rr ↓ ra ↓ 4 5 6

rr / ra UNTIMED Cutaway Rules The Cut-Away Notation ❄ rr ↓ ✲ ✲ lr ↑ lr ↑ Cheap version: LC max , shape 9599. 1. L1001 → shape 8598 10 ❄ rr ↓ ✲ ✲ . o o • o o o o o la ↓ la ↓ ❄ rr ↓ o o o o o o o o • o o o o o ✲ ✲ lr ↓ lr ↓ o o o o o o o o o o o o o o ❄ rr ↓ . o o o o o o o o ✲ ✲ la ↑ la ↑ o o o o o o o o o ❄ ❄ ❄ ❄ rr ↑ ra ↑ rr ↓ × o o o o o o o o o LC max minimised Left cut-aways constrain rr / ra . ✲ ✲ ✲ ✲ lr ↑ lr ↑ lr ↑ lr ↑ ❄ ❄ ❄ ❄ rr ↑ ra ↑ rr ↓ ✲ ✲ ✲ ✲ la ↓ la ↓ la ↓ la ↓ 1. • same state when quiescent lr / la 2. R0040 → shape 9559 25 ❄ ❄ ❄ ❄ rr ↑ ra ↑ rr ↓ ✲ ✲ ✲ ✲ o o o • o o o o o lr ↓ lr ↓ lr ↓ lr ↓ ❄ ❄ ❄ ❄ rr ↑ ra ↑ rr ↓ o o o o o 2. no holes in the state graph ✲ ✲ ✲ ✲ la ↑ la ↑ la ↑ la ↑ o o o o o . . . . ❄ ❄ ❄ ❄ rr ↑ ra ↑ rr ↓ o o o o o o o o o ✲ lr ↑ 3. always accept an input lr / ra ❄ ✲ Right cut-aways constrain la / lr . la ↓ ❄ 4. may delay an output la / rr ✲ lr ↓ ❄ ✲ la ↑ ❄ ra ↓ 7 8 9

Cut-Away Notation II Pipeline categories Protocol categories L1001 o R0040 → 8558 250 Is shape preserved when pipelined? L0000 L1001 L1111 L2002 L2112 L3003 L3113 L2222 L3223 L3333 L ◦ R . o o • o o o o o o o o o o R0000 R0020 R0040 o o o o o . . . . Basic shape SP d shape PP w,d shape . o o o o o o o o R0022 R0042 R2022 • L1001 o R0040 (Furber/Day, sect. 6) R2042 STABLE ••••••••• ••••••••• ••••••••• ••••• ••••• ••••• R2222 ••••••••• ••••••••• ••••••••• R2242 • ALL L o R → generates the whole fam- ••••••••• ••••••••• ••••••••• R2262 ily R0044 R2044 R4044 REGULAR • The cutaway options make it trivial to o •••••••• o •••••••• ••••••••• o •••• o •••• ••••• R2244 order the family into a lattice o •••••••• o •••••••• ••••••••• R2264 o •••••••• o •••••••• ••••••••• R4244 R4264 L 1 o R 1 ⊇ L 2 o R 2 2REGULAR IFF • 6 categories emerge: o •••••••• ••••••••• ••••••••• L 1 ⊇ L 2 AND R 1 ⊇ R 2 ••••• ••••• ••••• : STABLE regular: ••••• ooo o ••••• ooo o ••••••••• o •••••••• ••••••••• ••••••••• : 2reg O(8): as O(16) : linear dead: 10 11 12

Parallel pipelines PP w,d Single pipelines SP 2 Lattice of stable shapes L0000 L2002 L2222 L o R L0000 L1001 L1111 L2002 L2112 L3003 L3113 L2222 L3223 L3333 L o R L0000 L1001 L1111 L2002 L2112 L3003 L3113 L2222 L3223 L3333 L ◦ R 48 48 44 44 42 42 40 40 36 36 R0000 9599 7597 7377 R0000 • • • • • • • R0000 48 48 44 44 42 42 40 40 36 36 R0020 • • • • • • • • • • R0020 44 44 40 40 38 38 36 36 32 32 R0040 • • • • • • • • • • R0040 44 44 40 40 38 38 36 36 32 32 R0022 9577 7575 7355 R0022 42 42 38 38 36 36 34 34 30 30 R0042 • • • • • • • R0022 42 42 38 38 36 36 34 34 30 D R2022 • • • • • • • • • • R0042 40 40 36 36 34 34 32 32 28 D R2042 • • • • • • • • • R2022 7377 5375 5155 R2222 • • • • • • • • • R2042 40 40 36 36 34 34 32 32 28 D R2222 36 36 32 32 30 30 28 28 24 D R2242 9555 7553 7333 R0044 • • • • • • R2222 36 36 32 32 30 D D 28 D D R2262 • • • • • • • • • R2242 • • • • • • R2262 7355 5353 5133 R2244 40 40 36 36 34 34 32 32 28 28 R0044 36 36 32 30 D R2044 32 30 28 28 24 36 36 D 32 D 30 D D D D R4044 • • • • • • • R0044 5155 3153 D R4444 • • • • • • • • • R2044 • • • • R4044 28 28 24 24 22 22 20 20 16 D R2244 26 26 22 22 20 D D 18 D D R2264 7333 5331 5111 R2266 26 26 D 22 D 20 D D D D R4244 • • • • • • R2244 24 24 D 20 D D D D D D R4264 • • • • • • R2264 • • • • R4244 5133 3131 D R4466 • • • R4264 shapes behave as shapes for d ≥ 2. Many distinct shapes. Independent of w . 5111 D D R4488 Equivalences stay in the same box. Same result throughout each block. Maths says TL; Engineering BR? 13 14 15

Published circuit shapes What’s been done What’s to be done 1. Idea of an abstract design shape and 1. Circuit chrestomathy and shape cook book. L0000 L1001 L1111 L2002 L2112 L3003 L3113 L2222 L3223 L3333 L ◦ R how it composes. Any other circuits out there? R0000 Each shape may have many implemen- R0020 R0040 tations but each will maintain the piped 2. Including lt signals (goes exponential). behaviour of its shape R0022 R0042 3. Mixed parallel pipelines √ . R2022 R2042 2. Family of untimed, bundled protocols Single pipelines en route. R2222 derived by cutaways from the most state R2242 R2262 rich shape. 4. Maths/Engineering interplay and insights. Cutaways enable us to order the family. R0044 R2044 R4044 5. Timed disciplines. 3. We have classified pipeline behaviours R2244 R2264 for the whole family of shapes. R4244 R4264 6. Y (Ken is generating circuits). And can predict mixed parallel pipeline behaviour from their cutaways. Includes: Early; Broadish; Broad Un-/Semi-/Fully-Decoupled Normally open/normally closed 16 17 18