Logic Decomposition of Asynchronous Circuits Using STG Unfoldings

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Logic Decomposition of Asynchronous Circuits Using STG Unfoldings. Victor Khomenko School of Computing Science, Newcastle University, UK. Asynchronous circuits. The traditional synchronous (clocked) designs lack flexibility to cope with contemporary design technology challenges

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### Logic Decomposition of Asynchronous Circuits Using STG Unfoldings

Victor Khomenko

School of Computing Science,

Newcastle University, UK

Asynchronous circuits
• The traditional synchronous (clocked) designs

lack flexibility to cope with contemporary

design technology challenges

Asynchronous circuits – no clocks:

• Low power consumption and EMI
• Tolerant of voltage, temperature and

manufacturing process variations

• Modularity – no problems with the clock skew

and related subtle issues

[ITRS’09]: 22% of designs will be driven by ‘handshake clocking’ in 2013, and 40% in 2020

• Synthesis algorithms are complicated
• Computationally hard to synthesize efficient circuits
Motivation
• Logic decomposition is one of the most difficult tasks in logic synthesis
• The quality of the resulting circuit (in terms of area and latency) depends to a large extent on the way logic decomposition was performed

F

instant

evaluator

delay

Speed-independency assumptions
• Gates are atomic (so no internal hazards)
• Gates’ delays are positive and unbounded (and perhaps variable)
• Wire delays are negligible (SI) or, alternatively, wire forks are isochronic (QDI)

delay

delay

delay

delay

G

H1

Hk

Speed-independent decomposition

F

instant

evaluator

Data Transceiver

Device

Bus

d

lds

dtack-

dsr+

lds+

csc+

dsr

VME Bus

Controller

ldtack

dtack

d-

lds-

ldtack-

ldtack+

csc-

dsr-

dtack+

d+

VME Bus Controller

Unexpected!

Unexpected!

Naïve decomposition is hazardous

dtack-

dsr+

lds+

csc+

d-

lds-

ldtack-

ldtack+

csc-

dsr-

dtack+

d+

d

lds

dtack

dsr

csc

x

ldtack

Decompose at the PN level!

dtack-

dsr+

lds+

csc+

ldtack+

d-

lds-

ldtack-

dec+

dec-

csc-

dsr-

dtack+

d+

d

lds

dtack

Multiway acknowledgement

dsr

csc

dec

ldtack

d

lds

dtack

dsr

csc

ldtack

d

lds

dtack

C

dsr

csc

ldtack

Latch utilisation

Only possible because there is no globally reachable state at which dsr=ldtack=0 and csc=1

State Graphs vs. Unfoldings

State Graphs:

• Relatively easy theory
• Many algorithms
• Not visual
• State space explosion problem
State Graphs vs. Unfoldings

Unfoldings:

• Alleviate the state space explosion problem
• More visual than state graphs
• Proven efficient for model checking
• Quite complicated theory
• Not sufficiently investigated
• Relatively few algorithms

Function-guided signal insertion

Logic decomposition algorithm

forever do

for all non-input signals x do

S[x] ← ∅

for all G  {latches, gates} do

S[x] ← S[x]  decompositions(x,G)

bestH[x] ← best SI candidate in S[x]

if for each x, bestH[x] is implementable

Library matching

stop

if for each x, bestH[x]=UNDEFINED

fail

H ← the most complex bestH

Insert a new signal z implementing H into the STG

Function-guided signal insertion

Problem: given a Boolean function F, insert a new signal dec(i.e. a set of new transitions labelled dec+or dec-) with the implementation [dec]=F into the STG. Only unfolding prefix (rather than state graph) may be used.

Previous work: Transformations [PN’07]

Sequential pre-insertion

Sequential post-insertion

Concurrent insertion

Previous work: main results [PN’07]
• Validity criteria: safeness & bisimilarity
• can be checked before the transformation is performed, i.e. on the original prefix (to avoid backtracking)
• Perform the insertion directly on the prefix
• avoid re-unfolding
• good for visualization (re-unfolding can dramatically change the look of the prefix)
• Can transfer some information between the iterations of the algorithm
• The suite of transformations is good in practice for resolution of encoding conflicts
Motivation for more transformations

The suite of transformations is not sufficient for logic decomposition; intuitively:

only linear (in the PN size) number of sequential pre- and post-insertions (assuming that the pre- and postset sizes are bounded)

only quadratic (in the PN size) number of concurrent insertions

exponential number of ‘cuts’ in the PN where a Boolean expression can change its value

Example: imec-sbuf-ram-write

dec+

imec-sbuf-ram-write

prbar

req

wen

precharged

wsen

done

ack

wsldin

wsld

wenin

dec-

Implementation of prbar:

(csc2 req)  csc1  wsldin

dec

Generalised transition insertion [ICGT’10]

s1

d1

sources

s2

destinations

d2

s3

• All previously listed good points hold for GTIs as well 
• Exponentially many GTIs can exist:
• more likely that an appropriate transformation exists 
• no longer practical to enumerate them all 
• can enumerate only the ‘potentially useful’ (for logic decomposition) GTIs 

x

I

C

F=v

F=v

Compatible insertions

An insertion I is compatible with F if whenever an x can fire and trigger I, F’x=1, where

F’x= Fx=0  Fx=1

Intuitively, when x fires, the value of F must change, as I becomes enabled.

Compatible insertions

F=0

dtack-

dsr+

csc+

dsr+

lds+

ldtack+

dtack+

csc+

csc-

d+

dsr-

d-

lds-

ldtack-

F=1

F=1

F =ldtack csc

[ACSD’07]

Reduction to (incremental) SAT

Find an optimal w.r.t. a heuristic cost function SAT assignment of the Boolean formula

MUTEX SA  CUTOFF  FUN

depending on the variables I1, ..., Ik corresponding to the compatible insertions, and conveying that:

• no two insertions are non-commuting, or concurrent, or in auto-conflict, or one of them can trigger the other (MUTEX)
• consistent assignment of signs is possible in the prefix (SA) and beyond cut-offs (CUTOFF)
• F is a possible implementation of the newly inserted signal (FUN)
Cost function

Parameterised by the user; takes into account:

• the delay introduced by the insertion
• the number of syntactic triggers of all non-input signals
• the number of inserted transitions of a signal
• the number of signals which are not locked with the newly inserted signal

x

I

C

F=v

F=v

Building FUN

Let C be a configuration enabling some x, F’x=1, and I be the set of compatible insertions such that:

Then the clause VII I is in FUN.

One can build a Boolean formula FUNGEN depending on C and compatible insertions whose SAT assignments satisfy this condition.

Building FUN (cont’d)

Problem: it is infeasible to enumerate all configurations.

Idea 1: The same clause can be generated by many different configurations, and hence once one such configuration is found, the others can be excluded from the search.

Idea 2: Clauses subsumed by already generated ones can be excluded from the search.

It is enough to add a clause VIII to FUNGEN whenever a new clause VII I is computed.

C

C

Building FUN (example)

F=0

dtack-

dsr+

csc+

dsr+

lds+

ldtack+

dtack+

csc+

csc-

d+

dsr-

d-

lds-

ldtack-

F=1

F=1

F =ldtack csc

Experimental results
• Implemented in MPSAT (library matching not implemented yet) and compared with PETRIFY
• Assorted small benchmarks:
• Similar failure rates and the quality of circuits 
• structural insertions seem sufficient 
• The tests reflect the quality of heuristics in choosing the decomposition in each step rather than the quality of the signal insertion routine 
• Large benchmarks
• Tend to be non-decomposable by both tools 
• Only one series (scalable pipelines) was useful
• can be solved by a single insertion, hence minimizes the impact of heuristics and reflects the quality of the signal insertion routine
• huge reachability graphs, so unfoldings win 
Conclusions
• Unfolding-based decomposition algorithm
• alleviates state space explosion
• completes the design cycle based fully on unfoldings (i.e. state graphs are never built)
• All advantages of state-based decomposition are retained:
• multiway acknowledgement
• latch utilisation
• highly optimised circuits
Thank you!

Any questions?