1. Efficient Solution of Language Equations Using Partitioned Representations
Alan Mishchenko UC Berkeley, US
Robert Brayton UC Berkeley, US
Roland Jiang UC Berkeley, US
Tiziano Villa DIEGM, University of Udine, Italy
Nina Yevtushenko Tomsk State University, Tomsk, Russia

2. Overview Problem formulation Example: Traffic light controller
Partitioned representation
Solution based on partitioned representation
Experimental results
Conclusion

3. Problem Formulation Specification S (i,o) Fixed F (i,v,u,o)
FSM i1 i2 o1 o2
Specification S (i,o)
Fixed F (i,v,u,o)
Unknown X (u,v)
Fixed
Unknown i o u v
Spec
Problem: Given S and F, find the Most General Solution (MGS) X of
Solution:

4. Combinational and Sequential Flexibility
Network is a DAG
Node has single output
Node is a logic function
Sequential
Network is arbitrary
Node has many outputs
Node is an FSM
logic i1 i2 o1 o2
FSM i1 i2 o1 o2 X Y

5. Computing Most General FSM Solution
Input: Prefix-closed specification S(i,o) and fixed part F(i,v,u,o)
Output: Most general FSM (prefix-closed progressive) solution X
begin
01 X := Complete( S )
02 X := Determinize( X )
03 X := Complement( X )
04 X := Support( X, (i,v,u,o) )
05 X := Product( Complete(F), X )
06 X := Support( X, (u,v) )
07 X := Determinize( X )
08 X := Complete( X )
09 X := Complement( X )
10 X := PrefixClose( X )
11 X := Progressive( X, u )
12 return X
end

6. Example: Traffic Light Controller Synthesis
General Topology
This example
Specification
Specification S S I O
Fixed
Traffic Light z F F V U v
Unknown
Controller X X

7. Traffic Light Controller (Fixed Part)
.model fixed
.inputs v z
.outputs Acc
.mv v 2 wait go
.mv z 3 red green yellow
.mv CS, NS 3 Fr Fg Fy
.latch NS CS
.reset CS Fr
.table ->Acc 1
.table v z CS ->NS
wait red Fr Fr
go red Fr Fg
wait green Fg Fg
go green Fg Fy
wait yellow Fy Fy
go yellow Fy Fr
.end
z = {red, green, yellow}
v = {wait, go}
Specification S
Traffic Light z F v
Controller X

8. Traffic Light Controller (Specification)
.model spec
.inputs z
.outputs Acc
.mv z 3 red green yellow
.mv CS,NS 4 S1 S2 S3 S4
.table ->Acc 1
.latch NS CS
.reset CS S1
.table z CS ->NS
red S1 S2
red S2 S3
green S3 S4
yellow S4 S1
.end
z = {red, green, yellow}
v = {wait, go}
Specification S
Traffic Light z F v
Controller X

9. Traffic Light Controller (Synthesis Script)
echo "Synthesis ..."
determinize -lci spec.mva spec_dci.mva
support v(2),z(3) spec_dci.mva spec_dci_supp.mva
support v(2),z(3) fixed.mva fixed_supp.mva
product -l fixed_supp.mva spec_dci_supp.mva p.mva
support v(2) p.mva p_supp.mva
determinize -lci p_supp.mva p_dci.mva
progressive -i 0 p_dci.mva x.mva
echo "Verification ..."
support v(2),z(3) x.mva x_supp.mva
product x_supp.mva fixed_supp.mva prod.mva
support v(2),z(3) spec.mva spec_supp.mva
check prod.mva spec_supp.mva

10. Traffic Light Controller (Solution)
.model solution
.inputs v
.outputs Acc
.mv v 2 wait go
.mv CS, NS 4 \
FrS1 FrS2 FgS3 FyS4
.latch NS CS
.reset CS
FrS1
.table ->Acc 1
.table v CS ->NS
wait FrS1 FrS2
go FrS2 FgS3
go FgS3 FyS4
go FyS4 FrS1
.end
z = {red, green, yellow}
v = {wait, go}
Specification S
Traffic Light z F v
Controller X

11. Partitioned Representation
FSM i1 i2 o1 o2
Fixed part:
Latches u v o i ns cs
Fixed
Unknown i o u v
Spec
Output functions: oj = Oj(i,v,cs)
Transition functions: nsk = NSk(i,v,cs)
Communication functions: um= Um(i,v,cs)

12. Computing with Partitioned Representation
Completion
Complementation
Product computation
Changing support
Determinization (subset construction)

13. Completion The output relation is O(i,o,cs) = j[oj  Oj(i,cs)]
For each current state cs, the transitions are not defined iff
P(i,o) = cs[O(i,o,cs) & (cs)]
Using partitioned representation, this computation becomes
P(i,o) = cs[j[oj  Oj(i,cs)] & (cs)]
This is a partitioned image computation. Any image computation method can be used.

14. Complementation
For non-deterministic automata, requires determinization
For deterministic automata, make non-accepting states accepting, and vice versa
In the case of partitioned representation, the don’t-care state becomes acceptable, other states become non-acceptable

15. Product Computation
Combine the partitions of the two components

16. Changing Support (Lifting and Projecting)
Expanding support is trivial
Reducing support requires existential quantification
In general, cannot be done on the partitioned representation
Therefore, postponed until the determinization step

17. Determinization (Subset Construction)
Start with the subset containing only the initial state.
For each subset , compute the subsets reachable through (u, v).
Define the acceptance relation:
Compute the condition of the transition is into non-accepting states:
Restrict the transitions to those that are not contained in Q(u, v):
Direct the transition under Q(u, v) to DCN.
Complete and redirect the remaining transitions to DCA.

18. Experimental Results Name is the ISCAS benchmark name
i/o/cs is the number of input/output/state variables
Fcs/Xcs is the partitioning of the state variables
States(X) is the number of states in the solution
Mono is the runtime of the monolithic computation in seconds
Part is the runtime of partitioned computation is in seconds
Ratio is improvement due to the proposed method

19. Conclusions Introduced language solving problems
Showed a simple example
Discussed partitioned representation
Presented experimental results

20. Future Work
Developing methods for finding a particular solution contained in the most general solution
Developing efficient sequential synthesis methods without state space traversal