1. SAT-Based Logic Synthesis (yes, Logic Synthesis Is Everywhere!)
Robert Brayton Alan Mishchenko
Department of EECS, UC Berkeley
Mathias Soeken
Integrated Systems Laboratory, EPFL
Luca Amarù
Design Group, Synopsys Inc., Mountain View, CA

2. Overview Logic synthesis Boolean satisfiability
SAT-based logic synthesis
Canonical SOP computation
Boolean decomposition, resubstitution, ECO, etc
Agile SAT solving
Conclusions 2

3. Logic Synthesis
Given a Boolean function, logic synthesis derives a new circuit to represent it (or improves an old circuit)
Logic synthesis is used in hardware design and in such areas as
formal verification, artificial intelligence, risk analysis, etc
The original problem (deriving an optimal circuit structure for a given function) is very hard and is solved heuristically in most cases
The emphasis in logic synthesis computations is on scalability
handling larger problem and getting solutions faster
In recent years, traditional implementations based on SOPs and BDDs have been gradually replaced by SAT-based ones
both quality and runtime is often improved

4. Boolean Satisfiability
Boolean variable can take value 0 or 1 (for example, a)
Literal is Boolean variable with its value (for example, a)
Clause is a disjunction of literals: a + b
CNF is a conjunction of clauses: (a + b) & (a + c)
Given CNF representing a Boolean function, a SAT solver checks whether the function is constant 0
Example of a satisfiabile problem:
(a + b) & (a + c) one solution is a = 1, b = 1, c = 1
Example of an unsatisfiable problem:
(a + b) & (a + b) & (a + b) & (a + b) no solution exists

5. SAT Solver in Practical Applications
Netlist
Answer: “SAT” or “UNSAT”
CNF
SAT solver
CNF generator
CNF
Design constraints
If SAT, a counter-example
If UNSAT, a proof
(both counter-examples and proofs are very useful in practice)
User cost functions
Miter
Typical application: Equivalence Checking
Build circuit Miter = (Specification != Implementation)
Convert Miter into CNF and run SAT solver
- If UNSAT, equivalence checking succeeds
- If SAT, use the counter-example to debug the design
Output
Spec
Impl
Inputs

6. Incremental SAT Solver
Initial CNF
Round 1:
SAT solver
Initial assumptions
Additional CNF
Round 2:
SAT solver
New assumptions
Additional CNF
Round 3:
SAT solver
New assumptions
(assumptions are CNF clauses used only in the current round – they are handled differently from the rest)
Typical application: Bounded Model Checking
Load init and the first time frame
until (SAT solver returns “UNSAT”) {
add clauses for the next time frame
run SAT solver on the property output } P P P T1 T2 T3 …
Init

7. A Note on Counter-Examples
Counter-example (CEX) is an assignment of SAT variables that makes all CNF clauses satisfied
For this, we check whether ONSET(x) = 1 is true
CNF-based SAT solvers (in particular, MiniSAT), always return a complete assignment of variables (a minterm)
In practice incomplete assignments are often preferred
For this, we need to transform an assignment minterm by dropping (expanding) some of its literals, making it a cube
To expand an assignment, we use the off-set (similar to “expanding a cube against the off-set” in ESPRESSO)
We solve SAT instance: ONSET_minterm(x) & OFFSET(x) = 0
A proof of unsatisfiability expressed in terms of literals of the minterm, gives us literals remaining in the expanded cube

8. A Note on Proofs
Proof of unsatisfiability (POU), is a sequence of resolution steps, which derives empty clause (contradiction) from the original clauses
POU can be used to compute an UNSAT core (a subset of original clauses, which make the problem UNSAT)
Computing the full POU/core is often not necessary
A practical alternative is to compute an abstraction of the core in the form of a subset of assumptions needed for the problem to be UNSAT
This is achieved by a dedicated API of the SAT solver (analyze_final)
analyze_final is very fast because it simply performs conflict analysis one more time, on the top level
analyze_final is very practical because it allows UNSAT runs to be as useful as SAT ones, without recording and traversing the complete POU

9. ISOP Computation
ISOP is a Sum-of-Products (SOP) that is prime and irredundant
F = ab + cd is an ISOP
G= ab + a is not an ISOP
Logic synthesis often derives circuit structure from functional description or improves a given structure
Deriving ISOP and factoring it can be useful for this
Minato-Morreale ISOP computation is a BDD-based method widely used in practice
However, this method is not scalable; better methods are needed

10. SAT-Based ISOP Computation
Algorithm
Input Boolean function is given as two (shared) circuits (ON-set and OFF-set)
Convert these to CNF
Initialize two SAT solvers (S1 for ON-set; S2 for OFF-set)
Iterate until S1 returns “UNSAT”:
Get one minterm of the ON-set (satisfiable assignment of S1)
Expand it into a prime using the OFF-set (proof computed of S2)
Add it to the SOP and block it in the ON-set (add clause to S1)
Post-process the SOP by removing redundant cubes (done using new solver S3)

11. SAT-based ISOP Computation
one minterm
SAT solver S1 (ON-set)
SAT solver S2 (OFF-set)
SAT solver S3
(sat assignment)
redundant SOP
irredundant SOP
one prime cube
CNF
CNF
blocking clause
(analyze_final)
Factoring
Original circuit {ON-set, OFF-set}
Factored form
SOP generator
Improved circuit

12. Canonicity of the ISOP
Resulting ISOP is canonical because the same fixed variable order is used
to compute one satisfiable minterm
to expand minterm into a prime
to remove redundant primes if present
Canonicity (as in the case of BDDs) guarantees that the result is the same
for any structure of the circuit
for any CNF generation algorithm
for any SAT solver
for any operating system

13. How Canonicity is Achieved?
Minterm is selected canonically by making sure that it is the lexicographically smallest SAT assignment for the given variable order
Cube is expanded canonically by making sure literals are removed in the same order
Redundant cubes are removed canonically by trying to remove them in the same order

14. LEXSAT Algorithm
Efficiently computes the lexicographically smallest satisfying assignment
The main building block of the canonical ISOP computation
if canonicity is not required, we can use any assignment (on average about 2x faster)
LEXSAT is mentioned in Donald Knuth, Art of Computer Programming, Volume 4 (6A):
- Implementing LEXSAT by iterative SAT calls (Ex. 109, page 150) - LEXSAT as modification of another algorithm (Ex. 275, page 165)
This is joint work with Mathias Soeken and Giovanni De Micheli at EPFL

15. LEXSAT Algorithm Pseudocode
assignment LEXSAT_rec( assigned vars A, unassigned vars U, cnf F ) {
if ( F is UNSAT under assumptions in A )
return ‘none’; // no satisfiable assignment
if ( U ==  )
return Const1; // constant 1 Boolean function
v = next unassigned variable in U in the given order;
result = LEXSAT_rec( A  v=0, U / v, F );
if ( result != ‘none’ )
return !v & result; // found assignment with v == 0
return v & LEXSAT_rec( A  v=1, U / v, F ); // assignment with v == 1 }
assignment LEXSAT( ordered set of vars V, cnf F ) {
return LEXSAT_rec( , V, F );

16. Boolean Resubstitution
Given a functional specification
a logic network whose functionality has to be preserved (in synthesis) or achieved (in ECO)
Given a set of candidate divisors
these are nodes found in the specification network or nodes with desirable properties to be synthesized
Formulate a SAT problem, which
Selects the divisors among the candidates
Finds a new function of the node
When the node is updated, the functionality of the network is preserved (in synthesis) or restored (in ECO) while achieving the optimization goal (such as area minimization)

17. Resubstitution / ECO Illustrated
F(n, x)  S(x) …
Outputs
Implementation F(n, x)
Specification S(x) n
Node
Inputs x
Resubstitution / ECO of n in terms of x exists iff
I(x)
Boolean space
x n F(n, x) == S(x) is always true, that is
x n F(n, x)  S(x) is always false, that is
F(0, x)  S(x)
F(1, x)  S(x)
F(0, x)  S(x) & F(1, x)  S(x) is UNSAT

18. Resubstitution with Divisors
F(n, x)  S(x) …
Outputs
Implementation F(n, x)
Specification S(x)
Node n
Divisors d
Inputs x 1
M1 = 1
M2 = 1
Consider M(n, x) = [F(n, x)  S(x)] & [d == D(x)]
M1(x1) =
M2(x2)
Consider M(x) = n M(n, x) = M(0, x) & M(1, x) d1 d2
Resubstitution / ECO of n in terms of d exists iff x1 x2
M1(x1) & M2(x2) & [d1 == d2] is UNSAT

19. Using Don’t-Cares
This formulation of resubstitution/ECO exploits internal don’t-cares of the node in the network (can also use external don’t-cares)
However, the don’t-cares do not have to be computed explicitly before they are used
Instead, the circuit-based constraint is transformed into CNF and added to the SAT solver, resulting in a solution that is correct under the don’t-cares

20. Localizing Computation at a Node
Definition
A window for a node in the network is the context, in which its functionality is considered
A window includes
k levels of the TFI
m levels of the TFO
all re-convergent paths captured in this scope
Window POs
Window PIs
k = 3
m = 3

21. Agile SAT Solving
Use simulation and circuit structure analysis whenever possible to avoid unnecessary SAT calls
Simplify SAT-based formulations
avoid QBF if regular SAT can be used
experiment with different encodings
try to reduce the number of variables and clauses
reduce complexity of the constraints, even if it takes more variables
counter-intuitively, sometimes larger CNFs have shorter runtimes
Create formulations resulting in fast incremental runs
in most practical applications in the area of logic synthesis, a well-tuned formulation leads to SAT runtimes not exceeding 1 sec per call (in fact, it is a fraction of a second on average)
If constraints are complex, use lazy constraint addition
Whenever possible, give initial solution to the SAT solver

22. Conclusion Reviewed logic synthesis and Boolean satisfiability
Discussed a number of problems in logic synthesis that can be solved using Boolean satisfiability
Shared practical advice on developing efficient SAT based applications
In the rest of this tutorial, my collaborators will talk about
exact delay optimization
synthesis for emerging technologies

23. Abstract
In the first part, it is shown how SAT solvers can be used to implement classical logic synthesis algorithms, such as technology mapping, LUT mapping, collapsing, and logic rewriting, in order to obtain better quality and higher flexibility. Key is to tightly integrate one or more SAT solvers into the algorithm and use them in an agile and reliable way. It is shown how SAT solvers can be used to generate canonical solutions. Consequently, SAT solvers can be used as a replacement for decision diagrams, retaining all their benefits but avoiding drawbacks such as memory explosion.
The second part introduces new algorithms and data structures for delay optimization, a crucial optimization criteria. An enumeration based algorithm is introduced that allows to find optimum networks for given delay constraints. Majority-inverter graphs are introduced as a logic data structure that allows for significant delay optimization due to effective algebraic rewriting techniques.
The third part illustrates an emerging application of logic synthesis. It shows how classical logic synthesis algorithms can be applied for the scalable design of quantum computers. Recent progress in fabrication makes the practical application of quantum computers a tangible prospect. And as quantum computers scale up to tackle problems in computational chemistry, machine learning, and cryptoanalysis, logic synthesis and design automation will be necessary to fully leverage the power of this emerging computational model.