Logic Synthesis

1
Logic Synthesis

• CNF Satisfiability

2
CNF Formulas

• Product of Sum (POS) representation of Boolean
  function
• Describes solution using a set of constraints
• very handy in many applications because new
  constraints can just be added to the list of
  existing constraints
• very common in AI community
• Example
• j
• ( ab c)
• (a b c)
• ( abc)
• ( a b c)
• SAT on CNF (POS) Û Tautology on DNF (SOP)

3
Circuit versus CNF

• Naive conversion of circuit to CNF
• multiply out expressions of circuit until two
  level structure
• Example y x1Å x2 Å x2 Å ... Å xn (Parity
  function)
• circuit size is linear in the number of variables
• Å
• generated chess-board Karnaugh map
• CNF (or DNF) formula has 2n-1 terms (exponential
  in the vars)
• Better approach
• introduce one variable per circuit vertex
• formulate the circuit as a conjunction of
  constraints imposed on the vertex values by the
  gates
• uses more variables but size of formula is linear
  in the size of the circuit

4
Example
Single gate
a
(ab c)(ac)(bc)
c
b
Circuit of connected gates
(124)(14)(24) (235)(25)(35) (23
6)(26)(36) (457)(47)(57) (568)(5
8)(68) (789)(79)(89) (9)
Justify to 0
5
Basic Case Splitting Algorithm
a
b
b
c
c
c
d
d
d
d
d
Source Karem A. Sakallah, Univ. of Michigan
6
Implications in CNF

• Implications in a CNF formula are caused by unit
  clauses
• unit clause is a CNF term for which all variables
  except one are assigned
• the value of that clause can be implied
  immediately
• Example (abc) (a0)(b1)Þ(c1)
• No implications in circuit
• All clauses satisfied
• Not all satisfies (How do we avoid exploring that
  part of the circuit?)

(abc)(ac)(bc)
7
Example
a
(ab c) (ac) (bc)
c
b
Implications
8
Case Splitting with Implications
a
b
b
c
c
Source Karem A. Sakallah, Univ. of Michigan
9
Implementation

• Clauses are stores in array
• Track sensitivity of clauses for changes
• all literals but one assigned -gt implication
• all literals but two assigned -gt clause is
  sensitive to a change of either literal
• all other clauses are insensitive and do not need
  to be observed
• Learning
• learned implications are added to the CNF formula
  as additional clauses
• limit the size of the clause
• limit the lifetime of a clause, will be removed
  after some time
• Non-chronological back-tracking
• similar to circuit case

10
Conflict-based Learning
a
ab j ß j (a b)
a j ß j (a)
bc j ß j (b c)
b
b
c
Source Karem A. Sakallah, Univ. of Michigan
11
Conflict-based Learning

• Important detail for cut selection
• During implication processing, record decision
  level for each implication
• At conflict, select earliest cut such that
  exactly one node of the implication graph lies on
  current decision level
• Either decision variable itself
• Or UIP (unique implication point) that
  represents a dominator node in conflict graph
• By selecting such cut, implication processing
  will automatically flip decision variable (or UIP
  variable) to its complementary value

12
Further Improvements

• Random restarts
• stop after a given number of backtracks
• start search again with modified ordering
  heuristic
• keep learned structures !!!
• very effective for satisfiable formulas but often
  also effective for unsat formulas
• Learning of equivalence relations
• (a Þ b) Ù (b Þ a) Þ (a b)
• very powerful for formal equivalence checking