Logic Synthesis

Presentation Transcript

CNF Satisfiability

CNF Formula’s

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 =

( a+^b+ c)

(^a+ b+ c)

( a+^b+^c)

( a+ b+ c)

SAT on CNF (POS) Û Tautology on DNF (SOP)

Circuit versus CNF

Naive conversion of circuit to CNF:

multiply out expressions of circuit until two level structure
Example: y = x1Å x2Å x2Å ... 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

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Example

Single gate:

a

(^a+^b+ c)(a+^c)(b+^c)

c

b

Circuit of connected gates:

(^1+2+4)(1+^4)(^2+^4)

(^2+^3+5)(2+^5)(3+^5)

(2+^3+6)(^2+^6)(3+^6)

(^4+^5+7)(4+^7)(5+^7)

(5+6+8)(^5+^8)(^6+^8)

(7+8+9)(^7+^9)(^8+^9)

(^9)

Justify to “0”

(a+ b + c)

1

(a + b + c)

(a+ b + c)

(a + b + c)

(a+ b + c)

(a+ b + ¬c)

(a+ b + c)

(a+ b + ¬c)

(a+ b + c)

2

(a + b + ¬c)

(a+ b + ¬c)

(a + b + ¬c)

(a+ b + ¬c)

(a+ b + c)

(a + b + ¬c)

(a+ b + c)

(a+ b + ¬c)

(a+ b + c)

(a+ b + ¬c)

(¬a+ b + ¬c)

(a+ b + ¬c)

(¬a+ b + ¬c)

(a+ b + ¬c)

3

(¬a + b + ¬c)

(¬a+ b + ¬c)

(¬a + b + ¬c)

(¬a+ b + ¬c)

(a+ b + ¬c)

(¬a + b + ¬c)

(a+ b + ¬c)

(¬a+ b + ¬c)

(a+ b + ¬c)

(¬a+ b + ¬c)

(a+ c + d)

(¬a+ b + ¬c)

(a+ c + d)

(¬a+ b + ¬c)

(¬a + b + ¬c)

(¬a+ b + ¬c)

4

(a + c + d)

(a+ c + d)

(a + c + d)

(a+ c + d)

(¬a+ b + ¬c)

(a + c + d)

(a+ c + d)

(¬a+ b + ¬c)

(a+ c + d)

(¬a+ c + d)

(a+ c + d)

(a + c + d)

(a+ c + d)

5

(¬a + c + d)

(¬a+ c + d)

(¬a + c + d)

(¬a+ c + d)

(¬a + c + d)

(a+ c + d)

(¬a+ c + d)

(¬a + c + d)

(¬a+ c + d)

(a+ c + d)

(¬a+ c + ¬d)

(¬a+ c + d)

(¬a+ c + ¬d)

(¬a + c + d)

(¬a+ c + d)

6

(¬a + c + ¬d)

(¬a+ c + ¬d)

(¬a + c + ¬d)

(¬a+ c + ¬d)

(¬a+ c + d)

(¬a + c + ¬d)

(¬a+ c + ¬d)

(¬a+ c + d)

(¬a+ c + ¬d)

(¬a + c + ¬d)

(¬a+ c + ¬d)

(¬a + c + ¬d)

(¬a+ c + ¬d)

(¬b+ ¬c + ¬d)

(¬a+ c + ¬d)

(¬b+ ¬c + ¬d)

(¬a + c + ¬d)

(¬a+ c + ¬d)

7

(¬b + ¬c + ¬d)

(¬b+ ¬c + ¬d)

(¬b + ¬c + ¬d)

(¬b+ ¬c + ¬d)

(¬b + ¬c + ¬d)

(¬b+ ¬c + ¬d)

(¬b + ¬c + ¬d)

(¬b+ ¬c + ¬d)

(¬a+ c + ¬d)

(¬b + ¬c + ¬d)

(¬a+ c + ¬d)

(¬b+ ¬c + ¬d)

(¬a+ c + ¬d)

(¬b + ¬c + ¬d)

(¬b+ ¬c + ¬d)

(¬b+ ¬c + d)

(¬b+ ¬c + ¬d)

(¬b+ ¬c + d)

(¬b + ¬c + ¬d)

(¬b+ ¬c + ¬d)

(¬b + ¬c + ¬d)

(¬b+ ¬c + ¬d)

8

(¬b + ¬c + d)

(¬b+ ¬c + d)

(¬b + ¬c + d)

(¬b+ ¬c + d)

(¬b + ¬c + d)

(¬b+ ¬c + d)

(¬b + ¬c + d)

(¬b+ ¬c + ¬d)

(¬b + ¬c + d)

(¬b+ ¬c + d)

(¬b+ ¬c + ¬d)

(¬b + ¬c + d)

(¬b+ ¬c + d)

(¬b + ¬c + d)

(¬b+ ¬c + d)

(¬b + ¬c + d)

(¬b+ ¬c + d)

(¬b + ¬c + d)

(¬b+ ¬c + d)

Basic Case Splitting Algorithm

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d

Source: Karem A. Sakallah, Univ. of Michigan

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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: (a+^b+c) (a=0)(b=1)Þ(c=1)

No implications in circuit:

All clauses satisfied:
Not all satisfies (How do we avoid exploring that part of the circuit?)

(^a+^b+c)(a+^c)(b+^c)

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Example

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(^a+^b+ c) (a+^c) (b+^c)

c

b

Implications:

1

(a + b + c)

(a+ b + c)

2

(a + b + ¬c)

(a+ b + ¬c)

3

(¬a + b + ¬c)

(¬a+ b + ¬c)

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(a + c + d)

(a+ c + d)

5

(¬a + c + d)

(¬a+ c + d)

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(¬a + c + ¬d)

(¬a+ c + ¬d)

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(¬b + ¬c + ¬d)

(¬b+ ¬c + ¬d)

6

(¬b + ¬c + ¬d)

(¬b+ ¬c + ¬d)

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3

(¬b + ¬c + d)

(¬b+ ¬c + d)

(¬b + ¬c + d)

(¬b+ ¬c + d)

a

b

a

(¬b + ¬c + d)

(¬b+ ¬c + d)

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Case Splitting with Implications

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Implementation

Clauses are stores in array
Track sensitivity of clauses for changes:

all literals but one assigned -> implication
all literals but two assigned -> 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

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(a + b + c)

(a+ b + c)

(¬b+ ¬c)

(¬b + ¬c)

(¬b+ ¬c)

(¬b + ¬c)

(¬b+ ¬c)

(a+ b + c)

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(a + b + ¬c)

(a+ b + ¬c)

(¬a+ ¬b)

(¬a + ¬b)

(¬a+ ¬b)

(a+ b + ¬c)

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(¬a + b + ¬c)

(¬a+ b + ¬c)

(¬a)

(¬a+ b + ¬c)

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(a + c + d)

(a+ c + d)

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(a+ c + d)

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a

(¬a + c + d)

(¬a+ c + d)

a

(¬a+ c + d)

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(¬a + c + ¬d)

(¬a+ c + ¬d)

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b

5

(¬a+ c + ¬d)

d

(¬a+ c + ¬d)

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d

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(¬b + ¬c + ¬d)

(¬b+ ¬c + ¬d)

(¬b + ¬c + ¬d)

(¬b+ ¬c + ¬d)

(¬b + ¬c + ¬d)

(¬b+ ¬c + ¬d)

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(¬b + ¬c + d)

(¬b+ ¬c + d)

(¬b + ¬c + d)

b

(¬b + ¬c + d)

(¬b+ ¬c + d)

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(¬b + ¬c)

(¬a+ ¬b)

d

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Conflict-based Learning

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a® ¬j

ß

j ® (¬a)

ab® ¬j

ß

j ® (¬a + ¬b)

bc® ¬j

ß

j ® (¬b + ¬c)

b

c

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