Satisfiability problem of Boolean expressions in Conjunctive Normal Form

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Satisfiability problem of Boolean expressions in
Conjunctive Normal Form

• Alodeep Sanyal
• Electrical and Computer Engineering
• University of Massachusetts at Amherst

Source K. A. Sakallah (University of Michigan),
S. Malik (Princeton University) and M.
Ciesielski (University of Massachusetts)
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Boolean Satisfiability (SAT)

• Given a suitable representation for a Boolean
  function
• Find a truth assignment T appropriate to
  such that , or
• prove that such an assignment does not exist
• (i.e. for all possible
  assignments)
• Definitions A Boolean expression is
• satisfiable if there is a truth assignment T
  such that
• valid or tautology if for all T
• unsatisfiable if for any T

Figure adopted from C. Papadimitriou,
Computational Complexity, Addison Wesley
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Motivation

• SAT is a core computational engine for major
  applications
• VLSI CAD
• Testing and verification
• Logic synthesis
• FPGA routing
• Path delay analysis
• AI
• Knowledge based deduction
• Automatic theorem proving

Adopted from S. Malik, Princeton University
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Development in SAT research a brief history

• Main contributions in SAT research
• 1960 Davis and Putnam resolution based -
  dealing with 10 variables
• 1962 Davis, Logemann and Loveland DFS-based
  dealing with 10 variables
• Basic framework for many modern SAT solvers
• 1986 R. E. Bryant BDD-based dealing with
  100 variables
• 1996 Silva and Sakallah GRASP (conflict-driven
  learning and non-chronological backtracking)
  dealing with 1k variables
• 2001 Malik et al. Efficient BCP and decision
  making dealing with 10k vars

Adopted from S. Malik, Princeton University
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Satisfiability of Boolean expressions in
Conjunctive Normal Form (CNF)

• In a classical SAT problem, function is
  represented in conjunctive normal form (CNF)
• If the function involves n variables, there
  are 2n possible truth assignments to be checked
• As a matter of fact, SAT was the first
  established NP-Complete problem
• Many other decision (yes/no) problems can be
  formulated in terms of Boolean Satisfiability

S. A. Cook. The complexity of theorem proving
procedures, Third Annual ACM Symp. On Theory of
Computing, pp. 151-158, 1971
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Conjunctive Normal Form (CNF)
j ( a c ) ( b c ) (a b c )
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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)

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CNF Basics

• Implication
• x y x y
• (y) (x)
• y x (contra positive)
• Assignments a 0, b 1 a b
• Partial (some variables still unassigned)
• Complete (all variables assigned)
• Conflicting (imply j)
• Implications binate clauses
• SAT find any satisfying assignment
• Covering find an assignment optimizing some cost
  function

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CNF for Boolean gate
jd d (a b )d a b
d a bd a b
(a d)(b d)(a b d)
(a d)(b d)(a b d)
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Example
Single gate
a
(ab c)(ac)(bc)
c
b
Circuit network 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

• Note
• Node 1,2,3 PI
• Node 4,5,6,7,8,9 AND gate

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Circuit Satisfiability
j h d(ab) e(bc) fd gde hfg
h
(a d)(b d)(a b d)
(b e)(c e)(b c e)
(d f)(d f)
(d g)(e g)(d e g)
(f h)(g h)(f g h)
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Solving SAT - Basic Case Splitting Algorithm
(Davis-Putnam-Logemann-Loveland 1962)
a
b
b
c
c
c
d
d
d
d
d
Source Karem A. Sakallah, Univ. of Michigan
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Solving SAT Case Splitting with Implications
a
b
b
c
c
Source Karem A. Sakallah, Univ. of Michigan
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Implementation

• Clauses are stored in array
• Track sensitivity of clauses for changes
• all literals but one assigned (unit clause) -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

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Solving SAT Conflict-based Learning(GRASP
Silva and Sakallah 1996)
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
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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 unsatatisfiable formulas
• Learning of equivalence relations
• (a Þ b) Ù (b Þ a) Þ (a b)
• very powerful for formal equivalence checking

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SAT used in our research

• Kunal Ganeshpure, Alodeep Sanyal, and Sandip
  Kundu. A pattern generation technique for
  maximizing power supply currents. ICCD 2006
• apply SAT to set a maximal set of nodes (say S)
  to logic value 0
• apply SAT again to set S to find a maximal subset
  S ( ) to set it to logic value 1
• the switching occurring in set S gives
  an estimation on worst case power supply current
• Alodeep Sanyal, Kunal Ganeshpure, and Sandip
  Kundu. On accelerating soft error detection by
  targeted pattern generation. ISQED 2007
• identify a set of soft-error susceptible nodes
  using some filtering scheme
• apply SAT to obtain a truth assignment in the PIs
  such that the set of susceptible nodes are
  excited in their vulnerable state

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Acknowledgement

• This presentation liberally borrowed material
  from the following tutorials
• M. Ciesielski. Lecture on Boolean SAT CNF
  representation. Course Synthesis and
  Verification of Digital Systems
• A. Kuehlmann. SAT Tutorial. UC Berkeley 2003
• J. M. Silva and K. A. Sakallah. Boolean
  satisfiability in electronic design automation.
  DAC 2000
• S. Malik. The quest for efficient Boolean
  satisfiability solvers. CMU seminar on SAT, 2004

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Thank you!!!