Computation Engines: BDDs and SAT part 1

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Computation Engines BDDs and SAT(part 1)
290N The Unknown Component Problem Lecture 7
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Outline

• Formulation and computation
• Representations of Boolean functions
• Canonicity of a representation
• Binary decision diagrams (BDDs)
• Definition, properties, applications, etc
• Boolean operations using BDDs
• Deriving BDDs from the circuit
• Satisfiability (SAT)
• Search, implications, branch-and-bound, etc
• Boolean operations using SAT
• Deriving CNF from the circuit

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Formulation and Computation

• Formulation
• Uses formalisms (such as automata theory, etc)
• Requires some statements to be proved
• Is not always concerned with how practical it is
• Computation
• Relies on formulation
• Looks into algorithms and data structures
• Is important for practical applications

4
Computation in Discrete Domain

• Is performed by a variety of applications in
  computer science and engineering
• Represents and manipulates various discrete
  objects (functions, relations, sets, automata,
  FSMs, etc.)
• The most fundamental object seems to be a
  completely specified Boolean function
• Boolean functions can be represented and
  manipulated in a variety of ways
• There is no single best representation

5
Boolean Functions

• A completely specified Boolean function is a
  mapping Bn ? B, where B 0,1
• All other types of Boolean and multi-valued
  functions and relations can be represented using
  completely specified Boolean functions

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Representations of Boolean Functions

• Truth table

• Karnaugh Map

x1x2
x3
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Representations of Boolean Functions

• Sum-of-products (DNF)
• F x1x2x3 x1x2x3 x1x2x3
• Product-of-sums (CNF)
• F (x1x2x3) (x1x2x3)
• (x1x2x3) (x1x2x3)
• (x1x2x3)
• Exclusive sum-of-products
• F x3 ? x1x2x3
• Factored form
• F x3 (x1x2)

• AND/INV graph

• BDD

x1
x2
X3
x3
X1
X2
1
0
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Canonicity of a Representation

• A representation is canonical if for each
  function under certain conditions there exists
  only one representation
• Examples
• Truth table is canonical
• given the ordering of minterms
• BDD is canonical
• given the ordering of input variables
• SOP is not canonical
• but under some conditions it becomes canonical
• the set of all minterms
• the set of all primes
• ISOP computed using Minato-Morreale algorithm
  when the ordering of variables is fixed

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Binary Decision Diagrams

• Formal definition
• Informal definition
• Deriving BDD using the definition
• Deriving BDD from the truth table
• The effect of variable ordering
• Boolean operations on the BDD
• Computing BDD from the Circuit
• BDD package

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Formal Definition

• Definition. Let f 0,1n? 0,1 be a Boolean
  function where the arguments to f are denoted by
  a set of variables V, such that Vn. Let ? V
  ?1,,n be a bijection indicating a total
  ordering of these variables. That is, we consider
  variable x and y to be ordered x lt y when ?(x) lt
  ?(y). An Ordered Binary Decision Diagram (OBDD) P
  for f with respect to the given ordering ? is a
  directed acyclic graph consisting of nonterminal
  nodes labeled by the variables in V and terminal
  nodes labeled by the Boolean constants 1 and 0.
  Each nonterminal node has two outgoing edges the
  1-edge and the 0-edge. The OBDD has a starting
  node called the root. The computation of f(a)
  follows a path from the root to a terminal node,
  where at a node labeled by x, if a(x) 1, the
  path follows the 1-edge, and otherwise it follows
  the 0-edge. The value of the reached terminal
  node determines the value of f(a). On a path from
  the root to the sink, each variable occurs at
  most once. The variables on every path from the
  root to a terminal node respect ordering ?. That
  is, for an edge leading from a node labeled by x
  to one labeled by y, we must have ?(x) lt ?(y).
  
• R. E. Bryant, and C. Meinel, Ordered Binary
  Decision Diagrams,'' in Logic Synthesis and
  Verification, S. Hassoun and T. Sasao, eds.,
  Kluwer Academic Publishers, 2001.

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Informal Definition

• Negative (positive) cofactor of F(x,y,z) w.r.t. x
  is the result of substituting 0(1) into F(x,y,z)
  instead of variable x
• F0 F(0,y,z) F1 F(1,y,z)
• Binary decision diagram of function F is a direct
  acyclic graph, in which
• Each node stands for a function and two incoming
  edges of this node represent cofactors of this
  function w.r.t. a variable.
• The leaves of the graph represent constant
  functions, while the root represents function F
• The same variable order is used for all paths
• The graph is reduced

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Examples of BDDs
a
a1
a
a1
b
b1
b1
0
1
0
1
a2
F a
F ab
a2
b2
a
1
b2
a
F1
b
0
1
0
1
0
1
0
1
Fa1b1 a2b2
Fa1b1a2b2
F a?
F ab
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Shannon Expansion

• Shannon expansion is
• F(x, y, z) x F0 x F1
• Shannon expansion is canonical
• For the given function F and variable x, the
  cofactors F0 and F1 are uniquely determined
• Another informal definition of BDD
• The Shannon expansion is recursively applied to
  the function and its cofactors
• A new node is added to mark each expansion
• The same variable order is used for all paths
• The graph is reduced

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Deriving BDD using Shannon Expansion
F
F x3 (x1x2) F0 Fx10 x2x3 F1
Fx11 x3
F
x1
x1
F1
F0
x3
F0
x2
1
0
F01
x3
F0
F0 x2x3 F00x20 0 F01x21 x3
x2
F00
F00
F01
1
0
x3
0
1
0
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Deriving BDD from Truth Table
x1
1
0
x2
x2
0
1
1
0
x3
x3
x3
x3
1
0
0
1
0
1
0
1
1
1
0
1
0
0
0
0
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Reduction of a Decision Tree
Rule 1 Isomorphic nodes are merged
Rule 2 Redundant nodes are removed
a
a
a
a
b
b
b
b
b
b
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Example of Decision Tree Reduction
x1
x2
x2
x3
x3
x3
x3
1
0
1
1
0
0
0
0
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Example of Decision Tree Reduction
x1
x1
x1
x2
x2
x2
x2
x2
x3
x3
x3
x3
x3
x3
x3
1
0
1
0
1
0
reduction
Decision tree
BDD
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Parts of a BDD (and their meaning)

• Nodes (Boolean functions)
• Terminal nodes (constant Boolean functions)
• Edges (function/co-factor relationship)
• Paths (true/false variable assignments)
• Cuts (variable partitions)
• Nodes pointed to under a cut (the set of
  different cofactors of the function w.r.t.
  variables above the cut)
• Derived parameters
• Number of nodes (complexity of the function)
• Average path length (speed of evaluation of the
  function)

x1
x2
x3
1
0
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Effect of Variable Ordering on the BDD size
F(x1, x2, y1, y2) (x1 y1) (x2 y2)
x1 lt y1 lt x2 lt y2 x1 lt x2 lt y1 lt y2
x1
x1
y1
y1
x2
x2
x2
y1
y1
y1
y1
y2
y2
y2
y2
1
0
1
0
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Another Example
Fa1b1 a2b2
a1
a1
b1
a2
a2
a2
b1
b1
b2
b2
0
1
0
1
a1ltb1lta2lt b2
a1lta2ltb1lt b2
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Operations on BDDs

• Apply NOT, AND, OR, EXOR, etc.
• Quantification (existential, universal, unique)
• Substitute variables
• Compose
• Specialized operators
• Generalized cofactor (constrain)
• Restrict
• Compatible projection
• etc.

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IF-THEN-ELSE (ITE) Operator

• ITE operator
• ITE( F, G, H ) F G F H
• It can be shown that a cofactor of ITE is the ITE
  of cofactors
• ITE( F, G, H )x 0 ITE(F0, G0, H0)
• Computation of Boolean operations is based on the
  Shannon expansion
• ITE(F,G,H)
• ITE(x, ITE(F, G, H)x 0, ITE(F, G, G)x
  1 )
• ITE(x, ITE(F0, G0, H0), ITE(F1, G1, H1) )

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APPLY operator

• APPLY( F, G ) operator is a shorthand for any
  two-variable Boolean operation
• APPLY is reducible to ITE
• Example AND( F, G ) ITE( F, G, 0 )
• It follows that APPLY can be computed recursively
  just like ITE
• APPLY(F,G) x APPLY(F0, G0)
• x APPLY(F1, G1)

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APPLY Pseudocode

• procedure Apply( bdd F, bdd G )
• if ( IsAlreadyComputed( F, G ) ) return result
• if ( F ? 0,1 G ? 0,1 ) return APPLY_TABLE(
  F, G )
• if ( Var( F ) Var( G ) )
• u CreateNode( Var(F), Apply(Fx,Gx),
  Apply(Fx,Gx))
• else if ( Var( F ) lt Var( G ) )
• u CreateNode( Var(F) , Apply(Fx,G ),
  Apply(Fx,G ))
• else / if ( Var( F ) gt Var( G ) ) /
• u CreateNode( Var(G) , Apply(F,Gx ),
  Apply(F,Gx ))
• InsertComputed( F,G,u )
• return u

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Facbcd Gacd FG ?
A1,B1
A1
B1
a
a
A2,B2
A2
b
A6,B2
A6,B5

A6
B5
c
c
B2
A3,B2
A5,B2
A3
A3,B4
d
d
A4,B3
A5,B4
0
1
0
1
B3
B4
A4
A5
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Quantification

• Given a function F(x1, x2, x3)
• Existential quantification of F w.r.t. x1 is
• ?x1 F(x1, x2, x3) F(0, x2, x3) F(1, x2,
  x3)
• Universal quantification of F w.r.t. x1 is
• ?x1 F(x1, x2, x3) F(0, x2, x3) F(1, x2,
  x3)
• Unique quantification of F w.r.t. x1 is
• !x1 F(x1, x2, x3) F(0, x2, x3) ? F(1, x2,
  x3)
• Quantification is generalized to a set of
  variables by applying it w.r.t each variable in
  the set

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Example of Quantification
F(a,b,c,d) a?c cd abd
G(c,d)?abF(a,b,c,d)
H(c,d)?abF(a,b,c,d)
G
F
H
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Deriving BDDs from the Circuit

• The nodes of the circuit are visited recursively
  starting from the POs
• If the node is a PI, its global function is an
  elementary variable
• If the node is an internal node
• the computation is performed recursively for the
  fanins
• the global function of the node is computed by
  composing its local function with the global
  functions of the fanins

o2
o3
o1
Fi
F1
F2
g
h
b
a
c
d
e
f
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Properties of BDDs

• Small size
• for many practical functions
• Fast manipulation
• the smaller the faster
• Canonicity
• ease of caching
• useful for verification

• Large size
• for complex functions (i.e. multipliers)
• Slow manipulation
• the larger, the slower

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Implicit Computation

• With BDDs it is possible to construct and
  manipulate sets of discrete objects (cubes,
  states, etc.) in an implicit manner (without
  explicitly enumerating individual elements).
• As a result, BDDs may allow for an efficient
  computation when explicit methods fail
• Reachability analysis
• Symbolic model checking
• Sequential equivalence checking
• Exact SOP minimization
• Heuristic ESOP minimization
• Computation of symmetries of Boolean functions
• Computation of spectra (Walsh, Haar, Reed-Muller,
  etc)
• Manipulation of discrete matrices

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BDD Package

• Stores nodes in the hash table
• The cofactoring variable and the two cofactors
  are used as a key for hashing the node
• The computed tables stores the results of
  intermediate computations
• Reduces the complexity of computation from
  exponential to linear
• Periodically performs garbage collections and
  dynamic variable reordering

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Introduction to BDDs References

• R. E. Bryant. Symbolic Boolean Manipulation with
  Ordered Binary Decision Diagrams. ACM Computing
  Surveys, Vol. 24, No. 3 (September, 1992), pp.
  293-318.
• Henrik Reif Andersen. An Introduction to Binary
  Decision Diagrams. Dept. of Information
  Technology, Technical University of Denmark,
  1997.
• http//www.itu.dk/people/hra/notes-index.ht
  ml
• F. Somenzi. Binary Decision Diagrams (Tutorial),
  University of Colorado, 1999, http//citeseer.nj.n
  ec.com/somenzi99binary.html

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Boolean Satisfiability

• Definition
• Search for a satisfying assignment
• Computation using SAT
• Computing CNF from the Circuit
• SAT solver

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Definition

• Given a CNF formula ? representing a Boolean
  function f(x1,,xn), the satisfiability problem
  is
• identifying a assignment to the formula
  variables, x1 v1, x2 v2, , xn vn, such
  that all clauses are satisfied, i.e. f(v1,,vn)
  1,
• or proving that such assignment does not exist

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Example
CNF
ab
(a b c) (a b c) (a b c) (a c
d) (a c d) (a c d) (b c d) (b
c d)
cd
Cube bcd
Clause b c d
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Comment

• Such a simple problem and so much effort to solve
  it
• because the size of CNF used in practice is very
  large
• The best known solution is Davis-Logemann-Loveland
  (DLL) procedure, which perform exhaustive search
  with back-tracking
• This procedure is efficient because of a
  combination of good heuristics and smart data
  structures
• Conflict analysis with clause recording
• Non-chronological backtracking
• Variable selection heuristics
• Random restarts
• Two literal clause watching, etc

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Search for a Satisfying Assignment
a
b
b
c
c
c
d
d
d
d
d
Courtesy Karem Sakallah, University of Michigan
39
Computation using SAT

• The classical SAT is a yes/no thing
• It returns one satisfying assignment, or no
  assignment if the problem is UNSAT
• If a conflict occurs during search, SAT solver
  generates a conflict clause and continues
  exploring the search space
• It is possible to have SAT enumerate through the
  satisfying assignments of the problem
• for this, each satisfying assignment is treated
  similar to a conflict
• a new clause (blocking clause) is added and
  search continues
• For large boolean spaces, it is very important to
  generate satisfying assignments in the form of
  cubes rather than minterms
• There are several methods for doing this

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Boolean operations using SAT

• Complement
• Enumerate through the satisfying assignments and
  collect all blocking clauses
• Boolean AND
• put CNF clauses of arguments together
• Other Boolean operations
• reducible to complement and Boolean AND
• Composition
• renaming variables and appending clauses
• Universal quantification
• omitting the quantified variables in all CNF
  clauses
• Existential quantification
• reduced to universal and two complements

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Deriving CNF from the Circuit

• The CNF formula for each node is computed
• One way of computing a CNF for the node is
  applying de Morgan rule to the SOP of the off-set
  of the nodes function
• Another way is to use AND/INV graph
  representation of the nodes on-set, and add
  clauses for each gate in the graph
• The CNF of the network is derived by putting
  together (ANDing) the CNFs for each node
• For single output circuits, if only the positive
  (negative) phase of the circuit function is
  needed, the literal p (p) is added to the CNF,
  where p (p) is the positive (negative) phase of
  the output variable of the PO node

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SAT Solver

• Stores clauses as arrays of integers
• Makes decisions and propagates implications
• When conflict occurs, adds a conflict clause to
  the problem
• When a satisfying assignment is found, while
  enumerating through all satisfying assignments,
  adds a breaking clause to the problem
• Periodically removes inactive clauses
• Implements restarts
• Surprise A state-of-the-art SAT solver can be
  implemented in 600 lines of C code!!!

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Introduction to SAT References

• J.P. Marques-Silva, K.A. Sakallah GRASP A
  Search Algorithm for Propositional
  Satisfiability'' in IEEE Transactions on
  Computers, vol 48, pp. 506--521, 1999.
• W. Kunz, J. Marques-Silva, S. Malik. SAT and
  ATPG Algorithms for Boolean Decision Problems,
  in Logic Synthesis and Verification, S. Hassoun
  and T. Sasao, eds., Kluwer Academic Publishers,
  2001.
• N. Eén, N.Sörensson. An Extensible SAT-solver.
  SAT 2003. http//www.cs.chalmers.se/een/Satzoo/An
  _Extensible_SAT-solver.ps.gz