CS 4700: Foundations of Artificial Intelligence

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CS 4700Foundations of Artificial Intelligence

• Carla P. Gomes
• gomes_at_cs.cornell.edu
• Module
• Satisfiability
• (Reading RN Chapter 7)

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Proof methods

• Proof methods divide into (roughly) two kinds
• Application of inference rules
• Legitimate (sound) generation of new sentences
  from old
• Proof a sequence of inference rule
  applications Can use inference rules as
  operators in a standard search algorithm
• Different types of proofs
• Model checking
• truth table enumeration (always exponential in n)
• improved backtracking, e.g., Davis--Putnam-Logeman
  n-Loveland (DPLL) (including some inference rules)
  
• heuristic search in model space (sound but
  incomplete)
• e.g., min-conflicts-like hill-climbing
  algorithms

Previous module
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Satisfiability
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Propositional Satisfiability problem

• Satifiability (SAT) Given a formula in
  propositional calculus, is there a model
• (i.e., a satisfying interpretation, an assignment
  to its variables) making it true?
• We consider clausal form, e.g.
• ( a ? ?b ? ? c ) AND ( b ? ? c) AND ( a
  ? c)

possible assignments
SAT prototypical hard combinatorial search and
reasoning problem. Problem is NP-Complete. (Cook
1971) Surprising power of SAT for encoding
computational problems.
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Satisfiability as an Encoding Language
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Encoding Latin Square Problems in Propositional
Logic

• Variables
• Each variables represents a color assigned to a
  cell.
• Clauses
• Some color must be assigned to each cell
• No color is repeated in the same row
• No color is repeated in the same column

(clause of length n) n2
(sets of negative binary clauses) n(n-1)/2 n n
(example Row i Color k)
(sets of negative binary clauses) n(n-1)/2 n n
(example Colum j Color k)
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3D Encoding or Full Encoding

• This encoding is based on the cubic
  representation of the quasigroup each line of
  the cube contains exactly one true variable
• Variables
• Same as 2D encoding.
• Clauses
• Same as the 2 D encoding plus
• Each color must appear at least once in each row
• Each color must appear at least once in each
  column
• No two colors are assigned to the same cell

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Dimacs format

• At the top of the file is a simple header.
• p cnf ltvariablesgt ltclausesgt
• Each variable should be assigned an integer
  index. Start at 1, as 0 is used to indicate the
  end of a clause. The positive integer a positive
  literal, whereas a negative interger represents a
  negative literal.
• Example
• -1 7 0 ? (? x1 ? x7)

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Extended Latin Square 2x2
order 2 -1 -1 -1 -1

• p cnf 8 24
• -1 -2 0
• -3 -4 0
• -5 -6 0
• -7 -8 0
• -1 -5 0
• -2 -6 0
• -3 -7 0
• -4 -8 0
• -1 -3 0
• -2 -4 0
• -5 -7 0
• -6 -8 0
• 1 2 0
• 3 4 0
• 5 6 0
• 7 8 0
• 1 5 0
• 2 6 0

1/2 3/4 5/6 7/8
1 cell 11 is red 2 cell 11 is green 3 cell
12 is red 4 cell 12 is green 5 cell 21 is
red 6 cell 21 is green 7 cell 22 is red 8
cell 22 is green
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Significant progress in Satisfiability Methods
Software and hardware verification complete
methods are critical - e.g. for verifying the
correctness of chip design, using SAT encodings
Applications Hardware and Software
Verification Planning, Protocol Design, etc.

Going from 50 variable, 200 constraints to
1,000,000 variables and 5,000,000 constraints
in the last 10 years
Current methods can verify automatically the
correctness of gt 1/7 of a Pentium IV.
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Model Checking
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Turing Award
Source Slashdot
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A real world example
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Bounded Model Checking instance

i.e. ((not x1) or x7) and ((not x1) or
x6) and etc.
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10 pages later


(x177 or x169 or x161 or x153
or x17 or x9 or x1 or (not x185)) clauses /
constraints are getting more interesting
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4000 pages later

!!! a 59-cnf clause

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Finally, 15,000 pages later
Note that
!!!
MiniSAT solver solves this instance in less than
one minute.
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Effective propositional inference
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Effective propositional inference

• Two families of algorithms for propositional
  inference (checking satisfiability) based on
  model checking (which are quite effective in
  practice)
• Complete backtracking search algorithms
• DPLL algorithm (Davis, Putnam, Logemann,
  Loveland)
  
• Incomplete local search algorithms
• WalkSAT algorithm

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The DPLL algorithm

• Determine if an input propositional logic
  sentence (in CNF) is satisfiable.
  
• Improvements over truth table enumeration
• Early termination
• A clause is true if any of its literals is true.
• A sentence is false if any clause is false.
• Pure symbol heuristic
• Pure symbol always appears with the same "sign"
  in all clauses.
• e.g., In the three clauses (A ? ?B), (?B ? ?C),
  (C ? A), A and B are pure, C is impure.
• Make a pure symbol literal true.
• Unit clause heuristic
• Unit clause only one literal in the clause
• The only literal in a unit clause must be true.

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The DPLL algorithm
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DPLL

• Basic algorithm for state-of-the-art SAT
  solvers
• Several enhancements
• - data structures
• - clause learning
• - randomization and restarts

Check http//www.satlive.org/
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Learning in Sat
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The WalkSAT algorithm

• Incomplete, local search algorithm
• Evaluation function The min-conflict heuristic
  of minimizing the number of unsatisfied clauses
  
• Balance between greediness and randomness

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The WalkSAT algorithm
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Lots of solvers and information about SAT,
theory and practice
http//www.satlive.org/
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Computational Complexity of SAT
How does an algorithm scale?
Analyzable Realistic Spectrum of hardness
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SAT Complexity

• NP-Complete - worst-case complexity
• (2n possible assignments)
• Average Case Complexity (I)
• Constant Probability Model Goldberg 79
  Goldberg et al 82
• N variables L clauses
• p - fixed probability of a variable in a clause
  (literals 0.5 /-)
• (i.e., average clause length is pN)
• Eliminate empty and unit clauses
• Empirically, on average, SAT can be easily
  solved - O(n2)

Key problem easy distribution random guesses
find a solution in a constant number of tries
Franco 86 Franco and Ho 88
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Hard satisfiability problems

• Consider random 3-CNF sentences. e.g.,
• (?D ? ?B ? C) ? (B ? ?A ? ?C) ? (?C ? ?B ? E) ?
  (E ? ?D ? B) ? (B ? E ? ?C)
  
• m number of clauses
• n number of symbols

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SAT Complexity

• Average Case Complexity (II)
• Fixed-clause Length Model Random K-SAT Franco
  86
• N variables L clauses K number of literals per
  clause
• Randomly choose a set of K variables per clause
  (literals 0.5 /-)
• Expected time O(2n)

Can we provide a finer characterization beyond
worst-case results?
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Typical-Case Complexity

• Typical-case complexity a more detailed picture
• Characterization of the spectrum of hardness of
  instances as we vary certain interesting
  instance parameters
• e.g. for SAT clause-to-variable ratio.
• Are some regimes easier than others?
• What about a majority of the instances?

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Typical Case Analysis3 SATAll clauses have 3
literals
Median Runtime
Selman et al. 92,96
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Hard problems seem to cluster near m/n 4.3
(critical point)

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Intuition

• At low ratios
• few clauses (constraints)
• many assignments
• easily found
• At high ratios
• many clauses
• inconsistencies easily detected

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