UIUC CS 497: Section EA Lecture

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UIUC CS 497 Section EALecture 2

• Reasoning in Artificial Intelligence
• Professor Eyal Amir
• Spring Semester 2004

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Last Time

• Propositional logic as a language for
  representing knowledge
• Did not touch on reasoning procedures
• Defined language, signature, models
• From homeworks you should know
• Soundness Completeness theorem
• Deduction theorem
• De Morgan Laws

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Today

• Reasoning procedures for propositional logic
• Checking Satisfiability (SAT) using DPLL
• Proving entailment using Resolution
• Application du jour Formal Verification
• Applications we will not touch
• AI planning, graph algorithms, cryptography,

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SAT via Generate and Test

• If we have a truth table of KB, then we can check
  that KB satisfiable by looking at it.
• Problem n propositional symbols ? 2n rows in
  truth table
• Checking interpretation I takes time O(KB)
• Generating table is expensive O(2n KB) time
• Observation SAT requires us to look only for one
  model

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Clausal Form

• Every formula can be reformulated into an
  equivalent CNF formula (conjunction of clauses).
• Examples (using De Morgan Laws)

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Clausal Form

• Every formula can be reformulated into an
  equivalent CNF formula (conjunction of clauses).
• Examples

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Clausal Form

• Every formula can be reformulated into an
  equivalent CNF formula (conjunction of clauses).
• Examples

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Propagating a Truth-Value

• KB in CNF, and we observe pTRUE
• Then, removing clauses with p positive from KB
  gives an equivalent theory.
• Example

KB
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Propagating a Truth-Value

• KB in CNF, and we observe pTRUE
• Then, removing negative instances of p from KB
  gives an equivalent theory.
• Example

KB
Observe
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DPLL Search Procedure for CNF

• If no clauses in KB, return T
• If a clause in KB is empty (FALSE), return F
• If KB has a unit clause C with prop. p, then
  return DPLL(KB,p?polarity(p,C))
• Choose an uninstantiated variable p
• If DPLL(KB, p?TRUE) returns T, return T
• If DPLL(KB, p?FALSE) returns T, return T
• Return F

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DPLL in Action
On board
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DPLL in Action
On board
Note we could know without thorough
checking that this KB is satisfiable
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DPLL in Action
On board
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Related SAT Solving

• Order of selection of variables (lecture 5)
• Stochastic local search (paper 1)
• Binary Decision Diagrams (paper 2)
• Strategies other than unit (paper 23)
• 2-SAT is solvable in linear time
• Smart backtracking (paper 21)
• Clauses/Vars in Random SAT (paper 22)
• SAT via probabilistic models (paper 15)

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Take a Breath

• Until now SAT solving
• Search in the space of models
• From now Resolution theorem proving
• Search in the space of proofs
• Later Formal Verification

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Resolution Theorem Proving

• Given
• KB a set of propositional sentences
• Query Q a logical sentence
• Calling procedure
• Add ?Q to KB
• Convert KB into clausal form
• Run theorem prover. If we prove contradiction,
  return T. Otherwise, return F.

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Resolution Theorem Proving

• Add ?Q to KB
• Convert KB into clausal form
• Run theorem prover. If we prove contradiction,
  return T. Otherwise, return F.
• Deduction theorem
• KB Q iff KB ? ?Q
  FALSE

-
-
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Resolution Theorem Proving

• Add ?Q to KB
• Convert KB into clausal form
• Run theorem prover. If we prove contradiction,
  return T. Otherwise, return F.
• Deduction theorem
• KB Q iff KB ? ?Q
  FALSE

-
-
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Propositional Resolution

• Resolution inference rule
• C1 p1 ? C1
• C2 ?p1 ? C2
• --------------------
• C3 C1 ? C2

C1 ? ?p1 ? C1
C2 ? p1 ? C2
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Propositional Resolution

• Resolution algorithm (saturation)
• While there are unresolved C1,C2
• Select C1, C2 in KB
• If C1, C2 are resolvable, resolve them into a new
  clause C3
• Add C3 to KB
• If C3 (empty clause),
• we got a contradiction.
• STOP

C1 p1 ? C1 C2 ?p1 ? C2 --------------------
C3 C1 ? C2
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Resolution in Action
On board
C1 p1 ? C1 C2 ?p1 ? C2 --------------------
C3 C1 ? C2
KB
Negated Query
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Resolution in Action
On board
C1 p1 ? C1 C2 ?p1 ? C2 --------------------
C3 C1 ? C2
KB
Negated Query
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Properties of Resolution

• Running time for n variables, m clauses
• Resolving two clauses
• O(n)
• Finding two resolvable clauses
• O(1)
• Overall algorithm
• O(3nn)

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Properties of Resolution

• Theorem Resolution is sound
• Resolving clauses in KB generates valid
  consequences of KB
• Theorem Resolution is refutation complete
• Resolution of KB with ?Q yields the empty clause
  iff KB Q

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Properties of Resolution

• Resolution does not always generate Q
• KB a,b, ?a,b, b,c
• Q b ? ?c b,?c
• Theorem Resolution always generates a clause
  that subsumes Q iff KB Q
• Example Resolving KB generates b

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Simple Enhancements

• Remove subsumed clauses
• p subsumes p , q
• p , q subsumes p , q, r
• ?p does not subsume p , q
• Contract same literals
• p , p , q becomes p , q
• Unit resolution resolve unit clauses first

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Related to Prop. Resolution

• Clause selection for resolution (lecture 5,
  paper 19)
• Consequence finding (paper 3)
• Prime implicates/implicants (paper 4)

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Resolution vs SAT

• SAT solvers can find models
• Resolution sometimes better at finding
  contradictions
• With resolution it is easier to explain and
  provide a proof

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Summary So Far

• Finding models using DPLL
• Resolution theorem proving allows us to find
  contradictions and explanation.
• The deduction theorem tells us how to ask queries
  from either SAT solvers or Resolution

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Application Hardware Verification
f3
x1
f1
not
AND
x2
f5
AND
not
f2
OR
x3
f4
Question Can we set this boolean cirtuit to TRUE?
f5(x1,x2,x3) a function of the input signal
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Application Hardware Verification
f3
x1
f1
not
AND
x2
f5
AND
not
f2
OR
SAT(f5) ?
x3
f4
Question Can we set this boolean cirtuit to TRUE?
f5(x1,x2,x3) f3 ? f4 ?f1 ? (f2 ? x3)
?(x1 ? x2) ? (?x2 ? x3)
Mx1FALSE Mx2FALSE Mx3FALSE
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Hardware Verification

• Questions in logical circuit verification
• Equivalence of circuits
• Arrival of the circuit to a state (required a
  temporal model, which gets propositionalized)
• Achieving an output from the circuit

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Summary

• SAT checking using DPLL (instantiate, propagate,
  backtrack)
• Entailment/SAT checking using Resolution (create
  more and more clauses until KB is saturated)
• Formal verification uses mainly SAT checking such
  as DPLL, but also sometimes resolution

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Next Time

• FOL Resolution
• Homework
• Read readings (incl. application reading)
• Make sure you know
• Deduction theorem for FOL
• Language of FOL
• Soundness, completeness, and incompleteness
  theorems
• Models of first-order logic