A Partial Instantiation Method for Inference in FirstOrder Logic

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A Partial Instantiation Method for Inference in
First-Order Logic

• Presented by Eric McGregor
• Advisor Christopher Lynch

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Overview

• Automated Theorem Proving and Satisfiability
• First-Order Logic and Formula Convention
• Substitutions
• Viewing First-Order Formula As Propositional
  Formula
• General Form of the PI Algorithm
• Interpretations Restricted to the Herbrand
  Universe
• Implications of Unsatisfiability in Propositional
  Logic
• Implication of Satisfiability in Propositional
  Logic
• An Example
• Correctness and Completeness

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Automated Theorem Proving

• Given a set of hypothesis and a conclusion,
  Automated Theorem Proving, seeks a mechanized
  means to deduce if the hypothesis implies the
  conclusion.
• This is a decision problem.
• A theorem proving problem can be reduced to a
  satisfiability problem.

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Satisfiability

• To reduce it to a satisifability problem we
  transform our input to a first-order logic
  formula P-gtQ.
• Our original problem is true if P-gtQ is valid.
• For automated theorem proving we determine if
  is unsatisfiable.

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Common Strategies for Checking First-Order Logic
Satisfiability

• Resolution a method of repeatedly applying
  resolution rule. If empty clause is generated,
  then set of clauses is unsatisfiable.
• Resolution rule
• Partial-Instantiation (PI) a method which solves
  a series of propositional satisfiability problems
  each derived by partially instantiating one or
  more variables in the last.

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Renewed Interest In PI

• Advances in the computational performance of
  propositional logic satisfiability algorithms.
• Many problems are solved when only a few of the
  many possible instantiations have been generated.

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Source

• The PI method we will show today is given in the
  paper titled Partial Instantiation Methods for
  Inference in First-Order Logic by J.N. Hooker, G.
  Rago, V. Chandru, and A. Shrivastava, 2001
• Based on method by R. Jeroslow, 1988

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First-Order Logic

• Recall that a term is defined recursively by
• Variables x,y,z and constants a,b,c are
  terms.
• If f is a function symbol and v is a vector of
  terms then f(v) is a term.
• If P is a predicate symbol and v is a vector of
  terms then P(v) is an atom.
• A literal is an atom or the negation thereof.
• A clause is a disjunction of literals.

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Formula Convention

• When we consider a first-order logic
  satisfiability problem we convert the formula to
  one in the following form
• Formula is in prenex clausal form.
• All variables are universally quantified.
• Each clause is standardized apart.

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Substitutions

• A substitution replaces one or more variables of
  a formula F with terms, in such a way that each
  occurrence of a given variable is replaced by the
  same term.
• Example

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Instantiation

• The result of a substitution on a formula F
  is an instantiation of F.
• A formula F is ground if it has no variables.
• If is ground then it is a complete
  instantiation, otherwise it is a partial
  instantiation.

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Unifier, Most General Unifier

• A unifier of predicates and is a
  substitution such that
  .
• A unifier is a most general unifier (mgu) if
  for any unifier there is a substitution
  such that .
  

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Example of mgu

• Let and
• Let
• is a most general unifier of D and E.
• Consider
  
• is a unifier.
• Notice for that
  

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Renaming, Variant

• A renaming is a substitution which replaces
  variables with variables.
• We say that two formulas E and F are variants if
  they can be unified by a renaming of variables.
• Example Let E P(x,a) and F P(y,a).
  Let be a renaming .
  As E P(y,a) F ,
  then E and F are variants.

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Viewing First-Order Formula as Propositional
Formula

• A quanitifier-free first order logic formula can
  be viewed as a propositional formula in which
  variants of an atom are treated as the same atom.

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Viewing First-Order Formula as Propositional
Formula

• Example
  .
• and are variants
• and are not variants
• Thus F can be viewed as the propositional formula
  .
• We can then give this new formula to a
  propositional satisfiability checker.

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General Form Of The PI Algorithm

• Given a first-order logic formula F determine if
  F (F viewed as a propositional formula) is
  satisfiable.
• If F is unsatisfiable, F is unsatisfiable and
  we are done.
• Else F is satisfiable. Check for conflicts.
• If no conflicts exist, F is satisfiable and we
• are done.
• Else, add additional clauses to F in order to
• resolve conflicts and repeat from beginning.

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Satisfiability Obstacle

• What Universe do we consider?

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Herbrand Universe

• The Herbrand Universe, , of a formula F is
  the set of all terms built up by constants and
  functions in F. If no constants occur in F then
  we add a single element, say a.
• Example If
  then

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Herbrand Base

• The Herbrand Base, , of a formula F is the
  set of all atoms generated by predicate symbols
  in F and terms in the Herbrand Universe of F.
• Example If
  then
• and

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Herbrand Interpretations

• A Herbrand interpretation for F assigns a truth
  value to each atom in the Herbrand Base of F.
• F is true in a Herbrand interpretation I if every
  ground instance of F using terms in the Herbrand
  universe is a true propositional formula in I.
• F is satisfiable if it is true in some Herbrand
  interpretation.

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A Herbrand Interpretation Example

• Example If
  then
• and
• Consider the Herbrand interpretation I which maps
  all atoms containing predicate P to true and all
  others to false.
• Then all ground instances of F are true in I.
• Hence F is satisfiable.

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Unsatisfiability

• A formula is unsatisfiable iff it is false in all
  its Herbrand Interpretations.
• Furthermore, it is unsatisfiable iff some finite
  conjunction of ground instances is an
  unsatisfiable propositional formula.

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M-satisfiability

• F is M-satisfiable if there is some Herbrand
  interpretation I such that every ground instance
  of F is true in I or contains a term with depth
  greater than M.

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Example of M-satisfiability

• Suppose that F is
• Clearly F is unsatisfiable.
• Consider the Herbrand Interpretation which maps
  P(s(a)) to true and all other atoms to false.
• Then F is 1-satisfiable since there is a Herbrand
  interpretation in which every ground instance is
  true or contains a term of depth greater than 1.

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M-Satisfiability For All M

• A formula that is M-satisfiable for all M is
  satisfiable. Furthermore, an M-satisfiable
  formula is (M-1) satisfiable.

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General Form Of The PI Algorithm

• Given a first-order logic formula F determine if
  F (F viewed as a propositional formula) is
  satisfiable.
• If F is unsatisfiable, F is unsatisfiable and
  we are done.
• Else F is satisfiable. Check for conflicts.
• If no conflicts exist, F is satisfiable and we
• are done.
• Else, add additional clauses to F in order to
• resolve conflicts and repeat from beginning.

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Implications of Unsatisfiability as Propositional
Logic Formula

• Suppose F is a first-order formula.
• Let F be the propositional formula derived from
  F where variants are treated as the same atom.
• Suppose F was found to be unsatisfiable.
• How do we conclude F is unsatisfiable?

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Implications of Unsatisfiability as Propositional
Logic Formula

• Let F be a ground instance of F where each
  variable in F is replaced with a term in the
  Herbrand Universe not found in F.
• F is equivalent to F as propositional
  formulas.
• F is therefore unsatisfiable.
• As F is a finite conjunction of ground
  instances of F and F is unsatisfiable as a
  propositional formula, then F is unsatisfiable.

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General Form Of The PI Algorithm

• Given a first-order logic formula F determine if
  F (F viewed as a propositional formula) is
  satisfiable.
• If F is unsatisfiable, F is unsatisfiable and
  we are done.
• Else F is satisfiable. Check for conflicts.
• If no conflicts exist, F is satisfiable and we
• are done.
• Else, add additional clauses to F in order to
• resolve conflicts and repeat from beginning.

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Implication of Satisfiability as Propositional
Formula

• Recall, to show F is satisfiable we need to show
  that there is some interpretation I such that all
  ground instances of F are satisfiable as
  propostional formula under I.
• If some ground instance of F is satisfiable, are
  all ground instances of F satisfiable? Not
  necessarily.

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Implication of Satisfiability as Propositional
Formula

• Consider
• Let . F
  is a ground instance of F that is satisfiable.
• Let
  . F is a ground instance that is
  unsatisfiable.

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A Satisfiability Check

• If a formula is found satisfiable as a
  propositional formula, before claiming the
  formula is satisfiable as a first-order formula,
  we look for conflicts in the truth valuation.
• If a conflict is found we append more
  information to the formula to resolve the
  conflict and run the propositional satisfiability
  checker on the new formula.

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Satisfier Mapping

• Let v be a truth valuation which satisfies F.
• For each clause C in F, let L(C) be a literal of
  C for which v makes L(C) true.
• For each clause, let S(C) be the atom of L(C).
• S is called a satisfier mapping for F.
• S(C) is called a satisfier of C.
• If S(C) L(C) then S(C) is a true satisfier,
  otherwise S(C) is a false satisfier.

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Example Of A Satisfier Mapping

• T
• F
• F
• F

• In this example, the first literal of each clause
  contains the satisfier for the clause.
• i.e. P(s(a)), P(x), Q(s(y)) and R(s(a)) are
  satisfiers.
• P(s(a)) is a true satisfier.
• While P(x), Q(s(y)) and R(s(a)) are false
  satisfiers.

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Blocked

• Given a satisfier mapping S for a quantifier-free
  formula F, a pair of satisfiers P(t), P(t') is
  blocked if
• P(t) is a true satisfier
• P(t') is a false satisfier
• P(t) and P(t') have a most general unifier
  such that P(t) P(t') .
• There are clauses C, C' in F for which P(t) and
  P(t') are respectively satisfiers and for which
  either (a) C is not in F or (b) C' is not
  in F.

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Blocked Example

• Consider
• 
  T F
• is a true satisfier.
• is a false satisfier.
• and have a most
  general unifier
  such that
  
• is not in F.

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Resolving Blocked Pairs

• Suppose the satisfiers P(t) and P(t') are blocked
  in some formula F. Then P(t) is the satisfier
  for some clause C and P(t) is the satisfier for
  some clause C.
• Let be the mgu for P(t) and P(t).
• We attempt to resolve the blockage by conjoining
  the clauses and to F then checking
  for a new propositional valuation.

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Resolving Blocks

• Let where
• T
• 
  F
• We conjoin the following constraints to F

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M-Blocked

• A pair of satisfiers P(t) and P(t') are
  M-blocked if they are blocked and their most
  general unifier is such that P(t) and
  P(t) contains no terms of depth strictly
  greater than M.

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Satisfiability

• Given , let S be a
  satisfier mapping for F. Then
• (a) if S is not M-blocked then F is
• M-satisfiable, and
• (b) if S is unblocked then F is satisfiable.

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A Primal PI Algorithm

• Let F be a first-order formula of the proper
  form.
• 1. Initialization Set
  .
• 2. Ground Satisfiability Try to find a
  satisfier mapping S for that treats
  variants of the same atom as the same atom.
• 3. Termination Check
• - If S does not exist, then stop. F is
  unsatisfiable.
• - Otherwise, if S is unblocked, then stop. F is
  satisfiable.
• - Otherwise, if S is not M-blocked, then F is
  M-satisfiable. Let
• MM1, and repeat step 3.
• 4. Refinement (S is M-blocked) Let and
  be two clauses in whose satisfiers are
  M-blocked, and let be a most general
  unifier of and . Set
  after
  standardizing apart, set kk1 and repeat step 2.

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Example Of Satisfiability With Termination

• T
• F
• F
• F

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Example Of Satisfiability With Termination

• T
• F
• F
• F

• For M 0
• Blocked
• mgu
• is not 0-blocked
• F is 0-satisfiable

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Example Of Satisfiability With Termination

• T
• F
• F
• F

• For M 1
• Blocked
• mgu
• is 1-blocked

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Example Of Satisfiability With Termination

• T
• F
• F
• F
• 
  T

• For M 1
• Blocked
• is not 1-blocked
• F is 1-satisfiable

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Example Of Satisfiability With Termination

• T
• F
• F
• F
• 
  T

• For M 2
• Blocked
• is 2-blocked

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Example Of Satisfiability With Termination

• T
• F
• F
• F
• 
  T
• 
  T

• is unblocked
• PI terminates with
• satisfiability

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Correctness and Completeness of PI

• The algorithm PI indicates (a) unsatisfiability
  only if F is unsatisfiable, (b) satisfiability
  only if F is satisfiable, and (c)
  M-satisfiability only when F is M-satisfiable.
• If F is unsatisfiable, then PI terminates with an
  indication of unsatisfiability.

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Conclusion

• PI is one method to determine the validity of
  first-order logic formula.
• PI methods are appealing with advances in the
  computational performance of propositional logic
  satisfiability algorithms.
• In many cases PI terminates after only a few
  iterations thus providing a timely answer.

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Thank you