Artificial Intelligence 9' Resolution Theorem Proving

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Artificial Intelligence 9. Resolution Theorem
Proving

• Course V231
• Department of Computing
• Imperial College
• Jeremy Gow

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The Full Resolution Rule

• If Unify(Pj, Qk) ? ( makes them unifiable)
• P1 ? ? Pm, Q1 ? ? Qn
• Subst(?, P1 ? (no Pj) ? Pm ? Q1 ? (no Qk)
  ... ?Qn)
• Pj and Qk are resolved
• Arbitrary number of disjuncts
• Relies on preprocessing into CNF

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A More Concise Version

• E.g. for A 1, 2, 7 first clause is L1 ? L2 ?
  L7

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Resolution Proving

• Knowledge base of clauses
• Start with the axioms and negation of theorem in
  CNF
• Resolve pairs of clauses
• Using single rule of inference (full resolution)
• Resolved sentence contains fewer literals
• Proof ends with the empty clause
• Signifies a contradiction
• Must mean the negated theorem is false
• (Because the axioms are consistent)
• Therefore the original theorem was true

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Empty Clause means False

• Resolution theorem proving ends
• When the resolved clause has no literals (empty)
• This can only be because
• Two unit clauses were resolved
• One was the negation of the other (after
  substitution)
• Example q(X) and q(X) or p(X) and p(bob)
• Hence if we see the empty clause
• This was because there was an inconsistency
• Hence the proof by refutation

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Resolution as Search

• Initial State Knowledge base (KB) of axioms and
  negated theorem in CNF
• Operators Resolution rule picks 2 clauses and
  adds new clause
• Goal Test Does KB contain the empty clause?
• Search space of KB states
• We want proof (path) or just checking (artefact)

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Aristotles Example (Again)

• Socrates is a man and all men are mortal
  Therefore Socrates is mortal
• Initial state
• 1) is_man(socrates)
• 2) ?is_man(X) ? is_mortal(X)
• 3) is_mortal(socrates) (negation of theorem)
• Resolving (1) (2) gives new state
• (1)-(3) 4) is_mortal(socrates)

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Aristotles Example Search Space
1) is_man(socrates) 2) ?is_man(X) ?
is_mortal(X) 3) is_mortal(socrates)

• 1) is_man(socrates)
• 2) ?is_man(X) ? is_mortal(X)
• 3) is_mortal(socrates)
• 4) is_mortal(socrates)

1) is_man(socrates) 2) is_man(X) ?
is_mortal(X) 3) is_mortal(socrates) 4)
is_man(socrates)
1) is_man(socrates) 2) ?is_man(X) ?
is_mortal(X) 3) is_mortal(socrates) 4)
is_mortal(socrates) 5) False
1) is_man(socrates) 2) ?is_man(X) ?
is_mortal(X) 3) is_mortal(socrates) 4)
is_man(socrates) 5) False
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Resolution Proof Tree (Proof 1)
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Resolution Proof Tree (Proof 2)
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Reading Proof Tree 2

• You said that all men were mortal. That means
  that for all things X, either X is not a man, or
  X is mortal CNF step. If we assume that
  Socrates is not mortal, then, given your previous
  statement, this means Socrates is not a man
  first resolution step. But you said that
  Socrates is a man, which means that our
  assumption was false second resolution step, so
  Socrates must be mortal.

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Russell Norvig Example
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Reminder Kowalski NF

• Can reintroduce ? to CNF, e.g.
• A ? C ? B becomes (A ? C) ? B
• Kowalski normal form
• (A1 ?? An) ? (B1 ?? Bn)
• Resolve in KNF using KNF style rules
• e.g. Binary resolution
• A?B, B?C
• A?C

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RN Example Kowalski NF
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RN Example Proof Tree
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RN Example Prover9 Input
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RN Example Prover9 Proof
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Equality Axioms

• is_pres(obama) and is_pres(b_obama)
• will not unify (syntactically different)
• unification algorithm does not allow this
• Even if we add to the knowledge base
• obama b_obama
• Solution add equality axioms to KB
• XX, XY?YX, etc.
• Special axiom for every predicate/function
• X Y ? P(X) P(Y)

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Equality Demodulation

• Alternative solution rewrite with equalities
• Demodulation inference rule
• XY, AS
• Subst(?, AY)
• Two input clauses (one an equality XY)
• Unify X with a subterm S of other
• Apply unifier to clause with subterm Y (not S)
• Also works unifying with Y and putting in X

Unify(X, S) ?
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Heuristic Strategies

• Pure resolution search tends to be slow
• For interesting problems
• Many clauses in the initial knowledge base
• Each step adds a new clause (which can be used)
• Num. of possible resolution combinations explodes
• Selection Heuristics
• Intelligently choose which pair to resolve
• Pruning Heuristics
• Forbid certain pairs

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Unit Preference Strategy

• Prefer to resolve unit clauses
• Contain only a single literal
• Selection heuristic
• Searching for smallest (empty) clause
• Resolving with the unit clauses keeps small
• Very effective early on for simple problems
• Doesnt reduce branching rate for medium problems

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Set of Support Strategy

• Distinguished subset of KB clauses
• Set of support (SOS) clauses
• Every step must involve SOS (pruning heuristic)
• Must be careful not to lose completeness
• Example SOS strategy
• Initial SOS is negated theorem
• Add new clauses to SOS
• Hence False will be deduced (strategy is
  complete)
• Many provers use SOS, e.g. Prover9

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Input Resolution Strategy

• Special case of SOS strategy
• SOS clauses in the initial knowledge base
• Clearly reduces search space
• Every resolution must involve an original clause
• So number of possible resolutions grows slowly
• Not complete for first order logic
• But complete for Horn-clauses, e.g. Prolog

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Subsumption

• Clause C subsumes clause D
• if C is more general (D is more specific)
• Naive check for subsumption
• Select C2, a subset of literals of C
• Find Unify(C2, D) ?
• ? does not add anything to D (only renames vars)
• Example
• p(george) ? q(X) subsumed by p(A) ? q(B) ? r(C)
• Substitution A/george, X/B
• Second clause is more general

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Subsumption Strategy

• Check each new clause is not subsumed by KB
• Complete strategy
• Specific clauses can be inferred from general
  ones
• So we can throw specific clauses away
• Reduced search space still contains False
• Can be inefficient
• expense must be outweighed by the reduction in
  the search space

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Applications Axioms for Algebras

• Bill McCune and Larry Wos
• Argonne National Laboratories
• FO resolution provers EQP, Otter, Prover9
• Robbins Problem (axioms of Boolean algebras)
• Stated 60 years ago, mathematicians failed
• 1996 EQP solved in 8 days in 1996 (human work)
• General application to algebraic axiomatisations
• Generate possible axioms for algebras
• Prove new axioms equivalent to old

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Applications Theory Formation

• Simons HR system Automated Theory Formation
• Used in mathematical (and bioinformatics) domains
• Theories concepts, examples, conjectures,
  proofs
• HR uses Otter to prove conjectures it makes
• Effective in algebraic domains
• See notes for anti-associative algebra results
• Otter not so effective in number theory
• Used as a triviality filter (discard theorems
  it can prove)
• Example conjectures made by HR (and proved by
  Simon)
• Sum of divisors is prime ? number of divisors is
  prime
• Sum of divisors of a square is an odd number
• Perfect numbers are pernicious and many
  more..

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Inductive Theorem Proving

• Deduction by mathematical induction
• Induction over many different structures
• Allows reasoning about recursion/iteration
• Useful for hardware/software verification
• Dont confuse inductive learning (next lecture)

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Interactive Theorem Proving

• Necessary to interact with humans in order to
  prove theorems of any difficulty
• Mathematicians assistant
• Let a theorem prover do simple tasks while you
  develop a theory (e.g., Buchbergers Theorema)
• Guided theorem prover
• User follows and guides computer proof attempt
• Needs visualisation tools for proof trees

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Higher Order Theorem Proving

• Deduction in higher order logics
• See lecture 4
• Allows more natural and succinct statements
• Logics much less well-behaved
• HOL theorem prover
• Larry Paulsons group in Cambridge
• Has been used for verification tasks
• E.g. verification of crytographic protocols
• Uses induction and interactive control

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Proof Planning

• Initially Alan Bundys group in Edinburgh
• Human proofs often follow a similar structure
• Express this as a outline plan
• Methods represent a patterns of deduction
• Outline plan guides proof search
• Results in specific plan for theorem
• Critics deal with common problems
• Particularly useful for inductive theorems
• Proof of base case and step case follow pattern

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Databases Competitions

• TPTP library (Sutcliffe Suttner)
• Thousands of Problems for Theorem Provers
• Benchmarks for first order provers
• HR is only non-human to add to this library
• Annual CASC competition (Sutcliffe et al.)
• Which is fastest/most accurate FO prover on
  planet?
• Uses blind selection from the TPTP library
• 2002-08 champion Vampire (Voronkov Riazonov)