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Artificial Intelligence 9. Resolution Theorem Proving

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Title: Artificial Intelligence 9. Resolution Theorem Proving


1
Artificial Intelligence 9. Resolution Theorem
Proving
  • Course V231
  • Department of Computing
  • Imperial College
  • Jeremy Gow

2
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

3
A More Concise Version
  • E.g. for A 1, 2, 7 first clause is L1 ? L2 ?
    L7

4
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

5
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

6
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)

7
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)

8
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
9
Resolution Proof Tree (Proof 1)
10
Resolution Proof Tree (Proof 2)
11
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.

12
Russell Norvig Example
13
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

14
RN Example Kowalski NF
15
RN Example Proof Tree
16
RN Example Prover9 Input
17
RN Example Prover9 Proof
18
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)

19
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) ?
20
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

21
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

22
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

23
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

24
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

25
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

26
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

27
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..

28
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)

29
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

30
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

31
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

32
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)
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