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 Resolve pairs 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. Aristotle’s 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. Aristotle’s 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. ¬A  ¬C  B becomes (A  C)  B
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. R&N Example: Kowalski NF

15. R&N Example: Proof Tree

16. R&N Example: Prover9 Input

17. R&N 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
X=X, X=YY=X, etc.
Special axiom for every predicate/function:
X = Y  P(X) = P(Y)

19. Equality & Demodulation
Alternative solution: rewrite with equalities
Demodulation inference rule
X=Y, A[S]
Subst(, A[Y])
Two input clauses (one an equality X=Y)
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
Doesn’t 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 Naive check for subsumption
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
Simon’s 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
Don’t confuse inductive learning (next lecture)

29. Interactive Theorem Proving
Necessary to interact with humans in order to prove theorems of any difficulty
Mathematician’s assistant
Let a theorem prover do simple tasks while you develop a theory (e.g., Buchberger’s 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 Paulson’s 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 Bundy’s 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
champion: Vampire (Voronkov & Riazonov)