Automated Theorem Proving: Resolution and Davis-Putnam

1
Automated Theorem ProvingResolution and
Davis-Putnam

• Intermediate Logic

2
Resolution

• Resolution is, like the tree method, a method to
  check for the logical consistency of a set of
  statements.
• Resolution requires all sentences to be put into
  CNF.
• A set of sentences in CNF is made into a clause
  set a set of clauses, where a clause is a set of
  literals.
• Clauses are resolved using the resolution rule,
  and the resulting clause (the resolvent) is added
  to the clause set

L ? C1
L ? C2
CNEW C1/L ? C2/L
3
Putting into CNF
? (Equiv)
?(P ? Q)
?((P ? Q) ? (Q ? P))
? (Impl)
?((?P ? Q) ? (?Q ? P))
? (DeM)
?(?P ? Q) ? ?(?Q ? P)
? (DeM, DN)
(P ? ?Q) ? (Q ? ?P)
? (Dist)
((P ? ?Q) ? Q) ? ((P ? ?Q) ? ?P)
? (Dist)
(P ? Q) ? (?Q ? Q) ? (P ? ?P) ? (?Q ? ?P)
4
Resolution Graph
?(P ? Q)
?(Q ? R)
?(P ? R)
?
?
?
(P ? Q) ? (?P ? ?Q)
(Q ? R) ? (?Q ? ?R)
(P ? R) ? (?P ? ?R)
?
?
?
?P, ?Q
P, Q
Q, R
?Q, ?R
P, R
?P, ?R
?P, R
P, ?Q
?P
P

5
Satisfiability

• A clause is satisfied by a truth-value assignment
  if and only if that assignment makes at least one
  literal in that clause true.
• A clause set is satisfiable if and only if there
  is a truth-value assignment that satisfies all
  clauses in that clause set.
• Figuring out whether some clause set is
  satisfiable is the satisfiability problem.
• Note how the satisfiability problem relates
  directly to the consistency problem figuring out
  whether some set of statements is consistent.
• A set of sentences is consistent if and only if
  the corresponding clause set is satisfiable.

6
Soundness and Completeness of Resolution

• The rule of Resolution is sound, making the
  method of resolution sound as well (so, if the
  empty clause (which is a generalized disjunction
  of 0 disjuncts, which is a contradiction) can be
  resolved from a clause set, then that means that
  that clause set is indeed unsatisfiable.
• It can be shown that resolution is complete, i.e.
  that the empty clause can be resolved from any
  unsatisfiable clause set.

7
Resolutions as Derivations
A, B
1.
A ? (B ? C)
(A ? B) ? (A ? C)
?
?
A, C
2.
?A, D, E
3.
(A ? B) ? (D ? E)
(?A ? D ? E) ? (?B ? D ? E)
?
?B, D, E
4.
?
?
?E
?E
?
5.
?(A ? B) ? (D ? E)
(?A ? ?B) ? (D ? E)
?
?A
?A
?
6.
?C, ?D
?(C ? D)
?
?C ? ?D
?
7.
B
1,6
8.
17.42 from LPL A ? (B ? C) ?E (A ? B) ? (D ?
E) ?A ? C ? D
C
2,6
9.
D, E
4,8
10.
D
5,10
11.
?D
7,9
12.

11,12
13.
8
Resolutions as Decision Procedures

• Resolution can be made into a decision procedure
  by systematically exhausting all possible
  resolvents (of which there are finitely many).
• This will not be very efficient unless we add
  some resolution strategies.

9
Resolution Strategies

• Clause Elimination Strategies
• Tautology Elimination
• Subsumption Elimination
• Pure Literal Elimination
• Resolving Strategies
• Unit Preference Resolution
• Linear Resolution
• Ordered Resolution
• Etc.

10
Tautology Elimination

• A tautologous clause is a clause that contains an
  atomic statement as well as the negation of that
  atomic statement.
• Obviously, for any tautologous clause C, any
  truth-value assignment is going to satisfy C.
• Hence, with S any clause set, and with S the
  clause set S with all tautologous clauses
  removed S is satisfiable if and only if S is
  satisfiable.

11
Subsumption Elimination

• A clause C1 subsumes a clause C2 if and only if
  every literal contained in C1 is contained in C1,
  i.e. C1 ? C2.
• Obviously, if C1 subsumes C2 , then any
  truth-value assignment that satisfies C1 will
  satisfy C2.
• Hence, with S any clause set, and S the clause
  set S with all subsumed clauses removed S is
  satisfiable if and only if S is satisfiable.

12
Pure Literal Elimination

• A literal L is pure with regard to a clause set S
  if and only if L is contained in at least one
  clause in S, but L is not.
• A clause is pure with regard to a clause set S if
  and only if it contains a pure literal.
• Obviously, with S any clause set, and with S the
  clause set S with all pure clauses removed S is
  satisfiable if and only if S is satisfiable.

13
Unit Preference Resolution

• A unit clause is a clause that contains one
  literal.
• Unit preference resolution tries to resolve using
  unit clauses first.

14
Unit Literal Deletion and Splitting

• For any clause set S, SL is the clause set that
  is generated from S as follows
• Remove all clauses from S that contain L.
• Remove all instances of L from all other clauses
• Obviously, with C L ? S, S is satisfiable if
  and only if SL is satisfiable.
• It is also easy to see that for any clause set S,
  and any literal L S is satisfiable if and only
  if SL is satisfiable or SL is satisfiable.
• The last observation suggests a splitting
  strategy that forms the basis of Davis-Putnam.

15
Davis-Putnam

• Recursive routine Satisfiable(S) returns true iff
  S is satisfiable

boolean Satisfiable(S)
begin
if S return true
if S return false
select L ? lit(S)
return Satisfiable(SL) Satisfiable(SL)
end
16
Making Davis-Putnam Efficient Adding Bells and
Whistles

• The routine on the previous slide is not very
  efficient. However, we can easily make it more
  efficient
• return false as soon as ?S
• add the unit rule if L?S return
  Satisfiable(SL)
• strategically add deletion strategies
• strategically choose the literal on which to
  split
• As far as I have gathered from the ATP
  literature, such efficient Davis-Putnam routines
  are credited to do well in comparison to other
  ATP routines.

17
Davis-Putnam as Trees
P, Q
P, ?Q
?P, Q
?P, ?Q
(P)
(?P)
Q
Q
?Q
?Q
(?Q)
(Q)
(?Q)
(Q)




18
Davis-Putnam vs Truth-Trees

• How does Davis-Putnam differ from Truth-Trees?
• Davis-Putnam can be seen as an inside-out
  approach it assigns a truth-value to atomic
  statements and determines the consequences of
  that assignment for the complex statements
  involved.
• Truth-Trees are more of an outside-in approach
  it assigns truth-values to complex statements as
  a whole, and decomposes them accordingly until
  they have been decomposed to literals.
• How does the Davis-Putnam procedure stack up
  against the Truth-Tree procedure in terms of
  efficiency?
• No idea project!

19
Can we do DP without CNF?

• Sure, simply consider a set of statements, and
  see what happens to each of the statements when
  some atomic claim is set to true or false,
  respectively.
• For example, when we set A to True in (A ? B) ?
  (D ? E), the claim becomes (True ? B) ? (D ? E),
  which becomes True ? (D ? E), which becomes D ?
  E.
• Possible project Investigate efficiency of this
  method

20
Rules for DP without CNF
True ? P ? P
True ? P ? True
True ? P ? P
True ? P ? P
? True ? False
? False ? True
False ? P ? False
False ? P ? P
False ? P ? True
False ? P ? ?P
P ? True ? P
P ? True ? True
P ? True ? True
P ? True ? P
P ? False ? False
P ? False ? P
P ? False ? ?P
P ? False ? ?P
21
Example Doing DP without CNF
A ? (B ? C) ?E (A ? B) ? (D ? E) ?A ?(C ? D)
17.42 from LPL A ? (B ? C) ?E (A ? B) ? (D ?
E) ?A ? C ? D
?A
A
B ? C ?E B ? (D ? E) ?(C ? D)
?E D ? E False ?(C ? D)
E
?E

B ? C B ? D ?(C ? D)
B ? C False ?(C ? D)
Etc.

22
Can DP and TT be combined?

• DP really starts to look like a tree method, even
  though DP works inside-out and TT outside-in.
• Can these two methods be combined into one
  method?
• Sure! In fact, you can do some things to make
  this efficient, e.g
• Use inheritance to keep from copying statements
  that are unaffected.
• Project Investigate efficiency of this method
• Project Do the KE rules add any further
  efficiency?

23
Example DP and TT Combo
17.42 from LPL ?A ? B C ? (D ? E) D ? ?C A ?
?E ? C ? B
?A ? B C ? (D ? E) D ? ?C A ? ?E ?(C ? B) C ?B D
? E ?D ??A A E ?E
(youll probably have to reconstruct this one to
follow it it would help if it were annotated
with justifications!)

24
EG and ATP

• EGTT Applying Truth-Trees to EG
• EGDP Applying Davis-Putnam to EG
• EGTTDP Applying both TT and DP to EG

25
EGTT Applying Truth-Trees to EG

• Applying the idea of truth-trees to EG is simple
• Two decomposition rules

?
?
?1
?2
?
?
?1
?2

• Close any branch that contains a literal and its
  complement, or that contains an empty cut.
• Rest remains the same

26
EGTT A Satisfiability Procedure
boolean Satisfiable(G)
begin
if G
return false
?
if G
return false
?
L
L
if G
?
?
return Satisfiable(
?
?
)
return
if G
?1
?2
?
Satisfiable(
?1
?
)
?2
?
) Satisfiable(
else return true
end
27
EGDP Literal Reduction

• For any graph ?, ?L is the graph that is
  generated from ? as follows
• Remove all instances of L
• Replace all instances of L with an empty cut

28
Example Literal Reduction
D
E
A
E
With G
D
GE
A
29
EGDP Satisfiability Decision Procedure
boolean Satisfiable(G)
begin
if G
return true
if G
return false
select L ? lit(G)
return Satisfiable(GL) Satisfiable(GL)
end
30
Adding Bells and Whistles

• Again, this procedure can be made a lot more
  efficient by dealing with empty cuts, double
  cuts, and duplicates more efficiently, by various
  other tautology, subsumption, and pure literal
  deletion strategies, and by strategically picking
  the literal on which to split.

31
Tautology Elimination in EG

• Any subgraph of the following form (i.e. a cut
  containing an empty cut) is a tautology, and can
  therefore be eliminated

?

• Note This can be incorporated into the reduction
  routine. That is, rather than replacing any
  complement of a literal with an empty cut, we can
  remove any cut containing the complement.

32
Subsumption Elimination in EG
?
?
If a graph of the form
exists at a nested level with regard to
?
then the first graph is said to be subsumed by
the second (subsuming) graph, and can therefore
be removed.
33
Pure-Literal Elimination in EG

• A literal that exists at even levels only is said
  to be a pure literal, and can be eliminated for
  satisfiability purposes.
• A literal that exists at odd levels only is also
  a pure literal, and can be replaced with the
  empty cut for satisfiability purposes.

34
Satisfiability Example
E
A
C
D
A
B
C
C
D
) Sat(
)
B
C
Sat(
D
E
A
B
D
B
B
C
D
C
D
) Sat(
Sat(
D
) Sat(
B
) False
D
35
Advantage of EGDP over DP?

• The alleged advantage of EGDP over DP is that
  there is no need to put statements into clauses.
  This not only improves efficiency in that this
  preliminary step can eliminated, but by avoiding
  clauses, the algorithm actually becomes
  inherently more efficient in that 1) it leaves
  statements compact, and 2) it eliminates literals
  at any level.
• Project Investigate to see whether this is true!

36
Example Gaining Efficiency by Avoiding Clauses
A
B
B
DE (2x!)
DC (2x!)
Clausifying
A
B
C
C
A
C
No Clausifying!
DE
DC
A
B
C
A
B
C
B
C
So, by keeping statements compact, less steps
need to be taken!
37
EGTTDP

• Can EG be combined with both TT and DP?
• Yes, the characteristic rule of TT is the
  decomposition rule, and the characteristic rule
  for DP is the splitting rule. We can use both
  rules in a satisfiability procedure.
• Question When is a good time to use
  decomposition, and when is a good time to use
  splitting? Project!

38
EGTTDP A Satisfiability Procedure
boolean Sat(G)
begin
remove double cuts
if G
return true
if G
return false
?
if G
return Sat(
L
?
?L
)
Sat(
if G
?1
?2
?
) Sat(
?1
?
)
?2
?
return
(TT)
select L ? lit(G)
return Sat(GL) Sat(GL)
(DP)
end
39
HW 9

• Show the argument below to be valid using
• 1. Resolution
• 2. Davis-Putnam (on clauses)
• 3. Davis-Putnam (on original statements)
• 4. Davis-Putnam and Truth-Tree combo
• 5. EGTT, EGDP, and EGTTDP

Q ? ?S (P ? Q) ? R ?S ? R ---- ?P ? (Q ? S)