OSHL: A Propositional Prover with Semantics for First-Order Logic

1
OSHL A Propositional Prover with Semantics for
First-Order Logic

• David A. Plaisted
• UNC Chapel Hill

2
Current theorem provers

• Largely syntactic
• Resolution or ME (tableau) based
• First-order provers are often poor on non-Horn
  clauses
• Rarely can solve hard problems
• Human interaction needed for hard problems

3
Unit Resolution and General Resolution

• Resolution is efficient for Horn and renameable
  Horn problems.
• Resolution is efficient if the proof can be found
  by UR resolution.
• Hard problems tend not to be Horn, renameable
  Horn, or UR resolvable.
• Of 1697 TPTP problems provable by Otter in 30
  seconds, 1042 can be proved by UR resolution.

4
Unit Resolution and General Resolution

• Of the 1697 problems provable by Otter, only 297
  were both non Horn and had rating greater than
  zero.
• Of these 297, at most 215 are not UR resolvable.
• Otter can do hundreds of thousands of resolutions
  in 30 seconds on this machine.
• Resolution is inefficient on hard, non UR
  resolvable problems.
• Need for new approaches.

5
How do humans prove theorems?

• Semantics
• Case analysis
• Sequential search through space of possible
  structures
• Focus on the theorem

6
Systematic methods can now routinely solve
verification problems with thousands or tens of
thousands of variables, while local search
methods can solve hard random 3SAT problems with
millions of variables. (from a conference
announcement)
7
DPLL Example
p,r,?p,?q,r,p,?r
pT
pF
T,r,?T,?q,r,T,?r
F,r,?F,?q,r,F,?r
SIMPLIFY
SIMPLIFY
?q,r
r,?r
SIMPLIFY

8
Hyper Linking
9

• Eliminating Duplication with the Hyper-Linking
  Strategy, Shie-Jue Lee and David A. Plaisted,
  Journal of Automated Reasoning 9 (1992) 25-42.

10
Definition Detection
11

• Replacement Rules with Definition Detection,
  David A. Plaisted and Yunshan Zhu, in Caferra and
  Salzer, eds., Automated Deduction in Classical
  and Non-Classical Logics, LNAI 1761 (1998) 80-94.

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More DefinitionsS1 ? S2 ? ? SnSn ? Sn-1 ? ?
S1Left Associative
n OSHL OSHL OSHL Otter Otter Otter Vampire Vampire Vampire E-Setheo DCTP
n time Gen Kept time Gen Kept time Gen Kept time time
2 0.175 41 36 600 100303 24712 0.00 103 90 0.0 0.01
3 0.678 85 80 600 66753 31496 70.1 3606742 50382 0.3 300
4 2.107 141 136 600 47219 22119 300 25898955 68385 0.3 300
5 5.317 207 202 600 46054 20941 300 25298293 67864 2.6 300
6 12.02 283 278 600 60247 22923 300 25612105 68457 300 300
7 38.97 7 3 600 56299 19660 300 25641650 67977 300 300
8 77.94 7 3 600 56352 18932 300 25863117 68542 300 300
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More Definitions

• Similar results for other definitions
• S1 ? S2 ? ? SnSn ? Sn-1 ? ? S1, left side
  left associated, right side right associated
• S1 ? S2 ? ? Sn S1 ? S2 ? ? Sn ? S1 ? S2 ?
  ? Sn, both sides associated to the left
• S1 ? S2 ? ? Sn S1 ? S2 ? ? Sn ? S1 ? S2 ?
  ? Sn, left side left associated, right side right
  associated
• Similar results for n

14
Later propositional strategies

• Billons disconnection calculus, derived from
  hyper-linking
• Disconnection calculus theorem prover (DCTP),
  derived from Billons work
• FDPLL

15
Performance of DCTP on TPTP, 2003

• DCTP 1.3 first in EPS and EPR (largely
  propositional)
• DCTP 10.2p third in FNE (first-order, no
  equality) solving same number as best provers
• DCTP 10.2p fourth in FOF and FEQ (all first-order
  formulae, and formulae with equality)
• DCTP 1.3 is a single strategy prover.

16
Strategy Selection in E
17
Strategy Selection

• Schulz, Stephan, E-A Brainiac Theorem Prover,
  Journal of AI Communications 15(2/3)111-126,
  2002.

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Strategy Selection

• The Vampire kernel provides a fairly large number
  of features for strategy selection. The most
  important ones are
• Choice of the main saturation procedure (i)
  OTTER loop, with or without the Limited Resource
  Strategy, (ii) DISCOUNT loop.
• A variety of optional simplifications.
• Parameterised reduction orderings.
• A number of built-in literal selection functions
  and different modes of comparing literals.
• Age-weight ratio that specifies how strongly
  lighter clauses are preferred for inference
  selection.
• Set-of-support strategy.

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Strategy Selection

• The automatic mode of Vampire 7.0 is derived from
  extensive experimental data obtained on problems
  from TPTP v2.6.0. Input problems are classified
  taking into account simple syntactic properties,
  such as being Horn or non-Horn, presence of
  equality, etc. Additionally, we take into account
  the presence of some important kinds of axioms,
  such as set theory axioms, associativity and
  commutativity. Every class of problems is
  assigned a fixed schedule consisting of a number
  of kernel strategies called one by one with
  different time limits.

20
Various Provers

• PTTP solved 999 of 2200 tested problems.
• Otter proved 1595.
• leanCoP proved 745.
• Source
• Jens Otten and Wolfgang Bibel.leanCoP Lean
  Connection-Based Theorem Proving. Journal of
  Symbolic Computation, Volume 36, pages 139-161.
  Elsevier Science, 2003.
• Vampire 6.0 3286 refutations of 7267 problems,
  more solved

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DCTP Strategy Selection

• DCTP 1.31 has been implemented as a monolithic
  system in the Bigloo dialect of the Scheme
  language.
• DCTP 1.31 is a single strategy prover.
  Individual strategies are started by DCTP 10.21p
  using the schedule based resource allocation
  scheme known from the E-SETHEO system. Of course,
  different schedules have been precomputed for the
  syntactic problem classes. The problem classes
  are more or less identical with the sub-classes
  of the competition organisers.
• In CASC-J2 DCTP 10.21p performed substantially
  better.

22
Semantics

• Gelernter 1959 Geometry Theorem Prover
• Adapt semantics to clause form
• An interpretation (semantics) I is an assignment
  of truth values to literals so that I assigns
  opposite truth values to L and ?L for atoms L.
• The literals L and ?L are said to be
  complementary.

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Semantics
-

• We write I C (I satisfies C) to indicate
  that semantics I makes the clause C true.
• If C is a ground clause then I satisfies C if I
  satisfies at least one of its literals.
• Otherwise I satisfies C if I satisfies all ground
  instances D of C. (Herbrand interpretations.)
• If I does not satisfy C then we say I falsifies C.

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Example Semantics

• Specify I by interpreting symbols
• Interpret predicate p(x,y) as x y
• Interpret function f(x,y) as x y
• Interpret a as 1, b as 2, c as 3
• Then p(f(a,b),c) interprets to TRUE but p(a,b)
  interprets to FALSE
• Thus I satisfies p(f(a,b),c) but I falsifies
  p(a,b)

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Obtaining Semantics

• Humans using mathematical knowledge
• Automatic methods (finite models)
• Trivial semantics

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Goal of OSHL

• First-order logic
• Clause form
• Propositional efficiency
• Semantics
• Requires ground decidability

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Structure of OSHL

• Goal sensitivity if semantics chosen properly
• Choose initial semantics to satisfy axioms
• Use of natural semantics
• For group theory problems, can specify a group
• Sequential search through possible
  interpretations
• Thus similar to Davis and Putnams method
• Propositional Efficiency
• Constructs a semantic tree

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Ordered Semantic Hyperlinking (Oshl)

• Reduce first-order logic problem to propositional
  problem
• Imports propositional efficiency into first-order
  logic
• The algorithm
• Imposes an ordering on clauses
• Progresses by generating instances and refining
  interpretations

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OSHL

• I0 is specified by the user
• Di is chosen minimal so that Ii falsifies Di
• Di is an instance of a clause in S
• Ii is chosen minimal so that Ii satisfies Dj for
  all j lt i
• Let Ti be D0,D1, , Di-1.
• Ii falsifies Di but satisfies Ti
• When Ti is unsatisfiable OSHL stops and reports
  that S is unsatisfiable.

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Clause Ordering

• Llin
• P(f(x),g(x,c))lin 6
• Ldag
• P(f(x),f(x))dag 4
• Extend to clauses additively, ignoring negations
• OSHL chooses Di minimal in such an ordering

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Alternate version of OSHL

• Want to keep the size of T small
• Do this by throwing away clauses of T subject to
  the condition
• The minimal model of Ti1 is larger than the
  minimal model of Ti for all i.
• This guarantees completeness.
• Leads to a formulation using sequences of clauses
  and resolutions between clauses.

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Rules of OSHL Start with empty sequence (C1,C2,
, Cn), D minimal contradict I, I minimal
model (C1,C2, , Cn,D) (C1,C2, , Cn, D), Cn not
needed (C1,C2, , Cn-1,D) (C1,C2, , Cn,D), max
resolution possible (C1,C2, , Cn-1,res(Cn,D,L)) P
roof if empty clause derived
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-
Propositional Example (?p I0 p) () (-p1,
-p2, -p3) I0-p3 (-p1, -p2, -p3, -p4, -p5,
-p6) I0 -p3,-p6 (, , -p7) I0
-p3,-p6,-p7 (, , -p7, p3, p7) (,
-p4, -p5, -p6, p3) (-p1, -p2,
-p3,p3) (-p1, -p2 ) I0 -p2
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Semantics

• Trivial semantics
• Positive Choose I0 to falsify all atoms, first
  D is all positive. Forward chaining.
• Negative Choose I0 to satisfy all atoms, first
  D is all negative. Backward chaining.
• Natural semantics I0 chosen by user

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Semantics Ordering

• ltt a well founded ordering on atoms, extended to
  literals
• Extend ltt to interpretations as follows
• I and J agree on L if they interpret L the same
• Suppose I0 is given
• I ltt J if I and J are not identical, A is the
  minimal atom on which they disagree, and I agrees
  with I0 on A

36
Semantics Ordering

• ltt is not a well founded ordering on
  interpretations. But ltt minimal models of T
  always exist.
• Ii is always chosen as the ltt minimal model of T.
• Theorem Such Ii always has the form I0L1 Lm
  where Li are literals of clauses of T.
• I0L1 Lm L iff at(L) ? at(L1 Ln) and
  I0 L, or for some i L Li.

-
-
37
Instantiation Example

• Suppose I0 interprets arithmetic in the standard
  way.
• Suppose S contains axioms of arithmetic and the
  clause X3?5.
• Then the first instance chosen could be 23?5,
  (11)3?5, (3-1)3?5 et cetera but it could not
  be 33?5, nor could it be an instance of an axiom.

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Instantiation Example

• Suppose the first instance chosen is 23?5.
• Then I1 is I023?5, which interprets all atoms
  as in standard arithmetic except that the
  statement 23?5 is true.
• The next instance chosen might be 23-1 5-1 ?
  23 5. This contradicts I1. It is an instance
  of the clause X-1 Y-1 ? X Y and corresponds
  to generating the subgoal 23-1 5-1.

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U Rules

• Choose clauses instances to match existing
  literals. Look for a contradiction.
• Basic clauses and U clauses
• Basic clauses are used in three rules given
• Sequence can also have U clauses on the end
• U clauses have a selected literal
• In basic clauses the max. lit. is selected
• In U clauses other literals can be selected.
• Significant performance enhancement.

40
U Rules

• UR resolution Find C in S having a ground UR
  resolvent with selected literals. Let C' be the
  corresponding instance of C. Add C' to the end
  of the sequence of clauses and select the UR
  resolvent from it.
• Filtering Find C in S such that NIL is
  derivable by unit resolution from selected
  literals and C. Let C' be the corresponding
  instance of C. Add C' to the end of the sequence
  of clauses. Select a literal from it.

41
U Rules

• Case Analysis Find C in S and L in C such that
  L has all the variables of C. Find instance L'
  of L that is complementary to a selected literal
  of some clause in the sequence. Let C' be the
  corresponding instance of C. Add C' to the end
  of the sequence and select a literal from it.
• This rule expands definitions.

42
Examples of U Rules

• UR resolution Given the sequence (s(a), p(b),
  t(a), q(b)) and the clause not p(X), not q(X),
  r(X) create the sequence (s(a), p(b), t(a),
  q(b), not p(b), not q(b), r(b) )
• Filtering Given the sequence (s(a), p(b),
  t(a), q(b)) and the clause not p(X), not q(X)
  create the sequence (s(a), p(b), t(a), q(b),
  not p(b), not q(b) )

43
Examples of U Rules

• Case analysis Given the sequence (s(a), p(b),
  t(a), q(b)) and the clause not q(X), r(X),
  s(X) create the sequence (s(a), p(b), t(a),
  q(b), not q(b), r(b), s(b) )

44
Example Proof Using U Rules

• All positive semantics
• Clauses
• A1. ?X?Y, ?Y?X, XY
• A2. ?Z?X, ?X?Y, Z?Y
• A3. g(X,Y)?X, X?Y
• A4. ?g(X,Y)?Y, X?Y
• A5. ?Z?X, Z?X ? Y
• A6. ?Z?Y, Z?X ? Y
• A7. ?Z?X ? Y, Z?X, Z?Y
• T. ?A ? B B ? A

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Example Proof Using U Rules

• 1. ?A ? B B ? A (T)
• 2. ?A ? B ? B ? A, ?B ? A ? A ? B, A ? B B ?
  A (Case Analysis, A1)
• 3. ?g(A ? B, B ? A) ? B ? A, A ? B ? B ? A (UR
  resolution, A4)
• 4. g(A ? B, B ? A) ? B ? A, ?g() ? B (UR
  resolution, A5)
• 5. g(A ? B, B ? A) ? B ? A, ?g() ? A (UR
  resolution, A6)
• 6. g() ? B, g() ? A, ?g() ? A ? B (UR
  resolution, A7)
• 7. A ? B ? B ? A, g() ? A ? B (Filtering, A3)

46
Example Proof Using U Rules

• 1. ?A ? B B ? A
• 2. ?A ? B ? B ? A, ?B ? A ? A ? B, A ? B B ?
  A (Case Analysis)
• 3. ?g(A ? B, B ? A) ? B ? A, A ? B ? B ? A (UR
  resolution)
• 4. g(A ? B, B ? A) ? B ? A, ?g() ? B (UR
  resolution)
• 5. g(A ? B, B ? A) ? B ? A, ?g() ? A (UR
  resolution)
• 8. g() ? B, g() ? A, A ? B ? B ? A,
  (Resolution of 6. and 7.)

47
Example Proof Using U Rules

• 1. ?A ? B B ? A
• 2. ?A ? B ? B ? A, ?B ? A ? A ? B, A ? B B ?
  A (Case Analysis)
• 3. ?g(A ? B, B ? A) ? B ? A, A ? B ? B ? A (UR
  resolution)
• 4. g(A ? B, B ? A) ? B ? A, ?g() ? B (UR
  resolution)
• 9. g(A ? B, B ? A) ? B ? A, g() ? B, A ? B ? B
  ? A (Resolution of 8. and 5.)

48
Example Proof Using U Rules

• 1. ?A ? B B ? A
• 2. ?A ? B ? B ? A, ?B ? A ? A ? B, A ? B B ?
  A (Case Analysis)
• 3. ?g(A ? B, B ? A) ? B ? A, A ? B ? B ? A (UR
  resolution)
• 10. g(A ? B, B ? A) ? B ? A (Resolution of 9.
  and 4.)

49
Example Proof Using U Rules

• 1. ?A ? B B ? A
• 2. ?A ? B ? B ? A, ?B ? A ? A ? B, A ? B B ?
  A (Case Analysis)
• 11. A ? B ? B ? A (Resolution of 10. and 3.)

50
Example Proof Using U Rules

• 1. ?A ? B B ? A
• 12. ?B ? A ? A ? B, A ? B B ? A (Resolution
  of 11 and 2)

Now the other half of the proof will be done.
Note that there is only one ascending sequence of
clauses constructed by OSHL and we are only
indicating part of it.
51
Implementation Results

• Slower implementation speed of OSHL
• Uniform strategy versus strategy selection
• The choice of Otter
• Influence of U rules on an earlier version
• None 233 proofs in 30 seconds on TPTP problems
• Using them 900 proofs in 30 seconds
• All results for trivial semantics

52
Implementation Results

• OSHL has no special data structures.
• Implemented in OCaML
• No special equality methods
• Semantics was implemented but frequently only
  trivial semantics was used.
• Thus significant performance improvements are
  possible.

53
Implementation Results
P R O B S Otter Proofs Otter Proofs Otter Proofs Otter Proofs Otter Proofs OSHL Proofs OSHL Proofs OSHL Proofs OSHL Proofs OSHL Proofs
P R O B S All H O R N Non-Horn Non-Horn Non-Horn All H O R N Non-Horn Non-Horn Non-Horn
P R O B S All H O R N All R 0 R gt 0 All H O R N All R 0 R gt 0
All 4417 1697 764 933 297 636 1027 311 716 265 451
FLD 143 28 0 28 11 17 68 0 68 47 21
SET 604 168 2 166 40 126 211 2 209 93 116
Total Number of Proofs, 30 seconds
54
Implementation Results

• Shows that a prover working entirely at the
  ground level can come into the range of
  performance of a respectable resolution theorem
  prover.
• DCTP and FDPLL probably perform better than OSHL.
• DCTP and FDPLL do not work entirely at the ground
  level and do not use natural semantics.

55
Implementation Results
All Horn Non-Horn R0 Rgt0 Non Horn, Rgt0
Clauses, Otter 3483094 215290 3267804 915737 2567357 2460992
Clauses, OSHL 17212 8110 9102 14888 2324 2216
Ratio 202 26.6 359 61.5 1105 1111
For problems for which both provers found
proofs in 30 seconds..
56
Implementation Results

• In a given number of inferences OSHL finds more
  proofs than Otter for non Horn problems

57
Summary of theoretical results about semantics

• Several results show that OSHL with an
  appropriate semantics is implicitly performing
  unifications. Thus the choice of semantics has a
  profound effect on the operation of OSHL.
• OSHL has some features of propositional methods
  and some features of unification-based methods.
• Semantics might significantly improve OSHL.

58
Number of Clauses Generated

• Problem clauses, Otter
  Oshlsemantics
• GRP005-1 57 3
• GRP006-1 62 7
• GRO007-1 85 22
• GRP018-1 266 16
• GRP019-1 267 15
• GRP020-1 265 18
• GRP021-1 264 19
• GRP023-1 79 22
• GRP032-3 83 14
• GRP034-3 141 30
• GRP034-4 222 6
• GRP042-2 21 15
• GRP043-2 80 81
• GRP136-1 0 8
• GRP137-1 0 8

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Lifting Semantics