Encoding First Order Proofs in SAT

1
Encoding First Order Proofs in SAT

• Todd Deshane, Wenjin Hu, Patty Jablonski, Hai
  Lin, Christopher Lynch, and
• Ralph Eric McGregor
• Clarkson University

2
Introduction

• Novel Approach to First Order Logic Theorem
  Proving
• Method
• Encode proof of rigid unsatisfiability in SAT
• Use incrementally to solve general problem
• Proof in the Form of Rigid Connection Tableau
• Complete Sound

3
Rigidly Unsatisfiable

• A formula F is rigidly unsatisfiable if there
  exists a substitution s such that Fs is
  unsatisfiable.

• R(x)
• R(y)
• R(a) ? R(b)
• Rigidly Unsatisfiable

• R(x)
• R(a) ? R(b)
• Not Rigidly Unsatisfiable

4
Overview

• Preliminaries
• Closed Rigid Connection Tableaux
• Tableau Encodings Algorithms
• Rigid Horn Problems
• Rigid Non-Horn Problems
• General First Order Logic Algorithm
• An Implementation ChewTPTP
• Future Work

5
Setting Definitions

• Classical First Order Logic
• View substitution as a set of equations
• An assignment is a single equation contained in a
  substitution.
• Standard definition of unifier

6
Rigid Clausal Tableaux

• Rigid Clausal Tableaux are
• trees
• with all nodes (except the root node) labeled
  with literals
• with branches labeled with zero or more
  assignments
• with branches either open or closed

7
Rigid Clausal Tableaux

• Defined inductively as follows
• For a set of clauses S
• Base Case A single unlabeled root node is a
  rigid clausal tableau for S.

8
Rigid Clausal Tableaux Expansion Rule

• Given rigid clausal tableau for S
• A leaf node on an open branch
• Some clause C in S
• Construct new rigid clausal tableau for S
• Expand leaf with literals in C
• Label each new branch as open

9
Rigid Clausal Tableaux Closure Rule

• Given rigid clausal tableau for S
• A leaf node labeled L on an open branch
• An ancestor of the leaf labeled K
• s such that Ls Ks, and s is consistent with
  other assignments
• Construct new rigid clausal tableau for S
• Close branch
• Label branch with assignments of s

10
Rigid Connection Tableau

• A rigid connection tableau is a
• rigid clausal tableau
• each clause (except the root clause) in the
  tableau contains some literal which is unifiable
  with the negation of its parent.
• uses Extension Rule

11
Closed Rigid Connection Tableaux

• A rigid connection tableau is closed if each
  branch is closed.

12
Negative Clause at Root

• A negative clause is a clause which contains only
  negative literals.
• Theorem If there exists a closed rigid
  connection tableau for a set of clauses, S, then
  there exists a closed rigid connection tableau
  for S which has a negative clause off the root in
  the tableau.

13
Example
Figure 1

• R(a)
• Q(b) ? R(y)
• P(a) ? Q(x)
• P(w)

Figure 2
14
Example Continued

• R(a)
• Q(b) ? R(y)
• P(a) ? Q(x)
• P(w)

Figure 3
Figure 4
15
Proof for Rigid Unsatisfiability

• Theorem Let F be a first order logic formula.
  There exists a closed rigid connection tableau
  for F iff F is rigidly unsatisfiable.

16
Our Goal

• To generate a propositional logic encoding G for
  F such that G is propositionally satisfiable iff
  G is an encoding of a rigid closed connection
  tableau for F.

17
Tableau Encoding

• Let F be a first order formula.
• We start by enumerating each clause in F and each
  of the literals in each clause.
• We denote clause i by Ci and the jth literal in
  clause i by Lij.

• P(w)
• P(a) ? Q(x)
• Q(b) ? R(y)
• R(a)

• C1 L11
• C2 L21 ? L22
• C3 L31 ? L32
• C4 L41

18
Defining Propositonal Variables For Horn Problems

• Define cm T iff Cm appears in the tableau.
• Define lmn T iff Lmn is a negative literal in
  the tableau.
• Define emnq T iff Cq is an extension of Lmn.
• Define ua T iff a is an assignment implied by
  the substitutions used in the closure rules.
• Define pmq T iff there exists a path from Cm to
  Cq.

19
Defining Propositional Clauses For Horn Problems

• 1. \/ ci , where ci is a negative clause
• 2. cm ? lmn , where lmn is a negative literal in
  cm
• 3. lmn (\/ emnq) , where Lmn is unifiable with
  the postitive literal in Cq
• 4. emnq ? cq
• 5. emnq ? ust , where s t is an assignment in
  the unifier of Lmn and the positive literal in Cq

20
Encoding For Horn Problems, continued

• 6. uxs ? uxt , where s and t are not
  unifiable
• 7. uxs ? uxt ? uyr , where y r is an
  assignment in the unification of s
  and t
• 8. emnq ? pmq
• 9. pmq ? pqs ? pms , (transitivity of path
  relation)
• 10. pmm , (no cycles)

21
Example

• R(a)
• Q(b) ? R(y)
• P(a) ? Q(x)
• P(w)

• 1. c1
• 2. c1 ? l11
• 3. c2 ? l21
• 4. c3 ? l31
• 5. l11 ? e112
• 6. e112 ? c2
• 7. e112 ? uya
• 8. l21 ? e213
• 9. e213 ? c3
• 10. e213 ? uxb
• 11. l31 ? e314
• 12. e314 ? c4
• 13. e314 ? uwa
• And so on 105 total clauses

22
Algorithm For Rigid Horn Problems

• Input F, a set of FO formula in CNF.
• Output RIGIDLY UNSATISFIABLE or RIGIDLY
  SATISFIABLE
• generate set of encodings, S, for F
• loop
• run SAT solver on S
• if SAT solver returns SATISFIABLE
• run occurs-check procedure
• if no occurs-check return (RIGIDLY
  UNSATISFIABLE)
• else augment S and continue in loop
• else if SAT solver returns UNSATISFIABLE
• return (RIGIDLY SATISFIABLE)

23
Encoding For Non-Horn Problems

• Similar to Horn Encoding
• Closure property encoded differently

24
Sharing Expressed As Tree

• P ? Q
• P ? Q
• P ? Q
• P ? Q

25
Algorithm For Rigid Non-Horn Problem

• May Need Multiple Copies of Clauses

26
Algorithm For Rigid Non-Horn Problems

• Input F, a set of FO formula in CNF.
• Output RIGIDLY UNSATISFIABLE or RIGIDLY
  SATISFIABLE
• loop
• generate set of encodings, S, for F
• run SAT solver on S
• if SAT solver returns SATISFIABLE
• run occurs-check procedure
• if no occurs-check return (RIGIDLY
  UNSATISFIABLE)
• else augment S and continue in loop
• else if SAT solver returned UNSATISFIABLE
• augment F with additional copies of
  original clauses

27
General First Order Theorem Proving

• May Need Additional Instances

28
General Algorithm

• Input F, a set of FO formula in CNF.
• Output RIGIDLY UNSATISFIABLE or RIGIDLY
  SATISFIABLE
• loop
• generate set of encodings, S, for F
• run SAT solver on S
• if SAT solver returns SATISFIABLE
• run occurs-check
• if no occurs-check return (RIGIDLY
  UNSATISFIABLE)
• else augment S and continue in loop
• else if SAT solver returned UNSATISFIABLE
• augment F with additional instances of
  original clauses

29
ChewTPTP An Implementation

• Written in C on Linux
• Options
• TPTP Input
• Uses Minisat
• Experimental Results

30
ChewTPTP Example

• This is MiniSat 2.0 beta
• WARNING for repeatability, setting FPU to use
  double precision
• Problem Statistics
  
• 
  
• Number of variables 0
  
• Number of clauses 4
  
• Parsing time 0.00 s
  
• Search Statistics
  
• Conflicts ORIGINAL
  LEARNT Progress
• Vars Clauses Literals
  Limit Clauses Lit/Cl
• 
  
• 0 0 0 0
  0 0 nan 0.000
• 
  
• restarts 1
• conflicts 0 (nan /sec)
• decisions 1 (0.00
  random) (inf /sec)
• propagations 109 (inf /sec)
• conflict literals 0 ( nan
  deleted)
• Memory used 0.37 MB

31
Future Work

• Find good way to represent equality
• Find way to decide which clauses should have
  multiple instances
• Replace SAT solver with SAT solver modulo
  theories

32
Thank You