The Model Evolution Calculus
Peter Baumgartner, MPI Saarbruecken and U Koblenz
Cesare Tinelli, U Iowa
Background
• Recent research in propositional satisfiability (SAT)
has been very successful.
• An effective method for SAT was pioneered by
Davis, Putman, Logemann, and Loveland (DPLL).
• The best modern SAT solvers (MiniSat, zChaff,
Berkmin,…) are based on DPLL.
Baumgarter & Tinelli: The Model Evolution Calculus
The DPLL Procedure: Main Idea
{ p  q, q  r  s, p  q, p  r  s, p }
assert: p = T
{ q  r  s, q, r  s }
Baumgarter & Tinelli: The Model Evolution Calculus
The DPLL Procedure: Main Idea
{ p  q, q  r  s, p  q, p  r  s, p }
assert: p = T
{ q  r  s, q, r  s }
assert: q = F
{ r  s, r  s }
Baumgarter & Tinelli: The Model Evolution Calculus
The DPLL Procedure: Main Idea
{ p  q, q  r  s, p  q, p  r  s, p }
assert: p = T
{ q  r  s, q, r  s }
assert: q = F
{ r  s, r  s }
guess: r = T
{ s, s }
Baumgarter & Tinelli: The Model Evolution Calculus
The DPLL Procedure: Main Idea
{ p  q, q  r  s, p  q, p  r  s, p }
assert: p = T
{ q  r  s, q, r  s }
assert: q = F
{ r  s, r  s }
guess: r = T
{ s, s }
contradiction!
Baumgarter & Tinelli: The Model Evolution Calculus
The DPLL Procedure: Main Idea
{ p  q, q  r  s, p  q, p  r  s, p }
assert: p = T
{ q  r  s, q, r  s }
assert: q = F
{ r  s, r  s }
guess: r = F
{}
satisfiable!
Baumgarter & Tinelli: The Model Evolution Calculus
Correctness of DPLL method
Prop. A formula  is satisfiable iff
there is a sequence of guesses
such that DPLL() = 
Baumgarter & Tinelli: The Model Evolution Calculus
Research Questions
• Can we lift DPLL to the first-order level?
• Can we combine successful SAT techniques
(unit propagation, backjumping, learning,…)
with successful first-order techniques
(unification, subsumption, ...)?
Baumgarter & Tinelli: The Model Evolution Calculus
Previous Work
• Instance based methods
- (O)SHL [Plaisted],
- Disconnection method [Billon], [Letz, Stenz],
- Hyper Tableaux Next Generation [Baumgartner],
- Primal/Dual approach [Hooker et al],
- Ganzinger-Korovin method
• First-Order DPLL [Baumgartner]
- proper lifting of split rule
Baumgarter & Tinelli: The Model Evolution Calculus
This Work
The Model Evolution Calculus
≈
First-Order DPLL
+ DPLL’s simplification rules
+ Universal variables
The calculus is a direct lifting of the whole
DPLL to the first-order level.
Baumgarter & Tinelli: The Model Evolution Calculus
Overview
• The DPLL method as a sequent-style calculus
• A model generation view of DPLL
• The Model Evolution calculus as a lifting of the
DPLL calculus
• Properties of the ME calculus
• Further Work
Baumgarter & Tinelli: The Model Evolution Calculus
The DPLL Calculus
  {literals}
  {clauses}
  
Context
(asserted literals)
Baumgarter & Tinelli: The Model Evolution Calculus
The DPLL Calculus
  {literals}
L literal
  {clauses}
empty clause C clause
 subsume 
  , L  C
  
if L  
resolve
  , L  C
  , C
if L  
 close 
  ,
 
Baumgarter & Tinelli: The Model Evolution Calculus
The DPLL Calculus
  {literals}
L literal
  {clauses}
empty clause C clause
 subsume 
  , L  C
  
if L  
resolve
  , L  C
  , C
if L  
 close 
  , L1  ...  Ln
 
Baumgarter & Tinelli: The Model Evolution Calculus
if L1 ,..., Ln  
The DPLL Calculus (cont.)
  {literals}
L literal
 assert 
 split 
  {clauses}
empty clause C clause
  , L
, L  , L

L  
if 

L  
  L  C, 
, L  L  C , 
, L  L  C , 
Baumgarter & Tinelli: The Model Evolution Calculus
C 

if  L  

 L
The DPLL Calculus: Key Insight
  
 can be seen as a finite representation
of a Herbrand interpretation:
I   { L  | L is positive}
If I does not satisfy ,
“repair” it by adding literals to 
Baumgarter & Tinelli: The Model Evolution Calculus
Some Notation
I  ||  iff
I  |  and
there is no '   s.t. I  ' | 
permanently satisfies
Examples:
Note: at the prop. level
I{ p} | p  q but not I { p} || p  q
I  || L iff L  
I{p} | p  q and I {p} || p  q
Baumgarter & Tinelli: The Model Evolution Calculus
The DPLL Calculus Revisited:
A Model Evolution View
 subsume 
  , L  C
  
if I  || L
resolve 
  , L  C
  , C
if I  || L
 close 
  , C
 
Baumgarter & Tinelli: The Model Evolution Calculus
if I  || C
The DPLL Calculus Revisited:
A Model Evolution View
 split
  L  C, 
if (*)
, L  L  C , 
, L  L  C , 
1) not I  | L  C

(*)  2) L not contradictory with 

3) L not contradictory with 
Note:

 I  , L || L
(*) implies 

 I  , L || L
Baumgarter & Tinelli: The Model Evolution Calculus
I  , L || L  C
not I  , L | L  C
The DPLL Calculus Revisited:
A Model Evolution View
 assert 
  , L
if (*)
, L  , L
not I  || L
(*)  
 L not contradictory with 
Baumgarter & Tinelli: The Model Evolution Calculus
Lifting DPLL to First Order Logic
Main questions:
• How to use contexts to represent a FOL
Herbrand interpretation
• What is a contradictory context
• How to check |=
• How to check ||=
• How to repair an interpretation
Baumgarter & Tinelli: The Model Evolution Calculus
First-order Contexts
Sets  of parametric literals L(u,v,..) and
universal literals L(x,y,…)
• parameters (u,v, …) and variables (x,y,…)
both stand for ground terms
• (roughly) a parametric literal L in 
denotes all of its ground instances,
unless L’ for some instance L’ of L
• a universal literal denotes all of its
ground instances, unconditionally
Baumgarter & Tinelli: The Model Evolution Calculus
First-order Contexts: Examples
 = { p(u,v) }
p(u,v)
•  produces every instance of p(u,v)
Baumgarter & Tinelli: The Model Evolution Calculus
First-order Contexts: Examples
 = {p(u,v), p(u,u)}
p(u,v)
p(u,u)
•  produces every instance of p(u,v) except the
instances of p(u,u)
•  produces every instance of p(u,u)
Baumgarter & Tinelli: The Model Evolution Calculus
First-order Contexts: Examples
 = {p(u,v), p(u,u), p(f(u),f(u))}
p(u,v)
p(u,u)
p(f(u),f(u))
Baumgarter & Tinelli: The Model Evolution Calculus
First-order Contexts: Examples
 = {p(f(u),v), p(u,g(v))}
p(f(u),v)
OK
p(u,g(v))
p(f(u),g(v))
Baumgarter & Tinelli: The Model Evolution Calculus
First-order Contexts: Examples
 = {p(f(u),v), p(u,g(v)), p(b,g(v)) }
p(f(u),v)
OK
p(u,g(v))
p(b,g(v))
Baumgarter & Tinelli: The Model Evolution Calculus
First-order Contexts: Examples
 = {p(u,v), p(u,v)}
p(u,v)
p(u,v)
Baumgarter & Tinelli: The Model Evolution Calculus
Not OK!
Contradictory
First-order Contexts: Examples
 = {p(u,v), p(x,x)}
p(u,v)
p(x,x)
 produces every instance of p(x,x)
with no possible exceptions
Baumgarter & Tinelli: The Model Evolution Calculus
First-order Contexts: Examples
 = {p(u,v), p(x,x), p(f(u),f(u)}
p(u,v)
p(x,x)
p(f(u),f(u))
Baumgarter & Tinelli: The Model Evolution Calculus
Not OK!
Contradictory
Initial Context
 = {v}
v
• Lambda produces no positive literals
• We’ll consider only extensions of {v}
Baumgarter & Tinelli: The Model Evolution Calculus
Contexts and Interpretations
Let  be a non-contradictory context with
parametric literals and universal literals
 denotes a Herbrand interpretation:
 L is ground and positive,
I  L

 L is produced by 

Baumgarter & Tinelli: The Model Evolution Calculus
Checking |=
Let  be a non-contradictory context
Let L1  ...  Ln be a (parameter-free) clause
If not I  | L1  ...  Ln then
there are fresh variants K1 ,..., K n of literals in 
and a substitution 
such that
 is a simultaneous mgu of {K1 , L1},...,{K n , Ln }
•  is called a context unifier (of the clause against )
Baumgarter & Tinelli: The Model Evolution Calculus
Checking ||=
Let  be a non-contradictory context
Let L1  ...  Ln be a (parameter-free) clause
I  || ( L1  ...  Ln )
Example
iff there
are fresh variants K1 ,..., K n of literals in 
• I{p(u,v)} ||= (p(x,y)  p(x,x))
and a substitution 
• equivalently, match
such that
{p(x,y), p(x,x)} against {p($,$)}
1.  is a simultaneous mgu of {K1 , L1},...,{K n , Ln }
2. for each i, Pars( Ki )  Pars
Baumgarter & Tinelli: The Model Evolution Calculus
The Model Evolution Calculus:
Semantical View
 assert
 subsume 
not I  || L
  , L
if 
 , L  , L
 L not contr. with 
  , L  C
  
if I  || L
Exactly the same as in DPLL!
resolve 
 close 
  , L  C
  , C
  , C
 
if I  || L
if I  || C
Baumgarter & Tinelli: The Model Evolution Calculus
The Model Evolution Calculus:
Semantical View
 split
  , C  L
if (*)
sko
, L  , C  L
,( L )  , C  L
1)

2)

(*)  3)
4)

5)
 is a context unifier of (C  L) against 
 is admissible
L is not mixed
No Longer
L not contr. with 
( L )sko not contr. with 
Baumgarter & Tinelli: The Model Evolution Calculus
The Split Rule: Example
First, identify falsified clause instance:
Clauses:
Now, split with
x≥y  y≥x
abs(x)≥0
abs(x)≥u
| abs(c)≥u
(x≥y)
 (y≥z)
 (x≥z)
Context: abs(x)≥0
v≥u
v
(abs(x)≥0)  (0≥u)  (abs(x)≥u)
Permanenly
falsified
remainder
 admissible context unifier
Baumgarter & Tinelli: The Model Evolution Calculus

context
unifier
Example
abs(u)≥a  abs(x)≥a, ...
assert
abs(x)≥a, abs(u)≥a  abs(x)≥a, ...
compact
abs(x)≥a  abs(x)≥a, ...
subsume
abs(x)≥a  abs(f(x))≥a  p(x), ...
resolve
abs(x)≥a  p(x), ...
Baumgarter & Tinelli: The Model Evolution Calculus
Further Notions
• Derivation tree
• Exhausted/closed branch
• Derivation/refutation
• Limit tree
• Fair limit tree/derivation
Baumgarter & Tinelli: The Model Evolution Calculus
Main Results: Completeness
Let ( i  i )i  ,with   , be an exausted
branch in a fair limit tree of  0 .
Let  
i  i  j 
 j and  
If   then I  |  0 .
Baumgarter & Tinelli: The Model Evolution Calculus
i  i  j 
 j.
Main Results: Soundness and
Completeness
A clause set  0 is unsatisfiable
iff
it has a refutation.
Baumgarter & Tinelli: The Model Evolution Calculus
Main Results: Proof Convergence
If a clause set  0 is unsatisfiable then
every fair derivation of  0 extends
to a refutation.
Baumgarter & Tinelli: The Model Evolution Calculus
Making ME Efficient
Well-known DPLL improvements:
• Literal selection strategies
Model Elimination:
can exploit don’t care nondeterminism
for remainer literal to split on
• Learning (lemma generation)
not trivial – future work
• Intelligent backtracking (backjumping)
Baumgarter & Tinelli: The Model Evolution Calculus
Backjumping
, L 
 , L 
Baumgarter & Tinelli: The Model Evolution Calculus
Backjumping
, L 
S1
 , L 
Sn
Baumgarter & Tinelli: The Model Evolution Calculus
L not used to close
left subtree
Conclusions
• Full lifting of DPLL achieved
• Properties of DPLL preserved
–
–
–
–
–
sound and complete
proof convergent
simplification rules
model generation paradigm
(no Commit rule as in FDPLL)
• Abstract framework
– Wide range for fair strategies
– Semantically justified redundancy criteria
Baumgarter & Tinelli: The Model Evolution Calculus
Further Work
• Implement the calculus! (in progress)
• Lift DPLL optimizations (backjumping, lemma
generation, …)
• Add equality
• Study decidable fragments
• Add nonmonotonic features
• Build-in theories
• ...
Baumgarter & Tinelli: The Model Evolution Calculus