Compact Encoding of FOL

1
Compact Propositional Encoding ofFirst Order
Theories

• Eyal Amir
• University of Illinois at Urbana-Champaign
• (Joint with Deepak Ramachandran and Igor Gammer)

2
Example Pigeonhole Principle
p1
H1
p2
H2
p3
H3
p4
H4
3
Pigeonhole Classical Encoding

• H1(pi) ? H2(pi) ? H3(pi) ? H4(pi)
• ? H1(pi) ? H2(pi)
• ??p,q (p?q ? Hi(p) ? Hi(q))

Every i
O(n2) props
4
A Reformulation

• Hi(p) pigeon p is not in a hole in the subtree
  rooted at i.

H1
p1
p3
H3
H2
p4
p2
1
2
3
4
5
A Reformulation
Hi(p) pigeon p is not in a hole in the subtree
rooted at i.
?pH1(p) ?(H2(p)? H3(p))
H1
p1
p3
?pH2(p) ?(-H1(p)? -H2(p))
?pH1(p) ?(-H3(p)?-H4(p))
p4
p2
1
2
3
4
6
A Reformulation

• Hi(p) pigeon p is not in a hole in the subtree
  rooted at i.

?pH1(p) ? H2(p)
p1
p3
?p?H1(p) ??H2(p)
?p?H3(p)??H4(p)
p4
p2
1
2
3
4
7
A Reformulation

• Hi(p) pigeon p is not in a hole in the subtree
  rooted at i.

H1
p1
p3
H3
H2
?p,qHi(p)?Hi(q) ?(pq)
p4
p2
8
A Reformulation

• Hi(p) pigeon p is not in a hole in the subtree
  rooted at i.

?p?H1(p)
p1
p3
H3
H2
O(nlogn) props
p4
p2
9
Propositional Encodings Motivation

• How do we do this in general? scientific
  curiosity
• Applications
• Formal verification, BMC specification in FOL
• AI planning, diagnosis, spatial reasoning,
  question answering from knowledge specified in
  FOL
• Useful with many objects, predicates, time steps,
  etc.
• Can use efficient SAT solvers

10
Outline

• Algorithm for compact prop. encoding
• Analysis
• Interpolants correctness of algorithm
• A different Treewidth number of props.
• Towards automatic partitioning
• Related work open questions

11
Naïve Propositional Encoding

• Tasks converted into propositional logic
• P(a) ? Pa ?x P(x) ? Pa ? Pb ? Pc
• (Naive Propositionalization)
• Problem Very many propositions
• PCk for P predicates of arity k,
  and C constant symbols
• Note Assumes Domain Closure (DCA)
• constant symbols name all domain objects

12
Our Results

• New method for propositioal encoding
• Size depends on optimal tree decomposition (e.g.,
  O(PC) sometimes O(n logn) instead of O(n2)
  for Pigeonhole Principle)
• Applications
• Machine scheduling problem
• Board-Level Routing Problem for FPGAs

13
Our New Algorithm

• Rough algorithm (given theory T)
• Partition theory T into T1Tr (1)
• Encode every Ti in propositional logic Ti
  separately (create proposition Pa for constant a
  and predicate P in L(Ti)) (2)
• Return UirTi

14
Partitioned Theory
T

15
Partitioned Theory
T1
T2


16
Propositional Encoding
T1
T2


17
Propositional Encoding
T1
T2


18
Propositional Encoding
T1
T2


19
Propositional Encoding
T1
T2


20
Propositional Encoding
T1
T2


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Propositional Encoding
T1
T2


22
Partitioned Propositional Encoding

• Algorithm
• Partition theory T into T1Tr
• Propositionalize each Ti separately
• Return the propositional encoding UirTi

PiCi props.
? an exponential speed-up for SAT solver
How to make this soundcomplete?
23
Experiments Board-Level Routing
prop. vars per problem
SAT Running time (sec.) per problem
24
Experiments Machine Scheduling
prop. vars per problem
SAT Running time (sec.) per problem
25
Outline

• Algorithm for compact prop. encoding
• Analysis
• Interpolants correctness of algorithm
• A different Treewidth number of props.
• Towards automatic partitioning
• Related work open questions

26
Key to Exact Encoding

• Craigs interpolation theorem (First-Order
  Logic)
• If T1 T2, then there is a formula C including
  only symbols from L(T1) ? L(T2) such that T1 C
  and C T2

T1

T2
P(x)
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Key to Exact Encoding

• Craigs interpolation theorem (First-Order
  Logic)
• If T1 ?T2, then there is a formula C
  including only symbols from L(T1) ? L(T2) such
  that T1 C and C ?T2

T1

?T2
The only interpolants (the Cs) can be
?xP(x),?xP(x), (6 others)
P(x)
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Key to Exact Encoding

• Key idea If our prop. encoding of T1 T2
  implies all possible interpolants in L(T1)?L(T2),
  then the combined encoding is SAT iff T (in FOL)
  is SAT.

T1

?T2
The only interpolants (the Cs) can be
?xP(x),?xP(x), (6 others)
P(x)
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Propositional Encoding
T1
T2


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Correctness of Encoding Alg.

• Encoding of each partition needs to include
  encodings of all possible interpolants (detailed
  next)
• When there are more than two partitions, the
  partitions are arranged in a tree decomposition
  (in next section)

31
Outline

• Algorithm for compact prop. encoding
• Analysis
• Interpolants correctness of algorithm
• A different Treewidth number of props.
• Towards automatic partitioning
• Related work open questions

32
Propositional Encoding Size
T1
T2


33
Propositional encoding of Monadic FOL
Factor single-variable existentially quantified
formula Theorem Can convert any monadic FOL
formula to a conjunction of disjunctions of
factors Example
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Propositionalizing a Partition in Monadic FOL

• Convert Ti to factorized form
• Replace factors in ?i with prop. symbols
• Add meaning of new propositions

e(Ti)
This approach creates PiCi3Pi
propositional symbols.
35
Outline

• Algorithm for compact prop. encoding
• Analysis
• Interpolants correctness of algorithm
• A different Treewidth number of props.
• Towards automatic partitioning
• Related work open questions

36
Automatic Partitioning and Propositional Encoding

• Optimization criteria
• Tree of partitions of axioms that satisfies the
  running intersection property
• Minimize PiCi3Pi (contrast with
  treewidth, where we minimize Pi)
• Can use current algorithms for treewidth and
  decomposition as a starting point
• Sometime need reformulation of axioms

37
Automatic Partitioning

• Find a tree decomposition G of minimum width
• Every node in G is a set of symbols from T
• G satisfies the running intersection property if
  P appears in both v,u in G, then P appears in all
  the nodes on the path connecting them
• Choose G with minimum width (width(G) is the size
  of its largest node v)

38
Outline

• Algorithm for compact prop. encoding
• Analysis
• Interpolants correctness of algorithm
• A different Treewidth number of props.
• Towards automatic partitioning
• Related work open questions

39
Related Work

• McMillan 2005 some ground (QF) FOL theories
  have ground interpolants
• May use his results for similar results on
  compact prop. for those theories
• McMillans talk yesterday may use our compact
  prop. method to generate all interpolants

40
Related Work

• Pnueli, Rodeh, Strichman, Siegel 99, 02
  equational theories (no predicates), and
  weak(er?) bound
• where R resulting universe size, and others
  are problem dependent (worst case n!, for n
  variables/constants)

41
Conclusion

• Created efficient propositional encodings
• Monadic FOL Sound Complete
• Ramsey class Sound, Incomplete without DCA
• Applies to general FOL (with DCA)
• Speedup in SAT solving
• Exploits the structure of real world problems
• Future work Classes beyond Monadic FOL
• Right now ground FOL with equality
• 2-var fragment
• Some software on http//reason.cs.uiuc.edu/eyal

42
THE END
43
Problems with Naïve Prop.

• Problem 1 Very large number of propositions
• PCk for P predicates of arity k,
  and C constant symbols
• Problem 2 Assumes Domain Closure Assumption
  (DCA) only elements are those with constant
  symbols

44
General FOL (with DCA)

• Use DCA repeatedly until we get a formula in
  Monadic FOL
• Replace ?x1Qx2Qxn?(x1,, xn) with
• ?c?C Qx2Qxn?(x1,, xn)
• Replace ?x1Qx2Qxn?(x1,, xn) with
• ?c?C Qx2Qxn?(x1,, xn)

45
Automatic Partitioning

• Begin with logical Knowledge Base in FOL
• Construct symbol graph
• vertex symbol
• edge axiom uses both symbols
• Find a tree decomposition
• Partition axioms using clusters in tree
  decomposition

46
Automatic Partitioning
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47
Automatic Partitioning
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48
Automatic Partitioning
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49
Automatic Partitioning
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can_open
try_open
clean
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time
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let_dry
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50
Automatic Partitioning
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time
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51
Automatic Partitioning
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52
Automatic Partitioning
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53
Automatic Partitioning
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try_open ? open open Ù fetch ?
broom key ? open try_open ? open ? broom ? fetch
broom Ù dry ? can_clean can_clean ?
clean broom Ù let_dry ? dry time ?
let_dry drier ? let_dry time ? drier
broom