Nonclausal Reasoning

1
Non-clausal Reasoning

• Fahiem Bacchus, Christian Thiffault, Toronto
• Toby Walsh, UCC Uppsala
• (soon UNSW, NICTA, Uppsala)

2
Every morning

• I read the plaque on the wall of this house
• Dedicated to the memory of George Boole
• Professor of Mathematics at Queens College (now
  University College Cork)

3
George Boole (1815-1864)

• Boolean algebra
• The Mathematical Analysis of Logic, Cambridge,
  1847
• The Calculus of Logic, Cambridge and Dublin
  Mathematical journal, 1848
• Reduce propositional logic to algebraic
  manipulations

4
George Boole (1815-1864)

• Boolean algebra
• The Mathematical Analysis of Logic, Cambridge,
  1847
• The Calculus of Logic, Cambridge and Dublin
  Mathematical journal, 1848
• Reduce propositional logic to algebraic
  manipulations

Crater on the moon named after him!
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How do we automate reasoning with propositional
formulae?
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Propositional SATisfiability

• Rapid progress being made
• 10 years ago, lt 50 vars
• Today, gt 1000 vars
• Algorithmic advances
• Learning
• Watched literals
• ..
• Heuristic advances
• VSIDS branching

7
Propositional SATisfiability

• Efficient implementations
• Chaff, Berkmin, Forklift,
• SAT competition has new winner almost every year
• Practical applications
• Hardware verification
• Planning

8
SAT folklore

• Need to solve in CNF
• Everything is a clause
• Efficient reasoning
• Optimize code with simple data structures
• Effective reasoning
• Conversion into CNF does not hinder unit
  propagation

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Overturning SAT folklore

• Deciding arbitrary Boolean formulae
• Without converting into CNF
• Efficient reasoning
• Raw speed as good as optimized CNF solvers
• Effective reasoning
• More inference than unit propagation
• Exploit structure
• More exotic gates,

Similar ideas being explored in ATPG
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Davis Putnam procedure

• DPLL(S)
• if S empty then SAT
• if S contains then UNSAT
• if S contains unit, l then DPLL(S u l)
• else chose literal, l
• if DPLL(S u l) then SAT
• else DPLL(S u -l)

11
Unit Propagation

• If the formula has a unit clause then the literal
  in that clause must be true
• Set the literal to true and reduce the formula.
• Unit propagation is the most commonly used type
  of constraint propagation
• One of the most important parts of current SAT
  solvers

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Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
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Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
atrue
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Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
atrue
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Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
atrue
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Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
atrue
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Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
bfalse
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Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
bfalse
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Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
bfalse
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Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
c true
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Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
c true
22
Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
c true
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Implementing Unit Propagation

• UP is main (often only) inference rule applied at
  each search node.
• Performing UP occupies most of the time in these
  solvers.
• More efficient implementations of UP has been one
  of the recent advances.

24
Implementing Unit Propagation

• Most DPLL solvers do not build an explicit
  representation of the reduced formula
• Too expensive in time and space to do this.
• Rather they keep original formula and mark the
  changes made
• All changes generated by UP undone when we
  backtrack.

25
Tableau Crawford and Auton 95

• We number the variables and clauses.
• Each variable has
• a field to store its current value, true, false
  or unvalued
• the list of clauses it appears positively in
• the list of clauses it appears negatively in
• Each clause has
• a list of its literals
• a flag to indicate whether or not it is satisfied
• the number of unvalued literals it contains

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Tableau Crawford and Auton 95

• Unit propagated literal put on a stack
• pop the literal on top of the stack
• mark the variable with the appropriate value.
• mark each clause it appears positively in as
  satisfied.
• for each clause it appears negatively in
• if the clause is not already satisfied decrement
  the clauses counter
• if the counter is equal to 1, the clause is unit
• find the single unvalued literal in the clause
  and add that literal to the UP stack.
• remember all changes so that they can be undone
  on backtrack.

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Watch literals SATO, Chaff

• Tableaus technique requires visiting each clause
  a variable appears in when we value a variable.
• When clause learning is employed, and 100,000s
  of long new clauses are added to the original
  formula this becomes slow.
• The watch literal technique is more efficient.

28
Watch literals SATO, Chaff

• For each clause, pick two literals to watch.
• At least one of these literals must be false for
  the clause to be unit.
• For each variable instead of lists of all of the
  clauses it appears in positively and negatively,
  we only have lists of the clauses it is a watch
  for.
• reduces the total size of these lists from O(kn)
  to O(n)

29
Watch literals SATO, Chaff

• When we assign a value to a variable we
• Ignore the clauses it watches positively
• For each clause it watches negatively, we search
  the clause
• if we find an unvalued literal or a true literal
  not equal to the other watch we replace this
  literal the watch
• otherwise the clause is unit and we UP the other
  watch literal if it is not already true.
• On backtrack we do nothing!
• The new watch literals retain the property that
  at least one of them must become false if the
  clause is to become unit.

30
Solving non-CNF formulae

• Convert into CNF
• Use efficient DPLL solver like Chaff

• Adapt DPLL solver to reason with non-CNF
• Exploit structure
• Permit complex gates (eg counting, XOR, ..)

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Encoding into CNF

• Most common (and relatively efficient?) is that
  of Tseitin 1970.
• Recusively converts a formula by adding a new
  variable for every subformula.
• Linear space

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Tseitin Encoding

• A ? (C D)

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Tseitin Encoding

• A ? (C D)
• V1 ? (C D)
• (V1, C), (V1, D), (C,D,V1)

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Tseitin Encoding

• A ? (C D)
• V1 ? (C D)
• (V1, C), (V1, D), (C,D,V1)
• V2 ? (A ? V1)
• (V2,A,V1), (A, V2), (V1, V2)

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Tseitin Encoding

• A ? (C D)
• V1 ? (C D)
• (V1, C), (V1, D), (C,D,V1)
• V2 ? (A ? V1)
• (V2,A,V1), (A, V2), (V1, V2)

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Disadvantage of CNF

• Structural information is lost
• Flattens formulae into clauses.
• In a Boolean circuit
• Which variables are inputs?
• Which are internal wires?
• Additional variables are added.
• Potentially increases the size of the DPLL search.

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Structural Information

• Not all structural information can be recovered
  Lang Marquis, 1989.
• Recovering structural information can improve
  performance EqSatZ, LSAT.
• Why lose this information in the first place?
• In addition, we can exploit more complex gates

38
Extra Variables

• Potentially increase search space
• Do not branch on any on the newly introduced
  subformula variables.
• Theoretically this can increase exponentially the
  size of smallest DPLL proof Jarvisalo et al.
  2004
• Empirically solvers restricted in this way can
  perform poorly

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Extra Variables

• The alternative is unrestricted branching.
• However, with unrestricted branching, a CNF
  solver can waste a lot of time branching on
  variables that have become irrelevant.

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Irrelevant Variables

• A ? (C D) Afalse
• formula satisfied

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Irrelevant Variables
Solver must still determine that the remaining
clauses are SAT

• A ? (C D)
• V1 ? (C D)
• V2 ? (A ? V1)

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Converting to CNF is Unnecessary

• Search can be performed on the original formula.
• This has been noted in previous work on circuit
  based solvers, e.g. Ganai et al. 2002
• Reasoning with the original formula may permit
  other efficiencies
• E.g. exploiting structure, complex gates

43
DPLL on formulae

• View formulae as DAGs
• Every node has a label (True/ False/ Unassigned)
• Branch on the truth value of any unassigned node
• Use Boolean logic to propagate truth values to
  neighbouring nodes
• Contradiction when node is labeled both True and
  False
• Find consistent labeling with truth values that
  assigns True to root (SAT)
• Or exhaust all possibilities (UNSAT)

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True
\/
\/
False
?
xor
?
B
A


C
D
C
D
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Labeling ? unit propagation

• Labeling a node ? assigning a truth value to
  corresponding var in CNF encoding
• Propagating labels in the DAG ? unit propagation
  in the CNF encoding

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Learning

• Once a contradiction is detected a conflict
  clause can be learned
• set of impossible node assignments
• can use 1-UIP scheme (as in CNF solvers)
• Learned clauses stored and used to unit propagate
  node truth values

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Complex gates

• Gates can have arbitrary degree
• n-ary AND, n-ary OR,
• Gates can be complicated Boolean functions
• n-ary XOR (which requires exponential number of
  CNF clauses)
• cardinality gates (at least one, k out of n, ..)

48
Label propagation

• Use lazy data structures as in CNF solvers
• For example. assign one child as a true watch for
  an AND gate
• Dont check if AND gate can be labeled true until
  its true watch becomes true
• Some benchmarks have AND gates with thousands of
  children
• No intrinsic loss of efficiency in using the DAG
  over CNF.

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Structure based optimizations

• We can also exploit the extra structural
  information the DAG provides
• Two such optimizations
• Dont care propagation to deal with irrelevant
  subformulae
• Conflict clause reduction

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Dont Care labeling

• Add a third truth value to the DAG dont
  care
• A node C is dont care wrt a particular parent P
• If its truth value can no longer affect the truth
  value of P nor any of its P siblings.
• Or P is dont care.
• A node C is dont care if it is dont care wrt to
  all of its parents
• No need to branch on dont cares!

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Dont Care labeling

• Assign a dont care watch parent for each node.
• When P is labeled, C can becom dont care wrt to
  its watch parent P
• If C becomes dont care wrt to its dont care
  watch we look for another watch.
• If we cant find one we know, C has become dont
  care

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True
\/
\/
False
Dont care
?
xor
?
xor
B
B
A
A


C
D
C
D
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Conflict Clause Reductions

• If one learns (L1,L2,...) and one has (L1, L2)
  then we can reduce the conflict clause
• (L1,L2) resolves with (L1,L2,...) to give
  (L2,...)
• Result subsumes the original conflict clause
• In CNF, we would have to search the clause
  database to detect this situation
• Probably not going to be effective

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Conflict Clause Reductions

• Suppose P is an AND node, and C is a child
• Then C implies P
• If we have the conflict clause
• (P,C,X,)
• This reduces to
• (P,X,)
• Equivalent to a resolution step against (C,P)

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Conflict Clause Reductions

• When conflict clause generated
• Search neighbours in DAG for such reductions
• More useful on shorter clauses
• Experimentally found it only worth looking for
  such reductions on clauses of length 100 or less

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Empirical Results.

• We compared with Zchaff
• Tried to isolate impact of CNF v non-CNF
• Made the two solvers as close as possible
• Same magic numbers (e.g., clause database cleanup
  criteria, restart intervals etc.)
• Same branching heuristics
• Expect similar improvements could be obtained
  with others CNF solvers

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Empirical Results caveats

• Lack of non-clausal benchmarks
• Hope SAT-05 competition will include non-CNF
• Benchmarks we did obtain had already been
  transformed into simpler formulas
• No complex XOR or IFF gates

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FVP-UNSAT-2.0 (Velev) Time
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FVP-UNSAT-2.0 Decisions
60
FVP-UNSAT-2.0 Dont Cares
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FVP-UNSAT-2.0 Clause Reduction
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Other Series
63
Conclusions

• No intrinsic reason to convert to CNF
• Many other structure based optimizations remain
  to be investigated
• Branching heuristics
• Non-clausal conflicts
• More complex gates