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Nonclausal Reasoning

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Title: 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!
5
How do we automate reasoning with propositional
formulae?
6
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

9
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
10
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

12
Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
13
Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
atrue
14
Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
atrue
15
Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
atrue
16
Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
atrue
17
Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
bfalse
18
Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
bfalse
19
Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
bfalse
20
Unit Propagation
(a)(-a, b, c)(-b)(a, d, e)(-c, d, g)
c true
21
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
23
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

26
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.

27
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, ..)

31
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

32
Tseitin Encoding
  • A ? (C D)

33
Tseitin Encoding
  • A ? (C D)
  • V1 ? (C D)
  • (V1, C), (V1, D), (C,D,V1)

34
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)

35
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)

36
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.

37
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

39
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.

40
Irrelevant Variables
  • A ? (C D) Afalse
  • formula satisfied

41
Irrelevant Variables
Solver must still determine that the remaining
clauses are SAT
  • A ? (C D)
  • V1 ? (C D)
  • V2 ? (A ? V1)

42
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)

44

True
\/
\/
False
?
xor
?
B
A


C
D
C
D
45
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

46
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

47
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.

49
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

50
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!

51
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

52

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


C
D
C
D
53
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

54
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)

55
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

56
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

57
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

58
FVP-UNSAT-2.0 (Velev) Time
59
FVP-UNSAT-2.0 Decisions
60
FVP-UNSAT-2.0 Dont Cares
61
FVP-UNSAT-2.0 Clause Reduction
62
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
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