Transcript Non-clausal Reasoning Fahiem Bacchus, Christian Thiffault, Toronto Toby Walsh, UCC & Uppsala (soon UNSW, NICTA, Uppsala)

Non-clausal Reasoning
Fahiem Bacchus, Christian Thiffault, Toronto
Toby Walsh, UCC & Uppsala
(soon UNSW, NICTA, Uppsala)
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)
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
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
How do we automate reasoning
with propositional formulae?
Propositional SATisfiability

Rapid progress being made



Algorithmic advances




10 years ago, < 50 vars
Today, > 1000 vars
Learning
Watched literals
..
Heuristic advances

VSIDS branching
Propositional SATisfiability

Efficient implementations



Chaff, Berkmin, Forklift, …
SAT competition has new winner almost every year
Practical applications



Hardware verification
Planning
…
SAT folklore

Need to solve in CNF


Efficient reasoning


Everything is a clause
Optimize code with
simple data structures …
Effective reasoning

Conversion into CNF
does not hinder unit
propagation
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Overturning SAT folklore

Deciding arbitrary Boolean formulae


Efficient reasoning


Raw speed as good as optimized CNF solvers
Effective reasoning


Without converting into CNF
More inference than unit propagation
Exploit structure

More exotic gates, …
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})
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
Unit Propagation
(a)
(-a, b, c)
(-b)
(a, d, e)
(-c, d, g)
Unit Propagation
(a)
(-a, b, c)
(-b)
(a, d, e)
(-c, d, g)
a=true
Unit Propagation
(a)
(-a, b, c)
(-b)
(a, d, e)
(-c, d, g)
a=true
Unit Propagation
(a)
(-a, b, c)
(-b)
(a, d, e)
(-c, d, g)
a=true
Unit Propagation
(a)
(-a, b, c)
(-b)
(a, d, e)
(-c, d, g)
a=true
Unit Propagation
(a)
(-a, b, c)
(-b)
(a, d, e)
(-c, d, g)
b=false
Unit Propagation
(a)
(-a, b, c)
(-b)
(a, d, e)
(-c, d, g)
b=false
Unit Propagation
(a)
(-a, b, c)
(-b)
(a, d, e)
(-c, d, g)
b=false
Unit Propagation
(a)
(-a, b, c)
(-b)
(a, d, e)
(-c, d, g)
c = true
Unit Propagation
(a)
(-a, b, c)
(-b)
(a, d, e)
(-c, d, g)
c = true
Unit Propagation
(a)
(-a, b, c)
(-b)
(a, d, e)
(-c, d, g)
c = true
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.
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.
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
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 clause’s
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.
Watch literals [SATO, Chaff]



Tableau’s technique requires visiting
each clause a variable appears in when
we value a variable.
When clause learning is employed, and
100,000’s of long new clauses are added
to the original formula this becomes slow.
The watch literal technique is more
efficient.
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)
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.
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, ..)
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
Tseitin Encoding
A  (C & D)
Tseitin Encoding
A  (C & D)
V1  (C & D)
(~V1, C), (~V1, D), (~C,~D,V1)
1.
2.
3.
(~V1, C)
(~V1, D)
(~C,~D,V1)
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)
1.
2.
3.
(~V1, C)
(~V1, D)
(~C,~D,V1)
4. (~V2, ~A, V1)
5. (A, V2)
6. (~V1, V2)
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)
1.
2.
3.
(~V1, C)
(~V1, D)
(~C,~D,V1)
4.
5.
6.
7.
(~V2, ~A, V1)
(A, V2)
(~V1, V2)
(V2)
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.
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
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
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”.
Irrelevant Variables
A  (C & D)
A=false
formula satisfied
Irrelevant Variables
A  (C & D)
V1  (C & D)
V2  (A  V1)
1. (~V1, C)
2. (~V1, D)
3. (~C,~D,V1)
Solver must still
determine that the
remaining clauses are
SAT
4. (~V2, ~A,V1)
5. (A,V2)
6. (~V1,V2)
7. (V2)
8. (~A)
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
DPLL on formulae

View formulae as DAGs


Branch on the truth value of any unassigned
node



Every node has a label (True/ False/ Unassigned)
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)
True
False
\/

xor
B
A
&
C
D
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
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
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, ..)
Label propagation


Use lazy data structures as in CNF solvers
For example. assign one child as a true watch
for an AND gate



Don’t 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.
Structure based optimizations


We can also exploit the extra structural
information the DAG provides
Two such optimizations


Don’t care propagation to deal with irrelevant
subformulae
Conflict clause reduction
Don’t Care labeling


Add a third “truth” value to the DAG: “don’t care”
A node C is don’t 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 don’t care.
A node C is don’t care if it is don’t care wrt to all
of its parents
No need to branch on don’t cares!
Don’t Care labeling




Assign a don’t care watch parent for each node.
When P is labeled, C can becom don’t care wrt
to its watch parent P
If C becomes don’t care wrt to its don’t care
watch we look for another watch.
If we can’t find one we know, C has become
don’t care
True
False
Don’t care
\/

xor
B
A
&
C
D
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
Conflict Clause Reductions

Suppose P is an AND node, and C is a child


If we have the conflict clause:


(~P,~C,X,…)
This reduces to


Then ~C implies ~P
(~P,X,…)
Equivalent to a resolution step against (C,~P)
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
Empirical Results.

We compared with Zchaff


Made the two solvers as close as possible



Tried to isolate impact of CNF v non-CNF
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
Empirical Results caveats

Lack of non-clausal benchmarks


Hope SAT-05 competition will include nonCNF
Benchmarks we did obtain had already
been transformed into simpler formulas

No complex XOR or IFF gates
FVP-UNSAT-2.0 (Velev) Time
Problem
4pipe
4pipe_1
4pipe_2
4pipe_3
4pipe_4
5pipe
5pipe_1
5pipe_2
5pipe_3
5pipe_4
5pipe_5
6pipe
6pipe_6
7pipe
7pipe_bug
#Vars
5,237
4,647
4,941
5,233
5,525
9,471
8,441
8,851
9,267
9,764
10,113
15,800
17,064
23,910
24,065
Time
Imp/Sec
Zchaff
NoClause
Zchaff
NoClause
188.89
9.87 467,001
509,433
26.55
35.52 512,108
327,098
49.76
36.5 482,896
327,298
144.34
62.03 424,551
316,049
93.83
42.26 470,936
326,186
54.68
33.34 526,457
409,154
126.11
116.18 425,921
280,758
138.62
177.24 437,166
279,298
137.7
134.08 441,319
295,976
873.81
284.62 370,906
270,234
249.11
137.09 456,400
298,903
4,550.92
297.13 322,039
288,855
1,406.18
1,056.56 402,301
267,207
12,717.00
1,657.70 306,433
244,343
128.9
0.29 266,901
403,148
FVP-UNSAT-2.0 Decisions
Problem
4pipe
4pipe_1
4pipe_2
4pipe_3
4pipe_4
5pipe
5pipe_1
5pipe_2
5pipe_3
5pipe_4
5pipe_5
6pipe
6pipe_6
7pipe
7pipe_bug
#Vars
Zchaff
NoClause
5,237
541,195
41,637
4,647
131,223
114,512
4,941
210,169
112,720
5,233
392,564
169,117
5,525
295,841
122,497
9,471
334,761
102,077
8,441
381,921
255,894
8,851
397,550
362,840
9,267
385,239
292,802
9,764
1,393,529
503,128
10,113
578,432
283,554
15,800
5,232,321
435,781
17,064
2,153,346
1,326,371
23,910 12,437,654
1,276,763
24,065
1,075,907
481
FVP-UNSAT-2.0 Don’t Cares
Problem
DC Time No DC
DC Decision
4pipe
9.87
57.68
41,637
4pipe_1
35.52
62.65
114,512
4pipe_2
36.5
94.46
112,720
4pipe_3
62.03
213.27
169,117
4pipe_4
42.26
318.64
122,497
5pipe
33.34
246.93
102,077
5pipe_1
116.18
300.59
255,894
5pipe_2
177.24
360.67
362,840
5pipe_3
134.08
387.65
292,802
5pipe_4
284.62
2097.31
503,128
5pipe_5
137.09
379.19
283,554
6pipe
297.13 10,241.64
435,781
6pipe_6
1,056.56
3,455.35
1,326,371
7pipe
1,657.70 12,685.59
1,276,763
7pipe_bug
0.29
1
481
No DC
198,828
159,049
212,986
365,007
525,623
650,312
489,825
585,133
593,815
1,842,074
543,535
4,726,470
2,615,479
6,687,186
2,006
FVP-UNSAT-2.0 Clause
Reduction
Problem
4pipe
4pipe_1
4pipe_2
4pipe_3
4pipe_4
5pipe
5pipe_1
5pipe_2
5pipe_3
5pipe_4
5pipe_5
6pipe
6pipe_6
7pipe
7pipe_bug
Red. Time No Red. Red. Ave Cls. No
Size
Red
9.87
30.5
40
148
35.52
45.13
77
132
36.5
54.97
84
132
62.03
69.7
108
162
42.26
80.13
112
157
33.34
16.29
93
113
116.18
195.81
140
204
177.24
159.45
165
199
134.08
154.02
165
218
284.62
504.27
208
264
137.09
216.11
172
237
297.13
647.78
232
540
1,056.56 1,421.42
309
380
1,657.70 2,053.92
336
761
0.29
0.29
10
10
Other Series
Series (#probs)
sss-sat-1.0 (100)
vliw-sat-1.1 (100)
fvp-unsat-1.0 (4)
fvp-unsat-2.0 (22)
Series (#probs)
sss-sat-1.0 (100)
vliw-sat-1.1 (100)
fvp-unsat-1.0 (4)
fvp-unsat-2.0 (22)
Zchaff
Time
128
3,284
245
20,903
Dec.
Imp/Sec
Cls Size
2,970,794
728,144
70
154,742,779
302,302
82
3,620,014
322,587
326
26,113,810
327,590
651
NoClause
Time
Dec.
Imp/Sec
Cls Size
225 1,532,843
616,705
39
1,033 4,455,378
260,779
55
172
554,100
402,621
100
4,104 5,537,711
267,858
240
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
…