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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
— ICT Graduate School Course —Trento, May 2002
SAT: PropositionalSatisfiability and Beyond
Roberto SebastianiDept. of Information and Communication Technologies
University of Trento, [email protected] http://www.dit.unitn.it/˜rseba
c Roberto SebastianiICT Graduate School, Trento, May-June 2002 1
SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
PART 1:PROPOSITIONALSATISFIABILITY
ICT Graduate School, Trento, May-June 2002 2
SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Basics on SAT
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Basic notation & definitions
Boolean formulaare formulas
A propositional atom is a formula;if and are formulas, then , , ,
, are formulas.Literal: a propositional atom (positive literal) or itsnegation (negative literal)
: the set of propositional atomsoccurring in .a boolean formula can be represented as a tree or asa DAG
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Basic notation & definitions (cont)
Total truth assignment for :.
Partial Truth assignment for :, .
Set and formula representation of an assignment:can be represented as a set of literals:
EX:can be represented as a formula:
EX:
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Basic notation & definitions (cont)
( satisfies ):
...is satisfiable iff for some
( entails ):iff for every
( is valid):iff for every
is valid is not satisfiableICT Graduate School, Trento, May-June 2002 6
SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Equivalence and equi-satisfiability
and are equivalent iff, for every ,iff
and are equi-satisfiable iffexists s.t. iff exists s.t.
, equivalent
, equi-satisfiable
EX: and , not in, are equi-satisfiable but not equivalent.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Complexity
The problem of deciding the satisfiability of apropositional formula is NP-complete [14].The most important logical problems (validity,inference, entailment, equivalence, ...) can bestraightforwardly reduced to satisfiability, and are thus(co)NP-complete.
No existing worst-case-polynomial algorithm.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
NNF, CNF and conversions
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
POLARITY of subformulas
Polarity: the number of nested negations modulo 2.Positive/negative occurrences
occurs positively in ;if occurs positively [negatively] in ,then occurs negatively [positively] inif or occur positively [negatively] in ,then and occur positively [negatively] in ;if occurs positively [negatively] in ,then occurs negatively [positively] in andoccurs positively [negatively] in ;if occurs in ,then and occur positively and negatively in ;
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Negative normal form (NNF)
is in Negative normal form iff it is given only byapplications of to literals.every can be reduced into NNF:1. substituting all ’s and ’s:
2. pushing down negations recursively:
The reduction is linear if a DAG representation is used.Preserves the equivalence of formulas.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Conjunctive Normal Form (CNF)
is in Conjunctive normal form iff it is a conjunction ofdisjunctions of literals:
the disjunctions of literals are called clausesEasier to handle: list of lists of literals.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Classic CNF Conversion
Every can be reduced into CNF by, e.g.,1. converting it into NNF;2. applying recursively the DeMorgan’s Rule:
Worst-case exponential..
is equivalent to .Normal: if equivalent to , then identicalto modulo reordering.Rarely used in practice.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Labeling CNF conversion [43, 18]
Every can be reduced into CNF by, e.g.,1. converting it into NNF;2. applying recursively bottom-up the rules:
being literals and being a “new” variable.Worst-case linear.
.is equi-satisfiable w.r.t. .
Non-normal.More used in practice.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Labeling CNF conversion (improved)
As in the previous case, applying instead the rules:
Smaller in size.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
k-SAT and Phase Transition
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
The satisfiability of k-CNF (k-SAT) [21]
k-CNF: CNF s.t. all clauses have literalsthe satisfiability of 2-CNF is polynomialthe satisfiability of k-CNF is NP-complete forevery k-CNF formula can be converted into 3-CNF:
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Random K-CNF formulas generation
Random k-CNF formulas with variables and clauses:
DO
1. pick with uniform probability a set of atoms over
2. randomly negate each atom with probability
3. create a disjunction of the resulting literals
UNTIL different clauses have been generated;
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Random k-SAT plots
fix andfor increasing , randomly generate and solve(500,1000,10000,...) problems with k, L, Nplot
satisfiability percentagesmedian/geometrical mean CPU time/# of steps
against
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
The phase transition phenomenon: SAT % Plots [40, 38]
Increasing we pass from 100% satisfiable to100% unsatisfiable formulasthe decay becomes steeper withfor , the plot converges to a step in thecross-over point ( for k=3)Revealed for many other NP-complete problemsMany theoretical models [52, 22]
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
0
0.2
0.4
0.6
0.8
1
3 3.5 4 4.5 5 5.5 6CLAUSE # / VAR #
SAT%
N=50N=100N=200
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
The phase transition phenomenon: CPU times/step #
Using search algorithms (DPLL):
Increasing we pass from easy problems, to veryhard problems down to hard problemsthe peak is centered in the satisfiable pointthe decay becomes steeper withfor , the plot converges to an impulse in thecross-over point ( for k=3)easy problems ( ) increase polynomiallywith , hard problems increase exponentially withIncreasing , satisfiable problems get harder,unsatisfiable problems get easier.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
0
5000
10000
15000
20000
25000
30000
3 3.5 4 4.5 5 5.5 6CLAUSE # / VAR #
MEDIAN
N=50N=100N=200
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
0
5000
10000
15000
20000
25000
3 3.5 4 4.5 5 5.5 6CLAUSE # / VAR #
GEOMEAN
N=50N=100N=200
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Basic SAT techniques
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Truth Tables
Exhaustive evaluation of all subformulas:
Requires polynomial space.Never used in practice.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Semantic tableaux [51]
Search for an assignment satisfyingapplies recursively elimination rules to the connectivesIf a branch contains and , ( and ) for some, the branch is closed, otherwise it is open.
if no rule can be applied to an open branch , then;
if all branches are closed, the formula is not satisfiable;
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Tableau elimination rules
-elimination
-elimination
-elimination
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Tableau algorithm
Tableau/* branch closed */
False;/* -elimination */
Tableau ;/* -elimination */
Tableau ;/* -elimination */
TableauTableau ;
...True; /* branch expanded */
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Semantic Tableaux – summary
Branches on disjunctionsHandles all propositional formulas (CNF not required).Intuitive, modular, easy to extend
loved by logicians.Rather inefficient
avoided by computer scientists.Requires polynomial space
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
DPLL [17, 16]
Davis-Putnam-Longeman-Loveland procedure (DPLL)Tries to build recursively an assignment satisfying ;At each recursive step assigns a truth value to (allinstances of) one atom.Performs deterministic choices first.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
DPLL rules
( is a pure literal in iff it occurs only positively).
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
DPLL Algorithm
DPLL( )/* base */
True;/* backtrack */
False;a unit clause occurs in /* unit */
DPLL( );a literal occurs pure in /* pure */
DPLL( );l := choose-literal( ); /* split */
DPLL( )DPLL( );
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
DPLL – summary
Branches on truth values.Postpones branching as much as possible.Handles CNF formulas (non-CNF variant known[3, 27]).Mostly ignored by logicians.Probably the most efficient SAT algorithm
loved by computer scientists.Requires polynomial spaceChoose literal() critical!Many very efficient implementations [55, 50, 7, 42].A library: SIM [26]
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Ordered Binary Decision Diagrams (OBDDs) [11]
Normal representation of a boolean formula.variable ordering imposed a priory.Binary DAGs with two leaves: 1 and 0Paths leading to 1 represent modelsPaths leading to 0 represent counter-modelsOnce built, logical operations (satisfiability, validity,equivalence) immediate.Finds all models.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
(Implicit) OBDD structure
,,
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
OBDD - Examples
FT
b3
a3
b3
b2
a2
b2
b1b1
a1 a1
a2
a3 a3 a3
a2
a3
b1 b1 b1 b1 b1 b1
b2 b2 b2 b2
b3 b3
b1 b1
T F
Figure 1: BDDS of with differentvariable orderings
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Incrementally building an OBDD
,,
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
OBDD – summary
(Implicitly) branch on truth values.Handle all propositional formulas (CNF not required).Find all models.Factorize common parts of the search tree (DAG)Require setting a variable ordering a priori (critical!)Very efficient for some problems (circuits, modelchecking).Require exponential space in worst-caseUsed by Hardware community, ignored by logicians,recently introduced in computer science.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Incomplete SAT techniques: GSAT [48]
Hill-Climbing techniques: GSATlooks for a complete assignment;starts from a random assignment;Greedy search: looks for a better “neighbor”assignmentAvoid local minima: restart & random walk
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
GSAT algorithm
GSAT( )Max-tries
:= rand-assign( );Max-flips
( )True;
Best-flips := hill-climb( );:= rand-pick(Best-flips);
:= flip( );
“no satisfying assignment found”.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
GSAT – summary
Handle only CNF formulas.IncompleteExtremely efficient for some (satisfiable) problems.Require polynomial spaceUsed in Artificial Intelligence (e.g., planning)Variants: GSAT+random walk, WSATNon-CNF Variants: NC-GSAT [45], DAG-SAT [47]
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
SAT for non-CNF formulas
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Non-CNF DPLL [3]
NC DPLL( )/* base */
True;/* backtrack */
False;s.t. equivalent unit /* unit */
NC DPLL( );s.t. equivalent pure /* pure */
NC DPLL( );l := choose-literal( ); /* split */
NC DPLL( )NC DPLL( );
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Non-CNF DPLL (cont.)
:
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Non-CNF DPLL (cont.)
:
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Applying DPLL to [27, 25]
inapplicable in most cases.introduces new variables
size of assignment space passes from toIdea: values of new variables derive deterministicallyfrom those of original variables.Realization: restrict to split first onoriginal variables
DPLL assigns the other variables deterministically.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Applying DPLL to (cont)
If basic is used:
then B is deterministicaly assigned by unit propagationif and are assigned.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
If the improved is used:
then B is deterministically assigned:by unit propagation if and are assigned to .by pure literal if one of and is assigned to .
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Non-CNF GSAT [45]
NC GSAT( )Max-tries
:= rand-assign( );Max-flips
( )True;
Best-flips := hill-climb( );:= rand-pick(Best-flips);
:= flip( );
“no satisfying assignment found”.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Non-CNF GSAT (cont.)
computes directly in linear time.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
DPLL Heuristics &Optimizations
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Techniques to achieve efficiency in DPLL
Preprocessing: preprocess the input formula so that tomake it easier to solveLook-ahead: exploit information about the remainingsearch space
unit propagationpure literalforward checking (splitting heuristics)
Look-back: exploit information about search which hasalready taken place
BackjumpingLearning
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Variants of DPLL
DPLL is a family of algorithms.
different splitting heuristicspreprocessing: (subsumption, 2-simplification)backjumpinglearningrandom restarthorn relaxation...
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Splitting heuristics - Choose literal()
Split is the source of non-determinism for DPLLChoose literal() critical for efficiencymany split heuristics conceived in literature.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Some example heuristics
MOM heuristics: pick the literal occurring most often inthe minimal size clauses
fast and simpleJeroslow-Wang: choose the literal with maximum
estimates ’s contribution to the satisfiability ofSatz: selects a candidate set of literals, perform unitpropagation, chooses the one leading to smallerclause set
maximizes teh effects of unit propagation
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Some preprocessing techniques
Sorting+subsumption:
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Some preprocessing techniques (cont.)
2-simplifying [9]: exploiting binary clauses.
1. build the implication graph induced by literals2. detect strongly connected cycles
equivalence classes of literals3. perform substitutions4. perform unit and pure.
no more simplification possible.Very suseful for many application domains.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Conflict-directed backtracking (backjumping) [7, 50]
Idea: when a branch fails,1. reveal the sub-assignment causing the failure
(conflict set)2. backtrack to the most recent branching point in the
conflict seta conflict set is constructed from the conflict clause bytracking backwards the unit-implications causing it andby keeping the branching literals.when a branching point fails, a conflict set is obtainedby resolving the two conflict sets of the two branches.may avoid lots of redundant search.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Conflict-directed backtracking – example
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Conflict-directed backtracking – example (cont.)
(initial assignment)
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Conflict-directed backtracking – example (cont.)
(branch on )
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Conflict-directed backtracking – example (cont.)
(unit )ICT Graduate School, Trento, May-June 2002 63
SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Conflict-directed backtracking – example (cont.)
(unit )ICT Graduate School, Trento, May-June 2002 64
SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Conflict-directed backtracking – example (cont.)
(unit )ICT Graduate School, Trento, May-June 2002 65
SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Conflict-directed backtracking – example (cont.)
Conflict set: backtrack to
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Conflict-directed backtracking – example (cont.)
(branch on )
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Conflict-directed backtracking – example (cont.)
(unit )ICT Graduate School, Trento, May-June 2002 68
SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Conflict-directed backtracking – example (cont.)
conflict set: .
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Conflict-directed backtracking – example (cont.)
conflict set:backtrack to .
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Learning [7, 50]
Idea: When a conflict set is revealed, then canbe added to the clause set
DPLL will never again generate an assignmentcontaining .May avoid a lot of redundant search.Problem: may cause a blowup in space
techniques to control learning and to drop learnedclauses when necessary
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Learning – example (cont.)
Conflict set:learn
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
SOME APPLICATIONS
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Many applications of SAT
Many successful applications of SAT:Boolean circuits(Bounded) Planning(Bounded) Model CheckingCryptographyScheduling...
All NP-complete problem can be (polynomially)converted to SAT.Key issue: find an efficient encoding.
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Application #1:(Bounded) Planning
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
The problem [37, 36]
Problem Given a set of action operators , (arepresentation of) an initial state I and goal state G,and a bound n, find a sequence of operatorapplications , leading from the initial state tothe goal state.Idea: Encode it into satisfiability problem of a booleanformula
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Example
B B
GOAL
A C
A
INITIAL
C
T
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Encoding
Initial states:
Goal states:
Action preconditions and effects:
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Encoding: Frame axioms
Classic
“At least one action” axiom:
Explanatory
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Planning strategy
Sequential for each pair of actions and , addaxioms of the form for each odd time step.For example, we will have:
parallel for each pair of actions and , add axioms ofthe form for each odd time step if effectscontradict preconditions. For example, we will have
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Application #2:Bounded Model Checking
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Bounded Planning
Incomplete techniquevery efficient: competitive with state-of-the-artplannerslots of enhancements [37, 36, 19, 25]
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The problem [8]
Ingredients:A system written as a Kripke structure
S: set of statesI: set of initial statesT: transition relation
: labeling functionA property written as a LTL formula:
a propositional literal, , , , , and ,
, , , , “next”, “globally”, “eventually”, “until”and “releases”
an integer (bound)ICT Graduate School, Trento, May-June 2002 83
SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
The problem (cont.)
Problem:Is there an execution path of of length satisfying thetemporal property ?:
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The encoding
Equivalent to the satisfiability problem of a booleanformula defined as follows:
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
The encoding of and
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Example: (reachability)
: is there a reachable state in which holds?is:
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Example:
: is there a path where holds forever?is:
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SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Example: (fair reachability)
: is there a reachable state in whichholds provided that q holds infinitely often?
is:
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Bounded Model Checking
incomplete techniquevery efficient for some problemslots of enhancements [8, 1, 49, 53, 13]
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PART 2:BEYONDPROPOSITIONALSATISFIABILITY
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Goal
Extending SAT procedures to moreexpressive domains [30, 46, 5]
Two viewpoints:
(SAT experts) Export the efficiency of SAT techniquesto other domains(Logicians) Provide a new “SAT based” generalframework from which to build efficient decisionprocedures(alternative, e.g., to semantic tableaux)
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FORMAL FRAMEWORK
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Ingredients
A logic language extending boolean logic:Language-specific atomic expression are formulas(e.g., , , ,
)if and formulas, then , , ,
, are formulas.Nothing else is a formula(e.g., no external quantifiers!)
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Ingredients (cont.)
A semantic for extending standard boolean one:[definition specific for ]
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Ingredients (cont.)
A language-specific procedure able to decidethe satisfiability of lists of atomic expressions and theirnegationsE.g.:
( ) Sat( Unsat
( )Unsat
UnsatICT Graduate School, Trento, May-June 2002 96
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Definitions: atoms, literals
An atom is every formula in whose main connectiveis not a boolean operator.A literal is either an atom (a positive literal) or itsnegation (a negative literal).Examples:
,,
, ,,
: the set of top-level atoms in .
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Definitions: total truth assignment
We call a total truth assignment for a total function
We represent a total truth assignment either as a setof literals
or as a boolean formula
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Definitions: partial truth assignment
We call a partial truth assignment for a partialfunction
Partial truth assignments can be represented as setsof literals or as boolean functions, as before.A partial truth assignment for is a subset of a totaltruth assignment for .If , then we say that extends and thatsubsumes .a conflict set for is an inconsistent subsets.t. no strict subset of is inconsistent.
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Definitions: total and partial truth assignment (cont.)
Remark:Syntactically identical instances of the same atom inare always assigned identical truth values.E.g.,Equivalent but syntactically different atoms in maybe assigned different truth values.E.g.,
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Definition: propositional satisfiability in
A truth assignment for propositionally satisfies in ,written , iff it makes evaluate to :
A partial assignment propositionally satisfies iff alltotal assignments extending propositionally satisfy .
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Definition: propositional satisfiability in (cont)
Intuition: If is seen as a boolean combination of itsatoms, is standard propositional satisfiability.Atoms seen as (recognizable) blackboxesThe definitions of , is straightforward.
stronger than : if , then , but notvice versa.E.g., , but
.
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Satisfiability and propositional satisfiability in
Proposition: is satisfiable in iff there exists a truthassignment for s.t.
, and
is satisfiable in .
Search decomposed into two orthogonal components:Purely propositional: search for a truth assignments
propositionally satisfyingPurely domain-dependent: verify the satisfiability in
of .ICT Graduate School, Trento, May-June 2002 103
SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Example
, but is unsatisfiable, as contains conflict sets:
, and is satisfiable ( ).ICT Graduate School, Trento, May-June 2002 104
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Complete collection of assignments
A collection of (possibly partial)assignments propositionally satisfying is complete iff
for every total assignment s.t. , there iss.t. .
represents all assignments.“compact” representation of the whole set of total
assignments propositionally satisfying .
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Complete collection of assignments and satisfiability in
Proposition. Let be a completecollection of truth assignments propositionally satisfying
. Then is satisfiable if and only if is satisfiable forsome .
Search decomposed into two orthogonal components:Purely propositional: generate (in a lazy way) acomplete collection of truthassignments propositionally satisfying ;Purely domain-dependent: check one by one thesatisfiability in of the ’s.
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Redundancy of complete collection of assignments
A complete collection of assignmentspropositionally satisfying is
strongly non redundant iff, for every ,is propositionally unsatisfiable,
non redundant iff, for every , is nomore complete,redundant otherwise.
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If is redundant, then for some :
If is strongly non redundant, then is nonredundant:
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Redundancy: example
Let , , , atoms. Then
1.is the set of all total
assignments propositionally satisfying ;
2.is complete but redundant;
3. is complete, non redundant but notstrongly non redundant;
4. is complete and strongly nonredundant.
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A GENERALIZEDSEARCH PROCEDURE
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Truth assignment enumerator
A truth assignment enumerator is a total function() which takes as input a formula
in and returns a complete collection ofassignments propositionally satisfying .
A truth assignment enumerator isstrongly non-redundant ifis strongly non-redundant, for every ,non-redundant if isnon-redundant, for every ,redundant otherwise.
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Truth assignment enumerator w.r.t. SAT solver
Remark. Notice the difference:
A SAT solver has to find only one satisfyingassignment —or to decide there is none;A Truth assignment enumerator has to find a completecollection of satisfying assignments.
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A generalized procedure
( )
( ) /* next in */( )
( );((satifiable = False) )
( False)True; /* a satisf. assignment found */
False; /* no satisf. assignment found */
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( ) terminating, correct and complete( ) terminating, correct and complete.
depends on only forrequires polynomial space iff
requires polynomial space andis lazy
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Mandatory requirements for an assignment enumerator
An assignment enumerator must always:(Termination) terminate(Correctness) generate assignments propositionallysatisfying(Completeness) generate complete set of assignments
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Mandatory requirements for ()
() must always:(Termination) terminate(Correctness & completeness) return if issatisfiable in , otherwise
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Efficiency requirements for an assignent enumerator
To achieve the maximum efficiency, an assignentenumerator should:
(Laziness) generate the assignments one-at-a-time.(Polynomial Space) require only polynomial space(Strong Non-redundancy) be strongly non-redundant(Time efficiency) be fast[(Symbiosis with ) be able to tale benefitfrom failure & success information provided by
(e.g., conflict sets, inferred assignments)]
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Benefits of (strongly) non-redundant generators
Non-redundant enumerators avoid generating partialassignments whose unsatisfiability is a propositionalconsequence of those already generated.Strongly non-redundant enumerators avoid generatingpartial assignments covering areas of the searchspace which are covered by already-generated ones.Strong non-redundancy provides a logical warrant thatan already generated assignment will never begenerated again.
no extra control required to avoid redundancy.
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Efficiency requirements for ()
To achieve the maximum efficiency, () should:
(Time efficiency) be fast(Polynomial Space) require only polynomial space[(Symbiosis with ) be able toproduce failure & success information (e.g., conflictsets, inferred assignments)][(Incrementality) be incremental: ( )reuses computation of ( )]
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EXTENDING EXISTINGSAT PROCEDURES
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General ideas
Existing SAT procedures are natural candidates to beused as assignment enumerators.
Atoms labelled by propositional atomsSlight modifications(backtrack when assignment found)Completeness to be verified!(E.g., DPLL with Pure literal)Candidates: OBDDs, Semantic Tableaux, DPLL
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OBDDs
In an OBDDs, the set of paths from the root torepresent a complete collection of assignmentsSome may be inconsistent inReduction: [12, 41]1. inconsistent paths from the root to internal nodes
are detected2. they are redirected to the (0) node3. the resulting OBDD is simplified.
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OBDD: example
a
T F
F
{a}
{-a,b}
b
T
T F
(a) (a)
(b) (b)
OBDD
OBDD of .ICT Graduate School, Trento, May-June 2002 123
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OBDD reduction: example
T(b)F(b)
a
T F
F
{a}b
T (a) (a)a
T F
F
{a}
T (a) (a)a
T F
F
{a}
{-a,b}
b
T
T F
(a) (a)
(b) (b)
Reduced OBDD of ,, .
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OBDD: summary
strongly non-redundanttime-efficientfactor sub-graphsrequire exponential memorynon lazy[allow for early pruning][do not allow for backjumping or learning]
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Generalized semantic tableaux
General rules = propositional rules + -specific rules
-specificRules
Widely used by logicians
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Generalized tableau algorithm
-Tableau/* branch closed */
False;/* -elimination */
-Tableau ;/* -elimination */
-Tableau ;/* -elimination */
-Tableau-Tableau ;
...( ( )= satisfiable); /* branch expanded */
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General tableaux: example
Tableau Search Graph
a -g a -g a -g
a g
{a} {a,b}{a,-g} {a,b} {b} {b,-g} {a,g} {b,g}
b
b bb
Tableau search graph for .
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Generalized tableaux: problems
Two main problems [15, 29, 30]
syntactic branchingbranch on disjunctionspossible many duplicate or subsumed branches
redundantduplicates search (both propositional anddomain-dependent)
no constraint violation detectionincapable to detect when current branches violate aconstraint
lots of redundant propositional search.ICT Graduate School, Trento, May-June 2002 129
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Syntactic branching: example
T T
a
a a
-a -a -a
b b
-b
-b -b -b
G
Tableau search graph for .
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Detecting constraints violations: example
-a -b -a -b -a -b
a
f
f
T
T1
1
2
2b
. . . . .
T3
G
Tableau search graph for
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Generalized tableaux: summary
lazyrequire polynomial memoryredundanttime-inefficient[allow backjumping][do not allow learning]
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Remark.The word “Tableau” is a bit overloaded in literature. Someexisting (and rather efficient) systems, like FacT and DLP[34], call themselves “Tableau” procedures, although theyuse a DPLL-like technique to perform boolean reasoning.Same discourse holds for the boolean system KE [15]and its derived systems.
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Generalized DPLL
General rules = propositional rules + -specific rules
-specificRules
No Pure Literal Rule: Pure literal causes incompleteassignment sets!
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Pure literal and Generalized DPLL: Example
A satisfiable assignment propositionally satisfying is:
No satisfiable assignment propositionally satisfyingcontainsPure literal may assign as first step
return unsatisfiable.ICT Graduate School, Trento, May-June 2002 135
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Generalized DPLL algorithm
-DPLL( )/* base */
( ( )=satisfiable);/* backtrack */
False;a unit clause occurs in /* unit */
-DPLL( );l := choose-literal( ); /* split */
-DPLL( )-DPLL( );
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General DPLL: example
a -a
b -b{a}
{-a,b} g
DPLL search graph
DPLL search graph for .
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Generalized DPLL vs. generalized tableaux
Two big advantages: [15, 29, 30]
semantic vs. syntactic branchingbranch on truth valuesno duplicate or subsumed branches
strongly non redundantno search duplicates
constraint violation detectionbacktracks as soon as the current branch violates aconstraint
no redundant propositional search.
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Semantic branching: example
T
a -a
-b -b
Tableau search graph for .
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Detecting constraints violations: example
T1
a -a
-b
T23
DPLL search graph for
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Generalized DPLL: summary
lazyrequire polynomial memorystrongly non redundanttime-efficient[allow backjumping and learning]
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Optimizations
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Possible Improvements
Preprocessing atoms [28, 34, 5]Static learning [2]Early pruning [28, 12, 4]Enhanced Early pruning [4]Backjumping [34, 54]Memoizing [34, 24]Learning [34, 54]Triggering [54, 4]
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Preprocessing atoms [28, 34, 5]
Source of inefficiency: semantically equivalent butsyntactically different atoms are not recognized to beidentical [resp. one the negation of the other] theymay be assigned different [resp. identical] truth values.
Solution: rewrite trivially equivalent atoms into one.
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Preprocessing atoms (cont.)
Sorting: , ,);
Rewriting dual operators:, ,
Exploiting associativity:,
;Factoring , ,
;Exploiting properties of :
, if ;...
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Preprocessing atoms: summary
Very efficient with DPLLPresumably very efficient with OBDDsScarcely efficient with semantic tableaux
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Static learning [2]
Rationale: Many literals are mutually exclusive(e.g., )Preprocessing step: detect these literals and addbinary clauses to the input formula:(e.g., )(with DPLL) assignments including both literals arenever generated.requires steps.
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Static learning (cont.)
Very efficient with DPLLPossibly very efficient with OBDDs (?)Completely ineffective with semantic tableaux
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Early pruning [28, 12, 4]
rationale: if an assignment is unsatisfiable, then allits extensions are unsatisfiable.the unsatisfiability of detected during itsconstruction,
avoids checking the satisfiability of all the up toassignments extending .
Introduce a satisfiability test on incompleteassignments just before every branching step:
Likely-Unsatisfiable( ) /* early pruning */( ( ) )
False;
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DPLL+Early pruning
-DPLL( )/* base */
( ( )=satisfiable);/* backtrack */
False;a unit clause occurs in /* unit */
-DPLL( );Likely-Unsatisfiable( ) /* early pruning */
( ( ) )False;
l := choose-literal( ); /* split */-DPLL( )-DPLL( );
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Early pruning: example
Suppose it is built the intermediate assignment:
If is invoked on , it returns , andbacktracks without exploring any extension
of .ICT Graduate School, Trento, May-June 2002 151
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Early pruning: drawback
Reduces drastically the searchDrawback: possibly lots of useless calls to
to be used with care when callsrecursively (e.g., with modal logics)Roughly speaking, worth doing when each branchsaves at leastPossible solutions:
introduce a selective heuristic Likely-unsatisfiableuse incremental versions of
one split.
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Early pruning: Likely-unsatisfiable
Rationale: if no literal which may likely cause conflictwith the previous assignment has been added sincelast call, return false.Examples: return false if they are added only
boolean literalsdisequalitiesatoms introducing new variables...
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Early pruning: incrementality of
With early pruning, lots of incremental calls to:
( ) satisfiable( ) satisfiable( ) satisfiable
...incremental: ( ) reuses
computation of ( ) without restarting fromscratch lots of computation savedrequires saving the status of
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Early pruning: summary
Very efficient with DPLL & OBDDsPossibly very efficient with semantic tableaux (?)In some cases may introduce big overhead(e.g., modal logics)Benefits if is incremental
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Enhanced Early Pruning [4]
In early pruning, is not effective if it returns“satisfiable”.
( ) may be able to derive deterministically asub-assignment s.t. , and return it.The literals in are then unit-propagated away.
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Enhanced Early Pruning: Examples
(We assume that all the following literals occur in .)
If and , thencan derive from .
If and ,then can derive from .
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Enhanced Early Pruning: summary
Further improves efficiency with DPLLPresumably scarcely effective with semantic tableauxEffective with OBDDs?Requires a sophisticated
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Backjumping (driven by ) [34, 54]
Similar to SAT backjumpingRationale: same as for early pruningIdea: when a branch is found unsatisfiable in ,1. returns the conflict set causing the failure2. backtracks to the most recent branching
point in the conflict set
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Backjumping: Example
( ) returns false with the conflict set:
can jump back directly to the branching point, without branching on .
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Backjumping vs. Early Pruning
Backjumping requires no extra calls toEffectiveness depends on the conflict set , i.e., onhow recent the most recent branching point in is.Example: no pruning effect with the conflict set:
Same pruning effect as with Early Pruning only withthe best conflict setMore effective than Early Pruning only when theoverhead compensates the pruning effect (e.g., modallogics with high depths).
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Backjumping: summary
Very efficient with DPLLNever applied to OBDDsVery efficient with semantic tableauxAlternative to but less effective than early pruning.No significant overhead
must be able to detect conflict sets.
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Memoizing [34, 24]
Idea 1:When a conflict set is revealed, then can becached into an ad hoc data structure
( ) checks first if (any subset of) iscached. If yes, returns unsatisfiable.
Idea 2:When a satisfying (sub)-assignment is found,then can be cached into an ad hoc data structure
( ) checks first if (any superset of) iscached. If yes, returns satisfiable.
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Memoizing (cont.)
Can dramatically prune search.May cause a blowup in memory.Applicable also to semantic tableaux.Idea 1 subsumed by learning.
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Learning (driven by ) [34, 54]
Similar to SAT learningIdea: When a conflict set is revealed, then canbe added to the clause set
DPLL will never again generate an assignmentcontaining .May avoid a lot of redundant search.Problem: may cause a blowup in space
techniques to control learning and to drop learnedclauses when necessary
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Learning: example
returns the conflict set:
it is added the clause
Prunes up to assignmentsthe smaller the conflict set, the better.
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Learning: summary
Very efficient with DPLLNever applied to OBDDsCompletely ineffective with semantic tableauxMay cause memory blowup
must be able to detect conflict sets.
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Triggering [54, 4]
Proposition Let be a non-boolean atom occurring onlypositively [resp. negatively] in . Let be a completeset of assignments for , and let
Then is satisfiable if and only if there exist a satisfiables.t. .
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Triggering (cont.)
If we have non-boolean atoms occurring only positively[negatively] in , we can drop any negative [positive]occurrence of them from the assignment to bechecked byParticularly useful when we deal with equality atoms(e.g., ), as handling negative equalitieslike forces splitting:
.
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Application Fields
Modal LogicsDescription LogicsTemporal LogicsBoolean+Mathematical reasoning (Temporalreasoning, Resource Planning, Verification of TimedSystems, Verification of systems with arithmeticaloperators, verification of hybrid systems)QBF...
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CASE STUDY:MODAL LOGIC(S)
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Satisfiability in Modal logics
Propositional logics enhanced with modal operators, , etc.
Used to represent complex concepts like knowledge,necessity/possibility, etc.Based on Kripke’s possible worlds semantics [39]Very hard to decide [32, 31](typically PSPACE-complete or worse)Strictly related to Description Logics [44](ex: )Various fields of application: AI, formal verification,knowledge bases, etc.
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Syntax
Given a non-empty set of primitive propositionsand a set of modal operators, the modal language is the least set
of formulas containing , closed under the set ofpropositional connectives and the set ofmodal operators in .
depth( ) is the maximum number of nested modaloperators in .“ ” can be interpreted as “Agent knows ”
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Semantics
A Kripke structure for is a tuple, where
is a set of statesis a function ,
each is a binary relation on the states of .
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Semantics (cont)
Given s.t. , is defined as follows:
for everys.t. holds in .
for somes.t. holds in .
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Semantics (cont)
The (normal) modal logics vary with the properties of :
Axiom Property of DescriptionB symmetricD serialT reflexive4 transitive5 euclidean
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Normal Modal Logic Properties ofK —KB symmetricKD serialKT = KDT (T) reflexiveK4 transitiveK5 euclideanKBD symmetric and serialKBT = KBDT (B) symmetric and reflexiveKB4 = KB5 = KB45 symmetric and transitiveKD4 serial and transitiveKD5 serial and euclideanKT4 = KDT4 (S4) reflexive and transitiveKT5 = KBD4 = KBD5 = KBT4 = KBT5 = KDT5 =KT45 = KBD45 = KBT45 = KDT45 = KBDT4 =KBDT5 = KBDT45 (S5)
reflexive, transitive and symmetric(equivalence)
K45 transitive and euclideanKD45 serial, transitive and euclidean
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Axiomatic framework
Basic Axioms:
Specific Axioms:
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Axiomatic framework (cont.)
Inference rules:
modus ponens
necessitation
Correctness & completeness:is valid can be deduced
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Tableaux for modal K(m)/ [20]
Rules = tableau rules + -specific rules
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DPLL for K(m)/ : K-SAT [28, 29]
Rules = DPLL rules + -specific rules
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The K-SAT algorithm [28, 29]
( )( );
( )/* base */
( );/* backtrack */
False;a unit clause occurs in /* unit */
( );Likely-Unsatisfiable( ) /* early pruning */
( )False;
l := choose-literal( ); /* split */( )( );
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The K-SAT algorithm (cont.)
( )box index
(False;
True;
(conjunct “ ”
( )False;
True;
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: Example
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: Example (cont.)
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: Example (cont.)
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Example
Resulting Kripke Model:
11
11
( A v A v A ) 5 4 3( A v A v A ) 2 1 4
( A v A v A ) 3 1 2( A v A v A ) 4 2 3
2
A 2
2
1
1 1
3 1 2 A , A , A 4 2 A , A , A , A , A 5 13
4 5( A v A v A )
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Search in modal logic:
Two alternating orthogonal components of search:Modal search: model spanning
jumping among statesconjunctive branchingup to linearly many successors
Propositional search: local searchreasoning within the single statesdisjunctive branchingup to exponentially many successors
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ff41
42
mm
m
411412
413
f
mm
m
m
m
12
3
4
5
SearchDepth
ModalViewPropositionalView
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Some Systems
Kris [6], CRACK [10],Logics: & many description logicsBoolean reasoning technique: semantic tableauOptimizations: preprocessing
K-SAT [28, 23]Logics: K(m),Boolean reasoning technique: DPLLOptimizations: preprocessing, early pruning
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Some Systems (cont.)
FaCT & DLP [34]Logics: & many description logicsBoolean reasoning technique: DPLL-likeOptimizations: preprocessing, memoizing,backjumping + optimizations for description logics
ESAT &*SAT [24]Logics: non-normal modal logics, K(m),Boolean reasoning technique: DPLLOptimizations: preprocessing, early pruning,memoizing, backjumping, learning
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Some empirical results [23]
0 20 40 60 80 100 12010−2
10−1
100
101
102
103MEDIAN CPU TIME [SECS] (N=4, d=1, 100 samples/point)
# OF CLAUSES (L)
Kris (Tableau based)TA (Transl. based)KsatLisp (Sat based)KsatC (Sat based)% satisfiable
0 20 40 60 80 100 1200
1000
2000
3000
4000
5000
6000DPLL CALLS − PS1 (N=4, d=1, %p=0)
# OF CLAUSES (L)#
CALL
S
KsatLispKsatC% satisf
Left: KRIS, TA, K-SAT (LISP), K-SAT (C) median CPUtime, 100 samples/point.Right: K-SAT (LISP), K-SAT (C) median number ofconsistency checks, 100 samples/point.Background: satisfiability percentage.
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Some empirical results (cont.)
020
4060
80100
120
5060
7080
90100
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
# OF CLAUSES (L)
KsatC CPU TIME − PS2 (N=4, d=1, %p=0)
percentiles
CPU
TIM
E [S
EC]
050
100150
200
5060
7080
90100
0
5
10
15
20
25
30
35
# OF CLAUSES (L)
KsatC CPU TIME − PS3 (N=5, d=1, %p=0)
percentiles
CPU
TIM
E [S
EC]
0100
200300
400
5060
7080
90100
0
200
400
600
800
1000
# OF CLAUSES (L)
KsatC CPU TIME − PS4 (N=6, d=1, %p=0)
percentiles
CPU
TIM
E [S
EC]
020
4060
80100
120
5060
7080
90100
0
100
200
300
400
500
600
700
# OF CLAUSES (L)
TA CPU TIME − PS2 (N=4, d=1, %p=0)
percentiles
CPU
TIM
E [S
EC]
050
100150
200
5060
7080
90100
0
200
400
600
800
1000
# OF CLAUSES (L)
TA CPU TIME − PS3 (N=5, d=1, %p=0)
percentiles
CPU
TIM
E [S
EC]
0100
200300
400
5060
7080
90100
0
200
400
600
800
1000
# OF CLAUSES (L)
TA CPU TIME − PS4 (N=6, d=1, %p=0)
percentilesCP
U TI
ME
[SEC
]
K-SAT (up) versus TA (down) CPU times.
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Some empirical results [35]
Formulas of Tableau’98 competition [33]branch d4 dum grz lin path ph poly t4p
K p n p n p n p n p n p n p n p n p nleanK 2.0 1 0 1 1 0 0 0 21 21 4 2 0 3 1 2 0 0 0
KE 13 3 13 3 4 4 3 1 21 2 17 5 4 3 17 0 0 3LWB 1.0 6 7 8 6 13 19 7 13 11 8 12 10 4 8 8 11 8 7TA 9 9 21 18 21 21 21 21 21 21 20 20 6 9 16 17 21 19*SAT 1.2 21 12 21 21 21 21 21 21 21 21 21 21 8 12 21 21 21 21Crack 1.0 2 1 2 3 3 21 1 21 5 2 2 6 2 3 21 21 1 1Kris 3 3 8 6 15 21 13 21 6 9 3 11 4 5 11 21 7 5Fact 1.2 6 4 21 8 21 21 21 21 21 21 7 6 6 7 21 21 21 21DLP 3.1 19 13 21 21 21 21 21 21 21 21 21 21 7 9 21 21 21 21
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45 branch dum grz md path ph poly t4pKT p n p n p n p n p n p n p n p n p n
TA 17 6 13 9 17 9 21 21 16 20 21 16 5 12 21 1 11 0Kris 4 3 3 3 3 14 0 5 3 4 1 13 3 3 2 2 1 7FaCT 1.2 21 21 6 4 11 21 21 21 4 5 5 3 6 7 21 7 4 2DLP 3.1 21 21 19 12 21 21 21 21 3 21 16 14 7 21 21 12 21 21
45 branch dum grz md path ph poly t4pS4 p n p n p n p n p n p n p n p n p n
KT4 1 6 2 3 0 17 5 8 21 18 1 2 2 2 2 2 0 3leanS4 2.0 0 0 0 0 0 0 1 1 2 2 1 0 1 0 1 1 0 0
KE 8 0 21 21 0 21 6 4 3 3 9 6 4 3 1 21 3 1LWB 1.0 3 5 11 7 9 21 8 7 8 6 8 6 4 8 4 9 9 12TA 9 0 21 4 14 0 6 21 9 10 15 21 5 5 21 1 11 0FaCT 1.2 21 21 4 4 2 21 5 4 8 4 2 1 5 4 21 2 5 3DLP 3.1 21 21 18 12 21 21 10 21 3 21 15 15 7 21 21 21 21 21
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SAT techniques for modal logics: summary
SAT techniques have been successfully applied tomodal/description logicsMany optimizations applicable.Currently at the State-of-the-art.
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CASE STUDY:(LINEAR) MATHEMATICALREASONING
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MATH-SAT
Boolean combinations of mathematical propositions onthe reals or integers.Typically NP-completeVarious fields of application: temporal reasoning,scheduling, formal verification, resource planning, etc.
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Syntax
Let be the domain of either reals or integers withits set of arithmetical operators.Given a non-empty set of primitive propositions
and a set of (linear) mathematicalexpressions over , the mathematical language is theleast set of formulas containing and closed underthe set of propositional connectives .
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Syntax: math-terms and math-formulas
a constant is a math-term;a variable over is a math-term;
is a math-term, and being a constantand a variable over ;if and are math-terms, then and aremath-terms, .a boolean proposition over is amath-formula;if , are math-terms, then is amath-formula, ;if , are math-formulas, then , ,
, and , are math-formulas.ICT Graduate School, Trento, May-June 2002 200
SAT: Propositional Satisfiability and Beyond c Roberto Sebastiani
Interpretations
Interpretation: a map assigning real [integer] andboolean values to math-terms and math-formulasrespectively and preserving constants and operators:
, for every ;, for every constant ;, for every variable over ;
, for all math-terms , and;
, for all math-terms , and;
, for every math-formula ;, for all math-formulas , .
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DPLL for math-formulas [54, 2, 4, 5]
( )( );
( )/* base */
( );/* backtrack */
False;a unit clause occurs in /* unit */
( );Likely-Unsatisfiable( ) /* early pruning */
( )False;
l := choose-literal( ); /* split */( )( );
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: different algorithms for different kinds ofmath-atoms:
Difference expressions : Belman-Fordminimal path algorithm with negative cycle detectionEqualities : equivalent class building andrewriting.General linear expressions ( ): linearprogramming techniques (Symplex, etc.)Disequalities : postpone at the end. Expand( ) only if indispensable!
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Some SystemsTsat [2]
Logics: disjunctions of difference expressions(positive math-atoms only)Applications: temporal reasoningBoolean reasoning technique: DPLLOptimizations: preprocessing, static learning,forward checking
LPsat [54]Logics: MATH-SAT (positive math-atoms only)Applications: resource planningBoolean reasoning technique: DPLLOptimizations: preprocessing, backjumping,learning, triggering
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Some systems (cont.)
DDD [41]Logics: boolean + difference expressionsApplications: formal verification of timed systemsBoolean reasoning technique: OBDDOptimizations: preprocessing, early pruning
[4]Logics: MATH-SATApplications: resource planning, formal verificationof timed systemsBoolean reasoning technique: DPLLOptimizations: preprocessing, enhanced earlypruning, backjumping, learning, triggering
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SAT techniques for modal logics: summary
SAT techniques have been successfully applied toMATH-SATMany optimizations applicable.Currently competitive with state-of-the-art applicationsfor temporal reasoning, resource planning, formalverification of timed systems.
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[4] G. Audemard, P. Bertoli, A. Cimatti, A. Korniłowicz, and R. Sebastiani. A SATBased Approach for Solving Formulas over Boolean and Linear MathematicalPropositions. In Proc. CADE’2002., LNAI. Springer Verlag, February 2002. Toappear.
[5] G. Audemard, P. Bertoli, A. Cimatti, A. Korniłowicz, and R. Sebastiani.Integrating Boolean and Mathematical Solving: Foundations, Basic Algorithmsand Requirements. In Proc. CALCULEMUS’2002., LNAI. Springer Verlag, 2002.To appear.
[6] F. Baader, E. Franconi, B. Hollunder, B. Nebel, and H.J. Profitlich. An EmpiricalAnalysis of Optimization Techniques for Terminological Representation Systems
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[23] E. Giunchiglia, F. Giunchiglia, R. Sebastiani, and A. Tacchella. SAT vs.Translation based decision procedures for modal logics: a comparativeevaluation. Journal of Applied Non-Classical Logics, 10(2):145–172, 2000.
[24] E. Giunchiglia, F. Giunchiglia, and A. Tacchella. SAT Based Decision Proceduresfor Classical Modal Logics. Journal of Automated Reasoning. Special Issue:Satisfiability at the start of the year 2000, 2001.
[25] E. Giunchiglia, A. Massarotto, and R. Sebastiani. Act, and the Rest Will Follow:Exploiting Determinism in Planning as Satisfiability. In Proc. AAAI’98, pages948–953, 1998.
[26] E. Giunchiglia, M. Narizzano, A. Tacchella, and M. Vardi. Towards an EfficientLibrary for SAT: a Manifesto. In Proc. SAT 2001, Electronics Notes in DiscreteMathematics. Elsevier Science., 2001.
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[28] F. Giunchiglia and R. Sebastiani. Building decision procedures for modal logicsfrom propositional decision procedures - the case study of modal K. In Proc.CADE’13, LNAI, New Brunswick, NJ, USA, August 1996. Springer Verlag.
[29] F. Giunchiglia and R. Sebastiani. A SAT-based decision procedure for ALC. InProc. of the 5th International Conference on Principles of KnowledgeRepresentation and Reasoning - KR’96, Cambridge, MA, USA, November 1996.
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[30] F. Giunchiglia and R. Sebastiani. Building decision procedures for modal logicsfrom propositional decision procedures - the case study of modal K(m).Information and Computation, 162(1/2), October/November 2000.
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[46] R. Sebastiani and A. Villafiorita. SAT-based decision procedures for normalmodal logics: a theoretical framework. In Proc. 6th International Conference onArtificial Intelligence: Methodology, Systems, Applications - AIMSA’98, number1480 in LNAI, Sozopol, Bulgaria, September 1998. Springer Verlag.
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[54] S. Wolfman and D. Weld. The LPSAT Engine & its Application to ResourcePlanning. In Proc. IJCAI, 1999.
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The papers (co)authored by Roberto Sebastiani are availlable at:
.
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