SA T : Pr opositional SatisÞability and Be y ond · SA T : Pr opositional SatisÞability and Be y ond Rober to Sebastiani Dept. of Inf or mation and Comm unication T echnologies

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


PART 1:PROPOSITIONALSATISFIABILITY

ICT Graduate School, Trento, May-June 2002 2


Basics on SAT



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



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:



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


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.



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.



NNF, CNF and conversions



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 ;



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.



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.



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.



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.



Labeling CNF conversion (improved)

As in the previous case, applying instead the rules:

Smaller in size.



k-SAT and Phase Transition



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:



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;



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



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]



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



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.



0

5000

10000

15000

20000

25000

30000

3 3.5 4 4.5 5 5.5 6CLAUSE # / VAR #

MEDIAN

N=50N=100N=200



0

5000

10000

15000

20000

25000

3 3.5 4 4.5 5 5.5 6CLAUSE # / VAR #

GEOMEAN

N=50N=100N=200



Basic SAT techniques



Truth Tables

Exhaustive evaluation of all subformulas:

Requires polynomial space.Never used in practice.



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;



Tableau elimination rules

-elimination

-elimination

-elimination



Tableau algorithm

Tableau/* branch closed */

False;/* -elimination */

Tableau ;/* -elimination */

Tableau ;/* -elimination */

TableauTableau ;

...True; /* branch expanded */



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



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.



DPLL rules

( is a pure literal in iff it occurs only positively).



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( );



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]



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.



(Implicit) OBDD structure

,,



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



Incrementally building an OBDD

,,



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.



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



GSAT algorithm

GSAT( )Max-tries

:= rand-assign( );Max-flips

( )True;

Best-flips := hill-climb( );:= rand-pick(Best-flips);

:= flip( );

“no satisfying assignment found”.



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]



SAT for non-CNF formulas



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( );



Non-CNF DPLL (cont.)

:



Non-CNF DPLL (cont.)

:



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.



Applying DPLL to (cont)

If basic is used:

then B is deterministicaly assigned by unit propagationif and are assigned.



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 .



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”.



Non-CNF GSAT (cont.)

computes directly in linear time.



DPLL Heuristics &Optimizations



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



Variants of DPLL

DPLL is a family of algorithms.

different splitting heuristicspreprocessing: (subsumption, 2-simplification)backjumpinglearningrandom restarthorn relaxation...



Splitting heuristics - Choose literal()

Split is the source of non-determinism for DPLLChoose literal() critical for efficiencymany split heuristics conceived in literature.



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



Some preprocessing techniques

Sorting+subsumption:



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.



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.



Conflict-directed backtracking – example



Conflict-directed backtracking – example (cont.)

(initial assignment)




(branch on )




(unit )ICT Graduate School, Trento, May-June 2002 63









Conflict set: backtrack to




(branch on )







conflict set: .




conflict set:backtrack to .



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



Learning – example (cont.)

Conflict set:learn



SOME APPLICATIONS



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.



Application #1:(Bounded) Planning



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



Example

B B

GOAL

A C

A

INITIAL

C

T



Encoding

Initial states:

Goal states:

Action preconditions and effects:



Encoding: Frame axioms

Classic

“At least one action” axiom:

Explanatory



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



Application #2:Bounded Model Checking



Bounded Planning

Incomplete techniquevery efficient: competitive with state-of-the-artplannerslots of enhancements [37, 36, 19, 25]



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


The problem (cont.)

Problem:Is there an execution path of of length satisfying thetemporal property ?:



The encoding

Equivalent to the satisfiability problem of a booleanformula defined as follows:



The encoding of and



Example: (reachability)

: is there a reachable state in which holds?is:



Example:

: is there a path where holds forever?is:



Example: (fair reachability)

: is there a reachable state in whichholds provided that q holds infinitely often?

is:



Bounded Model Checking

incomplete techniquevery efficient for some problemslots of enhancements [8, 1, 49, 53, 13]



PART 2:BEYONDPROPOSITIONALSATISFIABILITY



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)



FORMAL FRAMEWORK



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!)



Ingredients (cont.)

A semantic for extending standard boolean one:[definition specific for ]



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


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 .



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



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.



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.,



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 .



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

.



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


Example

, but is unsatisfiable, as contains conflict sets:

, and is satisfiable ( ).ICT Graduate School, Trento, May-June 2002 104


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 .



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.



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.



If is redundant, then for some :

If is strongly non redundant, then is nonredundant:



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.



A GENERALIZEDSEARCH PROCEDURE



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.



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.



A generalized procedure

( )

( ) /* next in */( )

( );((satifiable = False) )

( False)True; /* a satisf. assignment found */

False; /* no satisf. assignment found */



( ) terminating, correct and complete( ) terminating, correct and complete.

depends on only forrequires polynomial space iff

requires polynomial space andis lazy



Mandatory requirements for an assignment enumerator

An assignment enumerator must always:(Termination) terminate(Correctness) generate assignments propositionallysatisfying(Completeness) generate complete set of assignments



Mandatory requirements for ()

() must always:(Termination) terminate(Correctness & completeness) return if issatisfiable in , otherwise



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)]



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.



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 ( )]



EXTENDING EXISTINGSAT PROCEDURES



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



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.



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


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 ,, .



OBDD: summary

strongly non-redundanttime-efficientfactor sub-graphsrequire exponential memorynon lazy[allow for early pruning][do not allow for backjumping or learning]



Generalized semantic tableaux

General rules = propositional rules + -specific rules

-specificRules

Widely used by logicians



Generalized tableau algorithm

-Tableau/* branch closed */

False;/* -elimination */

-Tableau ;/* -elimination */

-Tableau ;/* -elimination */

-Tableau-Tableau ;

...( ( )= satisfiable); /* branch expanded */



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 .



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


Syntactic branching: example

T T

a

a a

-a -a -a

b b

-b

-b -b -b

G




Detecting constraints violations: example

-a -b -a -b -a -b

a

f

f

T

T1

1

2

2b

. . . . .

T3

G

Tableau search graph for



Generalized tableaux: summary

lazyrequire polynomial memoryredundanttime-inefficient[allow backjumping][do not allow learning]



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.



Generalized DPLL

General rules = propositional rules + -specific rules

-specificRules

No Pure Literal Rule: Pure literal causes incompleteassignment sets!



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


Generalized DPLL algorithm

-DPLL( )/* base */

( ( )=satisfiable);/* backtrack */


-DPLL( );l := choose-literal( ); /* split */

-DPLL( )-DPLL( );



General DPLL: example

a -a

b -b{a}

{-a,b} g

DPLL search graph

DPLL search graph for .



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.



Semantic branching: example

T

a -a

-b -b




Detecting constraints violations: example

T1

a -a

-b

T23

DPLL search graph for



Generalized DPLL: summary

lazyrequire polynomial memorystrongly non redundanttime-efficient[allow backjumping and learning]



Optimizations



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]



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.



Preprocessing atoms (cont.)

Sorting: , ,);

Rewriting dual operators:, ,

Exploiting associativity:,

;Factoring , ,

;Exploiting properties of :

, if ;...



Preprocessing atoms: summary

Very efficient with DPLLPresumably very efficient with OBDDsScarcely efficient with semantic tableaux



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.



Static learning (cont.)

Very efficient with DPLLPossibly very efficient with OBDDs (?)Completely ineffective with semantic tableaux



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;



DPLL+Early pruning

-DPLL( )/* base */

( ( )=satisfiable);/* backtrack */


-DPLL( );Likely-Unsatisfiable( ) /* early pruning */

( ( ) )False;

l := choose-literal( ); /* split */-DPLL( )-DPLL( );



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


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.



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...



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



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



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.



Enhanced Early Pruning: Examples

(We assume that all the following literals occur in .)

If and , thencan derive from .

If and ,then can derive from .



Enhanced Early Pruning: summary

Further improves efficiency with DPLLPresumably scarcely effective with semantic tableauxEffective with OBDDs?Requires a sophisticated



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



Backjumping: Example

( ) returns false with the conflict set:

can jump back directly to the branching point, without branching on .



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).



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.



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.



Memoizing (cont.)

Can dramatically prune search.May cause a blowup in memory.Applicable also to semantic tableaux.Idea 1 subsumed by learning.



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



Learning: example

returns the conflict set:

it is added the clause

Prunes up to assignmentsthe smaller the conflict set, the better.



Learning: summary

Very efficient with DPLLNever applied to OBDDsCompletely ineffective with semantic tableauxMay cause memory blowup

must be able to detect conflict sets.



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. .



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:

.



Application Fields

Modal LogicsDescription LogicsTemporal LogicsBoolean+Mathematical reasoning (Temporalreasoning, Resource Planning, Verification of TimedSystems, Verification of systems with arithmeticaloperators, verification of hybrid systems)QBF...



CASE STUDY:MODAL LOGIC(S)



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.



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 ”



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 .



Semantics (cont)

Given s.t. , is defined as follows:

for everys.t. holds in .

for somes.t. holds in .



Semantics (cont)

The (normal) modal logics vary with the properties of :

Axiom Property of DescriptionB symmetricD serialT reflexive4 transitive5 euclidean



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



Axiomatic framework

Basic Axioms:

Specific Axioms:



Axiomatic framework (cont.)

Inference rules:

modus ponens

necessitation

Correctness & completeness:is valid can be deduced



Tableaux for modal K(m)/ [20]

Rules = tableau rules + -specific rules



DPLL for K(m)/ : K-SAT [28, 29]

Rules = DPLL rules + -specific rules



The K-SAT algorithm [28, 29]

( )( );

( )/* base */

( );/* backtrack */


( );Likely-Unsatisfiable( ) /* early pruning */

( )False;

l := choose-literal( ); /* split */( )( );



The K-SAT algorithm (cont.)

( )box index

(False;

True;

(conjunct “ ”

( )False;

True;



: Example



: Example (cont.)



: Example (cont.)



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 )



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



ff41

42

mm

m

411412

413

f

mm

m

m

m

12

3

4

5

SearchDepth

ModalViewPropositionalView



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



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



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.



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)


percentiles

CPU

TIM

E [S

EC]

0100

200300

400

5060

7080

90100

0

200

400

600

800

1000

# OF CLAUSES (L)


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)


percentiles

CPU

TIM

E [S

EC]

0100

200300

400

5060

7080

90100

0

200

400

600

800

1000

# OF CLAUSES (L)


percentilesCP

U TI

ME

[SEC

]

K-SAT (up) versus TA (down) CPU times.



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



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



SAT techniques for modal logics: summary

SAT techniques have been successfully applied tomodal/description logicsMany optimizations applicable.Currently at the State-of-the-art.



CASE STUDY:(LINEAR) MATHEMATICALREASONING



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.



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 .



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


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 , .



DPLL for math-formulas [54, 2, 4, 5]

( )( );

( )/* base */

( );/* backtrack */


( );Likely-Unsatisfiable( ) /* early pruning */

( )False;

l := choose-literal( ); /* split */( )( );



: 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!



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



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



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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The papers (co)authored by Roberto Sebastiani are availlable at:

.


Documents

SA T : Pr opositional SatisÞability and Be y ond · SA T : Pr opositional SatisÞability and Be y ond Rober to Sebastiani Dept. of Inf or mation and Comm unication T echnologies