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*
* function DPLL-SATISFIABLE?(s) returns true or false
* inputs: s, a sentence in propositional logic.
*
* clauses ← the set of clauses in the CNF representation of s
* symbols ← a list of the proposition symbols in s
* return DPLL(clauses, symbols, {})
*
* --------------------------------------------------------------------------------
*
* function DPLL(clauses, symbols, model) returns true or false
*
* if every clause in clauses is true in model then return true
* if some clause in clauses is false in model then return false
* P, value ← FIND-PURE-SYMBOL(symbols, clauses, model)
* if P is non-null then return DPLL(clauses, symbols - P, model ∪ {P = value})
* P, value ← FIND-UNIT-CLAUSE(clauses, model)
* if P is non-null then return DPLL(clauses, symbols - P, model ∪ {P = value})
* P ← FIRST(symbols); rest ← REST(symbols)
* return DPLL(clauses, rest, model ∪ {P = true}) or
* DPLL(clauses, rest, model ∪ {P = false})
*
*
*
* Figure 7.17 The DPLL algorithm for checking satisfiability of a sentence in
* propositional logic. The ideas behind FIND-PURE-SYMBOL and FIND-UNIT-CLAUSE
* are described in the test; each returns a symbol (or null) and the truth
* value to assign to that symbol. Like TT-ENTAILS?, DPLL operates over partial
* models.
*
* @author Ciaran O'Reilly
* @author Ravi Mohan
* @author Mike Stampone
*/
public class DPLLSatisfiable implements DPLL {
/**
* DPLL-SATISFIABLE?(s)
* Checks the satisfiability of a sentence in propositional logic.
*
* @param s
* a sentence in propositional logic.
* @return true if the sentence is satisfiable, false otherwise.
*/
@Override
public boolean dpllSatisfiable(Sentence s) {
// clauses <- the set of clauses in the CNF representation of s
Set clauses = ConvertToConjunctionOfClauses.convert(s)
.getClauses();
// symbols <- a list of the proposition symbols in s
List symbols = getPropositionSymbolsInSentence(s);
// return DPLL(clauses, symbols, {})
return dpll(clauses, symbols, new Model());
}
/**
* DPLL(clauses, symbols, model)
*
* @param clauses
* the set of clauses.
* @param symbols
* a list of unassigned symbols.
* @param model
* contains the values for assigned symbols.
* @return true if the model is satisfiable under current assignments, false
* otherwise.
*/
@Override
public boolean dpll(Set clauses, List symbols,
Model model) {
// if every clause in clauses is true in model then return true
if (everyClauseTrue(clauses, model)) {
return true;
}
// if some clause in clauses is false in model then return false
if (someClauseFalse(clauses, model)) {
return false;
}
// P, value <- FIND-PURE-SYMBOL(symbols, clauses, model)
Pair pAndValue = findPureSymbol(symbols,
clauses, model);
// if P is non-null then
if (pAndValue != null) {
// return DPLL(clauses, symbols - P, model U {P = value})
return dpll(clauses, minus(symbols, pAndValue.getFirst()),
model.union(pAndValue.getFirst(), pAndValue.getSecond()));
}
// P, value <- FIND-UNIT-CLAUSE(clauses, model)
pAndValue = findUnitClause(clauses, model);
// if P is non-null then
if (pAndValue != null) {
// return DPLL(clauses, symbols - P, model U {P = value})
return dpll(clauses, minus(symbols, pAndValue.getFirst()),
model.union(pAndValue.getFirst(), pAndValue.getSecond()));
}
// P <- FIRST(symbols); rest <- REST(symbols)
PropositionSymbol p = Util.first(symbols);
List rest = Util.rest(symbols);
// return DPLL(clauses, rest, model U {P = true}) or
// ...... DPLL(clauses, rest, model U {P = false})
return dpll(clauses, rest, model.union(p, true))
|| dpll(clauses, rest, model.union(p, false));
}
//
// SUPPORTING CODE
//
/**
* Determine if KB |= α, i.e. alpha is entailed by KB.
*
* @param kb
* a Knowledge Base in propositional logic.
* @param alpha
* a propositional sentence.
* @return true, if α is entailed by KB, false otherwise.
*/
@Override
public boolean isEntailed(KnowledgeBase kb, Sentence alpha) {
// AIMA3e p.g. 260: kb |= alpha, can be done by testing
// unsatisfiability of kb & ~alpha.
Set kbAndNotAlpha = new LinkedHashSet();
Sentence notQuery = new ComplexSentence(Connective.NOT, alpha);
Set symbols = new LinkedHashSet();
List querySymbols = new ArrayList(SymbolCollector.getSymbolsFrom(notQuery));
kbAndNotAlpha.addAll(kb.asCNF());
kbAndNotAlpha.addAll(ConvertToConjunctionOfClauses.convert(notQuery).getClauses());
symbols.addAll(querySymbols);
symbols.addAll(kb.getSymbols());
return !dpll(kbAndNotAlpha, new ArrayList(symbols), new Model());
}
//
// PROTECTED:
//
// Note: Override this method if you wish to change the initial variable
// ordering when dpllSatisfiable is called.
protected List getPropositionSymbolsInSentence(Sentence s) {
List result = new ArrayList(
SymbolCollector.getSymbolsFrom(s));
return result;
}
/**
* AIMA3e p.g. 260:
* Pure symbol heuristic: A pure symbol is a symbol
* that always appears with the same "sign" in all clauses. For example, in
* the three clauses (A | ~B), (~B | ~C), and (C | A), the symbol A is pure
* because only the positive literal appears, B is pure because only the
* negative literal appears, and C is impure. It is easy to see that if a
* sentence has a model, then it has a model with the pure symbols assigned
* so as to make their literals true, because doing so can never make a
* clause false. Note that, in determining the purity of a symbol, the
* algorithm can ignore clauses that are already known to be true in the
* model constructed so far. For example, if the model contains B=false,
* then the clause (~B | ~C) is already true, and in the remaining clauses C
* appears only as a positive literal; therefore C becomes pure.
*
* @param symbols
* a list of currently unassigned symbols in the model (to be
* checked if pure or not).
* @param clauses
* @param model
* @return a proposition symbol and value pair identifying a pure symbol and
* a value to be assigned to it, otherwise null if no pure symbol
* can be identified.
*/
protected Pair findPureSymbol(
List symbols, Set clauses, Model model) {
Pair result = null;
Set symbolsToKeep = new HashSet(symbols);
// Collect up possible positive and negative candidate sets of pure
// symbols
Set candidatePurePositiveSymbols = new HashSet();
Set candidatePureNegativeSymbols = new HashSet();
for (Clause c : clauses) {
// Algorithm can ignore clauses that are already known to be true
if (Boolean.TRUE.equals(model.determineValue(c))) {
continue;
}
// Collect possible candidates, removing all candidates that are
// not part of the input list of symbols to be considered.
for (PropositionSymbol p : c.getPositiveSymbols()) {
if (symbolsToKeep.contains(p)) {
candidatePurePositiveSymbols.add(p);
}
}
for (PropositionSymbol n : c.getNegativeSymbols()) {
if (symbolsToKeep.contains(n)) {
candidatePureNegativeSymbols.add(n);
}
}
}
// Determine the overlap/intersection between the positive and negative
// candidates
for (PropositionSymbol s : symbolsToKeep) {
// Remove the non-pure symbols
if (candidatePurePositiveSymbols.contains(s) && candidatePureNegativeSymbols.contains(s)) {
candidatePurePositiveSymbols.remove(s);
candidatePureNegativeSymbols.remove(s);
}
}
// We have an implicit preference for positive pure symbols
if (candidatePurePositiveSymbols.size() > 0) {
result = new Pair(
candidatePurePositiveSymbols.iterator().next(), true);
} // We have a negative pure symbol
else if (candidatePureNegativeSymbols.size() > 0) {
result = new Pair(
candidatePureNegativeSymbols.iterator().next(), false);
}
return result;
}
/**
* AIMA3e p.g. 260:
* Unit clause heuristic: A unit clause was defined
* earlier as a clause with just one literal. In the context of DPLL, it
* also means clauses in which all literals but one are already assigned
* false by the model. For example, if the model contains B = true, then (~B
* | ~C) simplifies to ~C, which is a unit clause. Obviously, for this
* clause to be true, C must be set to false. The unit clause heuristic
* assigns all such symbols before branching on the remainder. One important
* consequence of the heuristic is that any attempt to prove (by refutation)
* a literal that is already in the knowledge base will succeed immediately.
* Notice also that assigning one unit clause can create another unit clause
* - for example, when C is set to false, (C | A) becomes a unit clause,
* causing true to be assigned to A. This "cascade" of forced assignments is
* called unit propagation. It resembles the process of forward
* chaining with definite clauses, and indeed, if the CNF expression
* contains only definite clauses then DPLL essentially replicates forward
* chaining.
*
* @param clauses
* @param model
* @return a proposition symbol and value pair identifying a unit clause and
* a value to be assigned to it, otherwise null if no unit clause
* can be identified.
*/
protected Pair findUnitClause(
Set clauses, Model model) {
Pair result = null;
for (Clause c : clauses) {
// if clauses value is currently unknown
// (i.e. means known literals are false)
if (model.determineValue(c) == null) {
Literal unassigned = null;
// Default definition of a unit clause is a clause
// with just one literal
if (c.isUnitClause()) {
unassigned = c.getLiterals().iterator().next();
} else {
// Also, a unit clause in the context of DPLL, also means a
// clauseF in which all literals but one are already
// assigned false by the model.
// Note: at this point we already know the clause is not
// true, so just need to determine if the clause has a
// single unassigned literal
for (Literal l : c.getLiterals()) {
Boolean value = model.getValue(l.getAtomicSentence());
if (value == null) {
// The first unassigned literal encountered.
if (unassigned == null) {
unassigned = l;
} else {
// This means we have more than 1 unassigned
// literal so lets skip
unassigned = null;
break;
}
}
}
}
// if a value assigned it means we have a single
// unassigned literal and all the assigned literals
// are not true under the current model as we were
// unable to determine a value.
if (unassigned != null) {
result = new Pair(
unassigned.getAtomicSentence(),
unassigned.isPositiveLiteral());
break;
}
}
}
return result;
}
protected boolean everyClauseTrue(Set clauses, Model model) {
return model.satisfies(clauses);
}
protected boolean someClauseFalse(Set clauses, Model model) {
for (Clause c : clauses) {
// Only 1 needs to be false
if (Boolean.FALSE.equals(model.determineValue(c))) {
return true;
}
}
return false;
}
// symbols - P
protected List minus(List symbols,
PropositionSymbol p) {
List result = new ArrayList(
symbols.size());
for (PropositionSymbol s : symbols) {
// symbols - P
if (!p.equals(s)) {
result.add(s);
}
}
return result;
}
}