// Artificial Intelligence A Modern Approach (2nd Edition): page 288.
// COMPOSE(delta, tau) is the substitution whose effect is identical to
// the effect of applying each substitution in turn. That is,
// SUBST(COMPOSE(theta1, theta2), p) = SUBST(theta2, SUBST(theta1, p))
private Dictionary <Variable, Term> compose(FOLKnowledgeBase KB,
Dictionary <Variable, Term> theta1, Dictionary <Variable, Term> theta2)
{
Dictionary <Variable, Term> composed = new Dictionary <Variable, Term>();
// So that it behaves like:
// SUBST(theta2, SUBST(theta1, p))
// There are two steps involved here.
// See: http://logic.stanford.edu/classes/cs157/2008/notes/chap09.pdf
// for a detailed discussion:
// 1. Apply theta2 to the range of theta1.
foreach (Variable v in theta1.Keys)
{
composed.Add(v, KB.subst(theta2, theta1[v]));
}
// 2. Adjoin to delta all pairs from tau with different
// domain variables.
foreach (Variable v in theta2.Keys)
{
if (!theta1.ContainsKey(v))
{
composed.Add(v, theta2[v]);
}
}
return(cascadeSubstitutions(KB, composed));
}
// Artificial Intelligence A Modern Approach (2nd Edition): page 288.
// COMPOSE(delta, tau) is the substitution whose effect is identical to
// the effect of applying each substitution in turn. That is,
// SUBST(COMPOSE(theta1, theta2), p) = SUBST(theta2, SUBST(theta1, p))
private IMap <Variable, Term> compose(FOLKnowledgeBase KB, IMap <Variable, Term> theta1, IMap <Variable, Term> theta2)
{
IMap <Variable, Term> composed = CollectionFactory.CreateInsertionOrderedMap <Variable, Term>();
// So that it behaves like:
// SUBST(theta2, SUBST(theta1, p))
// There are two steps involved here.
// See: http://logic.stanford.edu/classes/cs157/2008/notes/chap09.pdf
// for a detailed discussion:
// 1. Apply theta2 to the range of theta1.
foreach (Variable v in theta1.GetKeys())
{
composed.Put(v, KB.subst(theta2, theta1.Get(v)));
}
// 2. Adjoin to delta all pairs from tau with different
// domain variables.
foreach (Variable v in theta2.GetKeys())
{
if (!theta1.ContainsKey(v))
{
composed.Put(v, theta2.Get(v));
}
}
return(cascadeSubstitutions(KB, composed));
}
// See:
// http://logic.stanford.edu/classes/cs157/2008/miscellaneous/faq.html#jump165
// for need for this.
private Dictionary <Variable, Term> cascadeSubstitutions(FOLKnowledgeBase KB,
Dictionary <Variable, Term> theta)
{
foreach (Variable v in theta.Keys)
{
Term t = theta[v];
theta.Add(v, KB.subst(theta, t));
}
return(theta);
}
// See:
// http://logic.stanford.edu/classes/cs157/2008/miscellaneous/faq.html#jump165
// for need for this.
private IMap <Variable, Term> cascadeSubstitutions(FOLKnowledgeBase KB,
IMap <Variable, Term> theta)
{
foreach (Variable v in theta.GetKeys())
{
Term t = theta.Get(v);
theta.Put(v, KB.subst(theta, t));
}
return(theta);
}
// See:
// http://logic.stanford.edu/classes/cs157/2008/miscellaneous/faq.html#jump165
// for need for this.
private Dictionary <Variable, Term> cascadeSubstitutions(FOLKnowledgeBase KB,
Dictionary <Variable, Term> theta)
{
var keys = theta.Keys.ToList();
foreach (Variable v in keys)
{
theta[v] = KB.subst(theta, theta[v]);
}
return(theta);
}
// END-InferenceProcedure
//
//
// PRIVATE METHODS
//
/**
* <code>
* function FOL-BC-ASK(KB, goals, theta) returns a set of substitutions
* input: KB, a knowledge base
* goals, a list of conjuncts forming a query (theta already applied)
* theta, the current substitution, initially the empty substitution {}
* </code>
*/
private List <List <ProofStepBwChGoal> > folbcask(FOLKnowledgeBase KB,
BCAskAnswerHandler ansHandler, List <Literal> goals,
Dictionary <Variable, Term> theta)
{
List <List <ProofStepBwChGoal> > thisLevelProofSteps = new List <List <ProofStepBwChGoal> >();
// local variables: answers, a set of substitutions, initially empty
// if goals is empty then return {theta}
if (goals.Count == 0)
{
thisLevelProofSteps.Add(new List <ProofStepBwChGoal>());
return(thisLevelProofSteps);
}
// qDelta <- SUBST(theta, FIRST(goals))
Literal qDelta = KB.subst(theta, goals[0]);
// for each sentence r in KB where
// STANDARDIZE-APART(r) = (p1 ^ ... ^ pn => q)
foreach (Clause r in KB.getAllDefiniteClauses())
{
Clause r2 = KB.standardizeApart(r);
// and thetaDelta <- UNIFY(q, qDelta) succeeds
Dictionary <Variable, Term> thetaDelta = KB.unify(r2.getPositiveLiterals()
[0].getAtomicSentence(), qDelta.getAtomicSentence());
if (null != thetaDelta)
{
// new_goals <- [p1,...,pn|REST(goals)]
List <Literal> newGoals = new List <Literal>(r2
.getNegativeLiterals());
newGoals.AddRange(goals.Skip(1));
// answers <- FOL-BC-ASK(KB, new_goals, COMPOSE(thetaDelta,
// theta)) U answers
Dictionary <Variable, Term> composed = compose(KB, thetaDelta, theta);
List <List <ProofStepBwChGoal> > lowerLevelProofSteps = folbcask(
KB, ansHandler, newGoals, composed);
ansHandler.addProofStep(lowerLevelProofSteps, r2, qDelta,
composed);
thisLevelProofSteps.AddRange(lowerLevelProofSteps);
}
}
// return answers
return(thisLevelProofSteps);
}
/**
* <code>
* function FOL-BC-ASK(KB, goals, theta) returns a set of substitutions
* input: KB, a knowledge base
* goals, a list of conjuncts forming a query (theta already applied)
* theta, the current substitution, initially the empty substitution {}
* </code>
*/
private ICollection <ICollection <ProofStepBwChGoal> > folbcask(FOLKnowledgeBase KB,
BCAskAnswerHandler ansHandler, ICollection <Literal> goals,
IMap <Variable, Term> theta)
{
ICollection <ICollection <ProofStepBwChGoal> > thisLevelProofSteps = CollectionFactory.CreateQueue <ICollection <ProofStepBwChGoal> >();
// local variables: answers, a set of substitutions, initially empty
// if goals is empty then return {theta}
if (goals.IsEmpty())
{
thisLevelProofSteps.Add(CollectionFactory.CreateQueue <ProofStepBwChGoal>());
return(thisLevelProofSteps);
}
// qDelta <- SUBST(theta, FIRST(goals))
Literal qDelta = KB.subst(theta, goals.Get(0));
// for each sentence r in KB where
// STANDARDIZE-APART(r) = (p1 ^ ... ^ pn => q)
foreach (Clause rIter in KB.getAllDefiniteClauses())
{
Clause r = rIter;
r = KB.standardizeApart(r);
// and thetaDelta <- UNIFY(q, qDelta) succeeds
IMap <Variable, Term> thetaDelta = KB.unify(r.getPositiveLiterals()
.Get(0).getAtomicSentence(), qDelta.getAtomicSentence());
if (null != thetaDelta)
{
// new_goals <- [p1,...,pn|REST(goals)]
ICollection <Literal> newGoals = CollectionFactory.CreateQueue <Literal>(r.getNegativeLiterals());
newGoals.AddAll(goals.subList(1, goals.Size()));
// answers <- FOL-BC-ASK(KB, new_goals, COMPOSE(thetaDelta,
// theta)) U answers
IMap <Variable, Term> composed = compose(KB, thetaDelta, theta);
ICollection <ICollection <ProofStepBwChGoal> > lowerLevelProofSteps = folbcask(
KB, ansHandler, newGoals, composed);
ansHandler.addProofStep(lowerLevelProofSteps, r, qDelta, composed);
thisLevelProofSteps.AddAll(lowerLevelProofSteps);
}
}
// return answers
return(thisLevelProofSteps);
}
// START-InferenceProcedure
/**
* <code>
* function FOL-FC-ASK(KB, alpha) returns a substitution or false
* inputs: KB, the knowledge base, a set of first order definite clauses
* alpha, the query, an atomic sentence
* </code>
*/
public InferenceResult ask(FOLKnowledgeBase KB, Sentence query)
{
// Assertions on the type of queries this Inference procedure
// supports
if (!(query is AtomicSentence))
{
throw new ArgumentException(
"Only Atomic Queries are supported.");
}
FCAskAnswerHandler ansHandler = new FCAskAnswerHandler();
Literal alpha = new Literal((AtomicSentence)query);
// local variables: new, the new sentences inferred on each iteration
List <Literal> newSentences = new List <Literal>();
// Ensure query is not already a know fact before
// attempting forward chaining.
List <Dictionary <Variable, Term> > answers = KB.fetch(alpha);
if (answers.Count > 0)
{
ansHandler.addProofStep(new ProofStepFoChAlreadyAFact(alpha));
ansHandler.setAnswers(answers);
return(ansHandler);
}
// repeat until new is empty
do
{
// new <- {}
newSentences.Clear();
// for each rule in KB do
// (p1 ^ ... ^ pn => q) <-STANDARDIZE-VARIABLES(rule)
foreach (Clause impl in KB.getAllDefiniteClauseImplications())
{
Clause impl2 = KB.standardizeApart(impl);
// for each theta such that SUBST(theta, p1 ^ ... ^ pn) =
// SUBST(theta, p'1 ^ ... ^ p'n)
// --- for some p'1,...,p'n in KB
foreach (Dictionary <Variable, Term> theta in KB.fetch(invert(new List <Literal>(impl2
.getNegativeLiterals()))))
{
// q' <- SUBST(theta, q)
Literal qPrime = KB.subst(theta, impl.getPositiveLiterals()
[0]);
// if q' does not unify with some sentence already in KB or
// new then do
if (!KB.isRenaming(qPrime) &&
!KB.isRenaming(qPrime, newSentences))
{
// add q' to new
newSentences.Add(qPrime);
ansHandler.addProofStep(impl, qPrime, theta);
// theta <- UNIFY(q', alpha)
Dictionary <Variable, Term> theta2 = KB.unify(qPrime.getAtomicSentence(), alpha
.getAtomicSentence());
// if theta is not fail then return theta
if (null != theta2)
{
foreach (Literal l in newSentences)
{
Sentence s = null;
if (l.isPositiveLiteral())
{
s = l.getAtomicSentence();
}
else
{
s = new NotSentence(l.getAtomicSentence());
}
KB.tell(s);
}
ansHandler.setAnswers(KB.fetch(alpha));
return(ansHandler);
}
}
}
}
// add new to KB
foreach (Literal l in newSentences)
{
Sentence s = null;
if (l.isPositiveLiteral())
{
s = l.getAtomicSentence();
}
else
{
s = new NotSentence(l.getAtomicSentence());
}
KB.tell(s);
}
} while (newSentences.Count > 0);
// return false
return(ansHandler);
}
