package sbt
package logic

	import scala.annotation.tailrec
	import Formula.{And, True}

Defines a propositional logic with negation as failure and only allows stratified rule sets (negation must be acyclic) in order to have a unique minimal model.

For example, this is not allowed:
  + p :- not q
  + q :- not p
but this is:
  + p :- q
  + q :- p
as is this:
  + p :- q
  + q := not r

 Some useful links:

/** Disjunction (or) of the list of clauses. */
final case class Clauses(clauses: List[Clause]) {
	assert(clauses.nonEmpty, "At least one clause is required.")

/** When the `body` Formula succeeds, atoms in `head` are true. */
final case class Clause(body: Formula, head: Set[Atom])

/** A literal is an [[Atom]] or its [[negation|Negated]]. */
sealed abstract class Literal extends Formula {
	/** The underlying (positive) atom. */
	def atom: Atom
	/** Negates this literal.*/
	def unary_! : Literal
/** A variable with name `label`. */
final case class Atom(label: String) extends Literal {
	def atom = this
	def unary_! : Negated = Negated(this)
/** A negated atom, in the sense of negation as failure, not logical negation.
* That is, it is true if `atom` is not known/defined. */
final case class Negated(atom: Atom) extends Literal {
	def unary_! : Atom = atom

/** A formula consists of variables, negation, and conjunction (and).
* (Disjunction is not currently included- it is modeled at the level of a sequence of clauses.
*  This is less convenient when defining clauses, but is not less powerful.) */
sealed abstract class Formula {
	/** Constructs a clause that proves `atoms` when this formula is true. */
	def proves(atom: Atom, atoms: Atom*): Clause = Clause(this, (atom +: atoms).toSet)

	/** Constructs a formula that is true iff this formula and `f` are both true.*/
	def && (f: Formula): Formula = (this, f) match {
		case (True, x) => x
		case (x, True) => x
		case (And(as), And(bs)) => And(as ++ bs)
		case (And(as), b: Literal) => And(as + b)
		case (a: Literal, And(bs)) => And(bs + a)
		case (a: Literal, b: Literal) => And( Set(a,b) )

object Formula {
	/** A conjunction of literals. */
   final case class And(literals: Set[Literal]) extends Formula {
		assert(literals.nonEmpty, "'And' requires at least one literal.")
	final case object True extends Formula

object Logic
	def reduceAll(clauses: List[Clause], initialFacts: Set[Literal]): Either[LogicException, Matched] =
		reduce(Clauses(clauses), initialFacts)

	/** Computes the variables in the unique stable model for the program represented by `clauses` and `initialFacts`.
	* `clause` may not have any negative feedback (that is, negation is acyclic)
	* and `initialFacts` cannot be in the head of any clauses in `clause`.
	* These restrictions ensure that the logic program has a unique minimal model. */
	def reduce(clauses: Clauses, initialFacts: Set[Literal]): Either[LogicException, Matched] =
		val (posSeq, negSeq) = separate(initialFacts.toSeq)
		val (pos, neg) = (posSeq.toSet, negSeq.toSet)

		val problem =
			checkContradictions(pos, neg) orElse
			checkOverlap(clauses, pos) orElse

			reduce0(clauses, initialFacts, Matched.empty)

	/** Verifies `initialFacts` are not in the head of any `clauses`.
	* This avoids the situation where an atom is proved but no clauses prove it.
	* This isn't necessarily a problem, but the main sbt use cases expects
	* a proven atom to have at least one clause satisfied. */
	private[this] def checkOverlap(clauses: Clauses, initialFacts: Set[Atom]): Option[InitialOverlap] = {
		val as = atoms(clauses)
		val initialOverlap = initialFacts.filter(as.inHead)
		if(initialOverlap.nonEmpty) Some(new InitialOverlap(initialOverlap)) else None

	private[this] def checkContradictions(pos: Set[Atom], neg: Set[Atom]): Option[InitialContradictions] = {
		val contradictions = pos intersect neg
		if(contradictions.nonEmpty) Some(new InitialContradictions(contradictions)) else None

	private[this] def checkAcyclic(clauses: Clauses): Option[CyclicNegation] = {
		val deps = dependencyMap(clauses)
		val cycle = Dag.findNegativeCycle(graph(deps))
		if(cycle.nonEmpty) Some(new CyclicNegation(cycle)) else None
	private[this] def graph(deps: Map[Atom, Set[Literal]]) = new Dag.DirectedSignedGraph[Atom] {
		type Arrow = Literal
		def nodes = deps.keys.toList
		def dependencies(a: Atom) = deps.getOrElse(a, Set.empty).toList
		def isNegative(b: Literal) = b match {
			case Negated(_) => true
			case Atom(_) => false
		def head(b: Literal) = b.atom

	private[this] def dependencyMap(clauses: Clauses): Map[Atom, Set[Literal]] =
		(Map.empty[Atom, Set[Literal]] /: clauses.clauses) {
			case (m, Clause(formula, heads)) =>
				val deps = literals(formula)
				(m /: heads) { (n, head) => n.updated(head, n.getOrElse(head, Set.empty) ++ deps) }

	sealed abstract class LogicException(override val toString: String)
	final class InitialContradictions(val literals: Set[Atom]) extends LogicException("Initial facts cannot be both true and false:\n\t" + literals.mkString("\n\t"))
	final class InitialOverlap(val literals: Set[Atom]) extends LogicException("Initial positive facts cannot be implied by any clauses:\n\t" + literals.mkString("\n\t"))
	final class CyclicNegation(val cycle: List[Literal]) extends LogicException("Negation may not be involved in a cycle:\n\t" + cycle.mkString("\n\t"))

	/** Tracks proven atoms in the reverse order they were proved. */
	final class Matched private(val provenSet: Set[Atom], reverseOrdered: List[Atom]) {
		def add(atoms: Set[Atom]): Matched = add(atoms.toList)
		def add(atoms: List[Atom]): Matched = {
			val newOnly = atoms.filterNot(provenSet)
			new Matched(provenSet ++ newOnly, newOnly ::: reverseOrdered)
		def ordered: List[Atom] = reverseOrdered.reverse
		override def toString ="Matched(", ",", ")")
	object Matched {
		val empty = new Matched(Set.empty, Nil)

	/** Separates a sequence of literals into `(pos, neg)` atom sequences. */
	private[this] def separate(lits: Seq[Literal]): (Seq[Atom], Seq[Atom]) = Util.separate(lits) {
		case a: Atom => Left(a)
		case Negated(n) => Right(n)

	/** Finds clauses that have no body and thus prove their head.
	* Returns `(<proven atoms>, <remaining unproven clauses>)`. */
	private[this] def findProven(c: Clauses): (Set[Atom], List[Clause]) =
		val (proven, unproven) = c.clauses.partition(_.body == True)
		(proven.flatMap(_.head).toSet, unproven)
	private[this] def keepPositive(lits: Set[Literal]): Set[Atom] =
		lits.collect{ case a: Atom => a}.toSet

	// precondition: factsToProcess contains no contradictions
	private[this] def reduce0(clauses: Clauses, factsToProcess: Set[Literal], state: Matched): Matched =
		applyAll(clauses, factsToProcess) match {
			case None => // all of the remaining clauses failed on the new facts
			case Some(applied) =>
				val (proven, unprovenClauses) = findProven(applied)
				val processedFacts = state add keepPositive(factsToProcess)
				val newlyProven = proven -- processedFacts.provenSet
				val newState = processedFacts add newlyProven
					newState // no remaining clauses, done.
				else {
					val unproven = Clauses(unprovenClauses)
					val nextFacts: Set[Literal] = if(newlyProven.nonEmpty) newlyProven.toSet else inferFailure(unproven)
					reduce0(unproven, nextFacts, newState)

	/** Finds negated atoms under the negation as failure rule and returns them.
	* This should be called only after there are no more known atoms to be substituted. */
	private[this] def inferFailure(clauses: Clauses): Set[Literal] =
		/* At this point, there is at least one clause and one of the following is the case as the result of the acyclic negation rule:
				i. there is at least one variable that occurs in a clause body but not in the head of a clause
				ii. there is at least one variable that occurs in the head of a clause and does not transitively depend on a negated variable
			In either case, each such variable x cannot be proven true and therefore proves 'not x' (negation as failure, !x in the code).
		val allAtoms = atoms(clauses)
		val newFacts: Set[Literal] = negated(allAtoms.triviallyFalse)
		else {
			val possiblyTrue = hasNegatedDependency(clauses.clauses, Relation.empty, Relation.empty)
			val newlyFalse: Set[Literal] = negated(allAtoms.inHead -- possiblyTrue)
			else // should never happen due to the acyclic negation rule
				error(s"No progress:\n\tclauses: $clauses\n\tpossibly true: $possiblyTrue")

	private[this] def negated(atoms: Set[Atom]): Set[Literal] = => Negated(a))

	/** Computes the set of atoms in `clauses` that directly or transitively take a negated atom as input.
	* For example, for the following clauses, this method would return `List(a, d)` :
	*  a :- b, not c
	*  d :- a
	def hasNegatedDependency(clauses: Seq[Clause], posDeps: Relation[Atom, Atom], negDeps: Relation[Atom, Atom]): List[Atom] =
		clauses match {
			case Seq() =>
				// because cycles between positive literals are allowed, this isn't strictly a topological sort
			case Clause(formula, head) +: tail =>
				// collect direct positive and negative literals and track them in separate graphs
				val (pos, neg) = directDeps(formula)
				val (newPos, newNeg) = ( (posDeps, negDeps) /: head) { case ( (pdeps, ndeps), d) =>
					(pdeps + (d, pos), ndeps + (d, neg) )
				hasNegatedDependency(tail, newPos, newNeg)

	/** Computes the `(positive, negative)` literals in `formula`. */
	private[this] def directDeps(formula: Formula): (Seq[Atom], Seq[Atom]) =
		Util.separate(literals(formula).toSeq) {
			case Negated(a) => Right(a)
			case a: Atom => Left(a)
	private[this] def literals(formula: Formula): Set[Literal] = formula match {
		case And(lits) => lits
		case l: Literal => Set(l)
		case True => Set.empty

	/** Computes the atoms in the heads and bodies of the clauses in `clause`. */
	def atoms(cs: Clauses): Atoms = => Atoms(c.head, atoms(c.body))).reduce(_ ++ _)

	/** Computes the set of all atoms in `formula`. */
	def atoms(formula: Formula): Set[Atom] = formula match {
		case And(lits) =>
		case Negated(lit) => Set(lit)
		case a: Atom => Set(a)
		case True => Set()

	/** Represents the set of atoms in the heads of clauses and in the bodies (formulas) of clauses. */
	final case class Atoms(val inHead: Set[Atom], val inFormula: Set[Atom]) {
		/** Concatenates this with `as`. */
		def ++ (as: Atoms): Atoms = Atoms(inHead ++ as.inHead, inFormula ++ as.inFormula)
		/** Atoms that cannot be true because they do not occur in a head. */
		def triviallyFalse: Set[Atom] = inFormula -- inHead

	/** Applies known facts to `clause`s, deriving a new, possibly empty list of clauses.
	* 1. If a fact is in the body of a clause, the derived clause has that fact removed from the body.
	* 2. If the negation of a fact is in a body of a clause, that clause fails and is removed.
	* 3. If a fact or its negation is in the head of a clause, the derived clause has that fact (or its negation) removed from the head.
	* 4. If a head is empty, the clause proves nothing and is removed.
	* NOTE: empty bodies do not cause a clause to succeed yet.
	*       All known facts must be applied before this can be done in order to avoid inconsistencies.
	* Precondition: no contradictions in `facts`
	* Postcondition: no atom in `facts` is present in the result
	* Postcondition: No clauses have an empty head
	* */
	def applyAll(cs: Clauses, facts: Set[Literal]): Option[Clauses] =
		val newClauses =
				cs.clauses.filter(_.head.nonEmpty) // still need to drop clauses with an empty head
			else => applyAll(c, facts)).flatMap(_.toList)
		if(newClauses.isEmpty) None else Some(Clauses(newClauses))

	def applyAll(c: Clause, facts: Set[Literal]): Option[Clause] =
		val atoms =
		val newHead = c.head -- atoms  // 3.
		if(newHead.isEmpty)  // 4. empty head
			substitute(c.body, facts).map( f => Clause(f, newHead) )  // 1, 2

	/** Derives the formula that results from substituting `facts` into `formula`. */
	def substitute(formula: Formula, facts: Set[Literal]): Option[Formula] = formula match {
		case And(lits) =>
			def negated(lits: Set[Literal]): Set[Literal] = => !a)
			if( lits.exists( negated(facts) ) )  // 2.
			else {
				val newLits = lits -- facts
				val newF = if(newLits.isEmpty) True else And(newLits)
				Some(newF)  // 1.
		case True => Some(True)
		case lit: Literal => // define in terms of And
			substitute(And(Set(lit)), facts)