/******************************************************************************

 * Top contributors (to current version):

 *   Andrew Reynolds, Mudathir Mohamed

 *

 * This file is part of the cvc5 project.

 *

 * Copyright (c) 2009-2021 by the authors listed in the file AUTHORS

 * in the top-level source directory and their institutional affiliations.

 * All rights reserved.  See the file COPYING in the top-level source

 * directory for licensing information.

 * ****************************************************************************

 *

 * An extension of the theory sets for handling cardinality constraints.

 */

#include "cvc5_private.h"

#ifndef CVC5__THEORY__SETS__CARDINALITY_EXTENSION_H

#define CVC5__THEORY__SETS__CARDINALITY_EXTENSION_H

#include "context/cdhashset.h"

#include "context/context.h"

#include "theory/sets/inference_manager.h"

#include "theory/sets/solver_state.h"

#include "theory/sets/term_registry.h"

#include "theory/type_set.h"

#include "theory/uf/equality_engine.h"

namespace cvc5 {

namespace theory {

namespace sets {

/**

 * This class implements a variant of the procedure from Bansal et al, IJCAR

 * 2016. It is used during a full effort check in the following way:

 *    reset(); { registerTerm(n,lemmas); | n in CardTerms }  check();

 * where CardTerms is the set of all applications of CARD in the current

 * context.

 *

 * The remaining public methods are used during model construction, i.e.

 * the collectModelInfo method of the theory of sets.

 *

 * The procedure from Bansal et al IJCAR 2016 introduces the notion of a

 * "cardinality graph", where the nodes of this graph are sets and (directed)

 * edges connect sets to their Venn regions wrt to other sets. For example,

 * if (A \ B) is a term in the current context, then the node A is

 * connected via an edge to child (A \ B). The node (A ^ B) is a child

 * of both A and B. The notion of a cardinality graph is loosely followed

 * in the procedure implemented by this class.

 *

 * The main difference wrt Bansal et al IJCAR 2016 is that the nodes of the

 * cardinality graph considered by this class are not arbitrary set terms, but

 * are instead representatives of equivalence classes. For more details, see

 * documentation of the inference schemas in the private methods of this class.

 *

 * This variant of the procedure takes inspiration from the procedure

 * for word equations in Liang et al, CAV 2014. In that procedure, "normal

 * forms" are generated for String terms by recursively expanding

 * concatentations modulo equality. This procedure similarly maintains

 * normal forms, where the normal form for Set terms is a set of (equivalence

 * class representatives of) Venn regions that do not contain the empty set.

 */

class CardinalityExtension

{

  typedef context::CDHashSet<Node> NodeSet;

 public:

  /**

   * Constructs a new instance of the cardinality solver w.r.t. the provided

   * contexts.

   */

  CardinalityExtension(SolverState& s,

                       InferenceManager& im,

                       TermRegistry& treg);

  /** reset

   *

   * Called at the beginning of a full effort check. This resets the data

   * structures used by this class during a full effort check.

   */

  void reset();

  /** register term

   *

   * Register that the term n exists in the current context, where n is an

   * application of CARD.

   */

  void registerTerm(Node n);

  /** check

   *

   * Invoke a full effort check of the cardinality solver. At a high level,

   * this asserts inferences via the inference manager object d_im. If no

   * inferences are made, then the current set of assertions is satisfied

   * with respect to constraints involving set cardinality.

   *

   * This method applies various inference schemas in succession until an

   * inference is made. These inference schemas are given in the private

   * methods of this class (e.g. checkMinCard, checkCardBuildGraph, etc.).

   */

  void check();

  /** Is model value basic?

   *

   * This returns true if equivalence class eqc is a "leaf" in the cardinality

   * graph.

   */

  bool isModelValueBasic(Node eqc);

  /** get model elements

   *

   * This method updates els so that it is the set of elements that occur in

   * an equivalence class (whose representative is eqc) in the model we are

   * building. Notice that els may already have elements in it (from explicit

   * memberships from the base set solver for leaf nodes of the cardinality

   * graph). This method is used during the collectModelInfo method of theory

   * of sets.

   *

   * The argument v is the Valuation utility of the theory of sets, which is

   * used by this method to query the value assigned to cardinality terms by

   * the theory of arithmetic.

   *

   * The argument mvals maps set equivalence classes to their model values.

   * Due to our model construction algorithm, it can be assumed that all

   * sets in the normal form of eqc occur in the domain of mvals by the order

   * in which sets are assigned.

   */

  void mkModelValueElementsFor(Valuation& v,

                               Node eqc,

                               std::vector<Node>& els,

                               const std::map<Node, Node>& mvals,

                               TheoryModel* model);

  /** get ordered sets equivalence classes

   *

   * Get the ordered set of equivalence classes computed by this class. This

   * ordering ensures the invariant mentioned above mkModelValueElementsFor.

   *

   * This ordering ensures that all children of a node in the cardinality

   * graph computed by this class occur before it in this list.

   */

  /**

   * get the slack elements generated for each finite type. This function is

   * called by TheorySetsPrivate::collectModelInfo to ask the TheoryModel to

   * exclude these slack elements from the members in all sets of that type.

   */

      const

  {

  }

  /**

   * get non-slack members in all sets of the given finite type. This function

   * is called by TheorySetsPrivate::collectModelInfo to specify the exclusion

   * sets for TheoryModel

   */

  const std::vector<Node>& getFiniteTypeMembers(TypeNode typeNode);

 private:

  /** constants */

  Node d_true;

  Node d_zero;

  /** the empty vector */

  std::vector<Node> d_emp_exp;

  /** Reference to the state object for the theory of sets */

  SolverState& d_state;

  /** Reference to the inference manager for the theory of sets */

  InferenceManager& d_im;

  /** Reference to the term registry */

  TermRegistry& d_treg;

  /** register cardinality term

   *

   * This method adds lemmas corresponding to the definition of

   * the cardinality of set term n. For example, if n is A^B (denoting set

   * intersection as ^), then we consider the lemmas card(A^B)>=0,

   * card(A) = card(A\B) + card(A^B) and card(B) = card(B\A) + card(A^B).

   *

   * The exact form of this lemma is modified such that proxy variables are

   * introduced for set terms as needed (see SolverState::getProxy).

   */

  void registerCardinalityTerm(Node n);

  /** check register

   *

   * This ensures that all (non-redundant, relevant) non-variable set terms in

   * the current context have been passed to registerCardinalityTerm. In other

   * words, this ensures that the cardinality graph for these terms can be

   * constructed since the cardinality relationships for these terms have been

   * elaborated as lemmas.

   *

   * Notice a term is redundant in a context if it is congruent to another

   * term that is already in the context; it is not relevant if no cardinality

   * constraints exist for its type.

   */

  void checkRegister();

  /** check minimum cardinality

   *

   * This adds lemmas to the argument of the method of the form

   *   distinct(x1,...,xn) ^ member(x1,S) ^ ... ^ member(xn,S) => card(S) >= n

   * This lemma enforces the rules GUESS_LOWER_BOUND and PROPAGATE_MIN_SIZE

   * from Bansal et. al IJCAR 2016.

   *

   * Notice that member(x1,S) is implied to hold in the current context. This

   * means that it may be the case that member(x,T) ^ T = S are asserted

   * literals. Furthermore, x1, ..., xn reside in distinct equivalence classes

   * but are not necessarily entailed to be distinct.

   */

  void checkMinCard();

  /** check cardinality cycles

   *

   * The purpose of this inference schema is to construct two data structures:

   *

   * (1) d_card_parent, which maps set terms (A op B) for op in { \, ^ } to

   * equivalence class representatives of their "parents", where:

   *   parent( A ^ B ) = A, B

   *   parent( A \ B ) = A

   * Additionally, (A union B) is a parent of all three of the above sets

   * if it exists as a term in the current context. As exceptions,

   * if A op B = A, then A is not a parent of A ^ B and similarly for B.

   * If A ^ B is empty, then it has no parents.

   *

   * We say the cardinality graph induced by the current set of equalities

   * is an (irreflexive, acyclic) graph whose nodes are equivalence classes and

   * which contains a (directed) edge r1 to r2 if there exists a term t2 in r2

   * that has some parent t1 in r1.

   *

   * (2) d_oSetEqc, an ordered set of equivalence classes whose types are set.

   * These equivalence classes have the property that if r1 is a descendant

   * of r2 in the cardinality graph, then r1 must come before r2 in d_oSetEqc.

   *

   * This inference schema may make various inferences while building these

   * two data structures if the current equality arrangement of sets is not

   * as expected. For example, it will infer equalities between sets based on

   * the emptiness and equalities of sets in adjacent children in the

   * cardinality graph, to give some examples:

   *   (A \ B = empty) => A = A ^ B

   *   A^B = B => B \ A = empty

   *   A union B = A ^ B => A \ B = empty AND B \ A = empty

   * and so on.

   *

   * It will also recognize when a cycle occurs in the cardinality graph, in

   * which case an equality chain between sets can be inferred. For an example,

   * see checkCardCyclesRec below.

   *

   * This method is inspired by the checkCycles inference schema in the theory

   * of strings.

   */

  void checkCardCycles();

  /**

   * Helper function for above. Called when wish to process equivalence class

   * eqc.

   *

   * Argument curr contains the equivalence classes we are currently processing,

   * which are descendants of eqc in the cardinality graph, where eqc is the

   * representative of some equivalence class.

   *

   * Argument exp contains an explanation of why the chain of children curr

   * are descedants of . For example, say we are in context with equivalence

   * classes:

   *   { A, B, C^D }, { D, B ^ C,  A ^ E }

   * We may recursively call this method via the following steps:

   *   eqc = D, curr = {}, exp = {}

   *   eqc = A, curr = { D }, exp = { D = B^C }

   *   eqc = A, curr = { D, A }, exp = { D = B^C, A = C^D }

   * after which we discover a cycle in the cardinality graph. We infer

   * that A must be equal to D, where exp is an explanation of the cycle.

   */

  void checkCardCyclesRec(Node eqc,

                          std::vector<Node>& curr,

                          std::vector<Node>& exp);

  /** check normal forms

   *

   * This method attempts to assign "normal forms" to all set equivalence

   * classes whose types have cardinality constraints. Normal forms are

   * defined recursively.

   *

   * A "normal form" of an equivalence class [r] (where [r] denotes the

   * equivalence class whose representative is r) is a set of representatives

   * U = { r1, ..., rn }. If there exists at least one set in [r] that has a

   * "flat form", then all sets in the equivalence class have flat form U.

   * If no set in U has a flat form, then U = { r } if r does not contain

   * the empty set, and {} otherwise. Notice that the choice of representative

   * r is determined by the equality engine.

   *

   * A "flat form" of a set term T is the union of the normal forms of the

   * equivalence classes that contain sets whose parent is T.

   *

   * In terms of the cardinality graph, the "flat form" of term t is the set

   * of leaves of t that are descendants of it in the cardinality graph induced

   * by the current set of assertions. Notice a flat form is only defined if t

   * has children. If all terms in an equivalence class E with flat forms have

   * the same flat form, then E is added as a node to the cardinality graph with

   * edges connecting to all equivalence classes with terms that have a parent

   * in E.

   *

   * In the following inference schema, the argument intro_sets is updated to

   * contain the set of new set terms that the procedure is requesting to

   * introduce for the purpose of forcing the flat forms of two equivalent sets

   * to become identical. If any equivalence class cannot be assigned a normal

   * form, then the resulting intro_sets is guaranteed to be non-empty.

   *

   * As an example, say we have a context with equivalence classes:

   *   {A, D}, {C, A^B}, {E, C^D}, {C\D}, {D\C}, {A\B}, {empty, B\A},

   * where assume the first term in each set is its representative. An ordered

   * list d_oSetEqc for this context:

   *   A, C, E, C\D, D\C, A\B, empty, ...

   * The normal form of {empty, B\A} is {}, since it contains the empty set.

   * The normal forms for each of the singleton equivalence classes are

   * themselves.

   * The flat form of each of E and C^D does not exist, hence the normal form

   * of {E, C^D} is {E}.

   * The flat form of C is {E, C\D}, noting that C^D and C\D are terms whose

   * parent is C, hence {E, C\D} is the normal form for class {C, A^B}.

   * The flat form of A is {E, C\D, A\B} and the flat form of D is {E, D\C}.

   * Hence, no normal form can be assigned to class {A, D}. Instead this method

   * will e.g. add (C\D)^E to intro_sets, which will force the solver

   * to explore a model where the Venn regions (C\D)^E (C\D)\E and E\(C\D) are

   * considered while constructing flat forms. Splitting on whether these sets

   * are empty will eventually lead to a model where the flat forms of A and D

   * are the same.

   */

  void checkNormalForms(std::vector<Node>& intro_sets);

  /**

   * Called for each equivalence class, in order of d_oSetEqc, by the above

   * function.

   */

  void checkNormalForm(Node eqc, std::vector<Node>& intro_sets);

  /**

   * Add cardinality lemmas for the univset of each type with cardinality terms.

   * The lemmas are explained below.

   */

  void checkCardinalityExtended();

  /**

   * This function adds the following lemmas for type t for each S

   * where S is a (representative) set term of type t, and for each negative

   * member x not in S:

   * 1- (=> true (<= (card (as univset t)) n) where n is the

   * cardinality of t, which may be symbolic

   * 2- (=> true (subset S (as univset t)) where S is a set

   * term of type t

   * 3- (=> (not (member negativeMember S))) (member

   * negativeMember (as univset t)))

   */

  void checkCardinalityExtended(TypeNode& t);

  /** element types of sets for which cardinality is enabled */

  std::map<TypeNode, bool> d_t_card_enabled;

  /**

   * This maps equivalence classes r to an application of cardinality of the

   * form card( t ), where t = r holds in the current context.

   */

  std::map<Node, Node> d_eqc_to_card_term;

  /**

   * User-context-dependent set of set terms for which we have called

   * registerCardinalityTerm on.

   */

  NodeSet d_card_processed;

  /** The ordered set of equivalence classes, see checkCardCycles. */

  std::vector<Node> d_oSetEqc;

  /**

   * This maps set terms to the set of representatives of their "parent" sets,

   * see checkCardCycles. Parents are stored as a pair of the form

   *   (r, t)

   * where t is the parent term and r is the representative of equivalence

   * class of t.

   */

  std::map<Node, std::vector<std::pair<Node, Node>>> d_cardParent;

  /**

   * Maps equivalence classes + set terms in that equivalence class to their

   * "flat form" (see checkNormalForms).

   */

  std::map<Node, std::map<Node, std::vector<Node> > > d_ff;

  /** Maps equivalence classes to their "normal form" (see checkNormalForms). */

  std::map<Node, std::vector<Node> > d_nf;

  /** The local base node map

   *

   * This maps set terms to the "local base node" of its cardinality graph.

   * The local base node of S is the intersection node that is either S itself

   * or is adjacent to S in the cardinality graph. This maps

   *

   * For example, the ( A ^ B ) is the local base of each of the sets (A \ B),

   * ( A ^ B ), and (B \ A).

   */

  std::map<Node, Node> d_localBase;

  /**

   * a map to store proxy nodes for the universe sets

   */

  std::map<Node, Node> d_univProxy;

  /**

   * collect the elements in all sets of finite types in model, and

   * store them in the field d_finite_type_elements

   */

  void collectFiniteTypeSetElements(TheoryModel* model);

  /**

   * a map to store the non-slack elements encountered so far in all finite

   * types

   */

  std::map<TypeNode, std::vector<Node> > d_finite_type_elements;

  /**

   * a map to store slack elements in all finite types

   */

  std::map<TypeNode, std::vector<TNode> > d_finite_type_slack_elements;

  /** This boolean determines whether we already collected finite type constants

   *  present in the current model.

   *  Default value is false

   */

  bool d_finite_type_constants_processed;

  /*

   * a map that stores skolem nodes n that satisfies the constraint

   * (<= (card t) n) where t is an infinite type

   */

  std::map<TypeNode, Node> d_infiniteTypeUnivCardSkolems;

}; /* class CardinalityExtension */

}  // namespace sets

}  // namespace theory

}  // namespace cvc5

#endif