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diff --git a/src/theory/uf/eq_proof.h b/src/theory/uf/eq_proof.h new file mode 100644 index 000000000..c396da313 --- /dev/null +++ b/src/theory/uf/eq_proof.h @@ -0,0 +1,340 @@ +/********************* */ +/*! \file eq_proof.h + ** \verbatim + ** Top contributors (to current version): + ** Haniel Barbosa + ** This file is part of the CVC4 project. + ** Copyright (c) 2009-2020 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.\endverbatim + ** + ** \brief A proof as produced by the equality engine. + ** + **/ + +#include "cvc4_private.h" +#include "expr/node.h" +#include "theory/uf/equality_engine_types.h" + +namespace CVC4 { + +class CDProof; + +namespace theory { +namespace eq { + +/** + * An equality proof. + * + * This represents the reasoning performed by the Equality Engine to derive + * facts, represented in terms of the rules in MergeReasonType. Each proof step + * is annotated with the rule id, the conclusion node and a vector of proofs of + * the rule's premises. + **/ +class EqProof +{ + public: + /** A custom pretty printer used for custom rules being those in + * MergeReasonType. */ + class PrettyPrinter + { + public: + virtual ~PrettyPrinter() {} + virtual std::string printTag(unsigned tag) = 0; + }; + + EqProof() : d_id(MERGED_THROUGH_REFLEXIVITY) {} + /** The proof rule for concluding d_node */ + unsigned d_id; + /** The conclusion of this EqProof */ + Node d_node; + /** The proofs of the premises for deriving d_node with d_id */ + std::vector<std::shared_ptr<EqProof>> d_children; + /** + * Debug print this proof on debug trace c with tabulation tb and pretty + * printer prettyPrinter. + */ + void debug_print(const char* c, + unsigned tb = 0, + PrettyPrinter* prettyPrinter = nullptr) const; + /** + * Debug print this proof on output stream os with tabulation tb and pretty + * printer prettyPrinter. + */ + void debug_print(std::ostream& os, + unsigned tb = 0, + PrettyPrinter* prettyPrinter = nullptr) const; + + /** Add to proof + * + * Converts EqProof into ProofNode via a series of steps to be stored in + * CDProof* p. + * + * This method can be seen as a best-effort solution until the EqualityEngine + * is updated to produce ProofNodes directly, if ever. Note that since the + * original EqProof steps can be coarse-grained (e.g. without proper order, + * implicit inferences related to disequelaties) and are phrased over curried + * terms, the transformation requires significant, although cheap (mostly + * linear on the DAG-size of EqProof), post-processing. + * + * An important invariant of the resulting ProofNode is that neither its + * assumptions nor its conclusion are predicate equalities, i.e. of the form + * (= t true/false), modulo symmetry. This is so that users of the converted + * ProofNode don't need to do case analysis on whether assumptions/conclusion + * are either t or (= t true/false). + * + * @param p a pointer to a CDProof to store the conversion of this EqProof + * @return the node that is the conclusion of the proof as added to p. + */ + Node addToProof(CDProof* p) const; + + private: + /** + * As above, but with a cache of previously processed nodes and their results + * (for DAG traversal). The caching is in terms of the original conclusions of + * EqProof. + * + * @param p a pointer to a CDProof to store the conversion of this EqProof + * @param visited a cache of the original EqProof conclusions and the + * resulting conclusion after conversion. + * @param assumptions the assumptions of the original EqProof (and their + * variations in terms of symmetry and conversion to/from predicate + * equalities) + * @return the node that is the conclusion of the proof as added to p. + */ + Node addToProof( + CDProof* p, + std::unordered_map<Node, Node, NodeHashFunction>& visited, + std::unordered_set<Node, NodeHashFunction>& assumptions) const; + + /** Removes all reflexivity steps, i.e. (= t t), from premises. */ + void cleanReflPremises(std::vector<Node>& premises) const; + + /** Expand coarse-grained transitivity steps for disequalities + * + * Currently the equality engine can represent disequality reasoning in a + * rather coarse-grained manner with EqProof. This is always the case when the + * transitivity step contains a disequality, (= (= t t') false) or its + * symmetric. + * + * There are two cases. In the simplest one the general shape of the EqProof + * is + * (= t1 t2) (= t3 t2) (= (t1 t3) false) + * ------------------------------------- EQP::TR + * false = true + * + * which is expanded into + * (= t3 t2) + * --------- SYMM + * (= t1 t2) (= t2 t3) + * ------------------- TRANS + * (= (= t1 t3) false) (= t1 t3) + * --------------------- SYMM ------------------ TRUE_INTRO + * (= false (= t1 t3)) (= (= t1 t3) true) + * ----------------------------------------------- TRANS + * false = true + * + * by explicitly adding the transitivity step for inferring (= t1 t3) and its + * predicate equality. Note that we differentiate (from now on) the EqProof + * rules ids from those of ProofRule by adding the prefix EQP:: to the former. + * + * In the other case, the general shape of the EqProof is + * + * (= (= t1 t2) false) (= t1 x1) ... (= xn t3) (= t2 y1) ... (= ym t4) + * ------------------------------------------------------------------- EQP::TR + * (= (= t4 t3) false) + * + * which is converted into + * + * (= t1 x1) ... (= xn t3) (= t2 y1) ... (= ym t4) + * ------------------------ TR ------------------------ TR + * (= t1 t3) (= t2 t4) + * ----------- SYMM ----------- SYMM + * (= t3 t1) (= t4 t2) + * ---------------------------------------- CONG + * (= (= t3 t4) (= t1 t2)) (= (= t1 t2) false) + * --------------------------------------------------------------------- TR + * (= (= t3 t4) false) + * --------------------- MACRO_SR_PRED_TRANSFORM + * (= (= t4 t3) false) + * + * whereas the last step is only necessary if the conclusion has the arguments + * in reverse order than expected. Note that the original step represents two + * substitutions happening on the disequality, from t1->t3 and t2->t4, which + * are implicitly justified by transitivity steps that need to be made + * explicity. Since there is no sense of ordering among which premisis (other + * than (= (= t1 t2) false)) are used for which substitution, the transitivity + * proofs are built greedly by searching the sets of premises. + * + * If in either of the above cases then the conclusion is directly derived + * in the call, so only in the other cases we try to build a transitivity + * step below + * + * @param conclusion the conclusion of the (possibly) coarse-grained + * transitivity step + * @param premises the premises of the (possibly) coarse-grained + * transitivity step + * @param p a pointer to a CDProof to store the conversion of this EqProof + * @param assumptions the assumptions (and variants) of the original + * EqProof. These are necessary to avoid cyclic proofs, which could be + * generated by creating transitivity steps for assumptions (which depend on + * themselves). + */ + bool expandTransitivityForDisequalities( + Node conclusion, + std::vector<Node>& premises, + CDProof* p, + std::unordered_set<Node, NodeHashFunction>& assumptions) const; + + /** Builds a transitivity chain from equalities to derive a conclusion + * + * Given an equality (= t1 tn), and a list of equalities premises, attempts to + * build a chain (= t1 t2) ... (= tn-1 tn) from premises. This is done in a + * greedy manner by finding one "link" in the chain at a time, updating the + * conclusion to be the next link and the premises by removing the used + * premise, and searching recursively. + * + * Consider for example + * - conclusion: (= t1 t4) + * - premises: (= t4 t2), (= t2 t3), (= t2 t1), (= t3 t4) + * + * It proceeds by searching for t1 in an equality in the premises, in order, + * which is found in the equality (= t2 t1). Since t2 != t4, it attempts to + * build a chain with + * - conclusion (= t2 t4) + * - premises (= t4 t2), (= t2 t3), (= t3 t4) + * + * In the first equality it finds t2 and also t4, which closes the chain, so + * that premises is updated to (= t2 t4) and the method returns true. Since + * the recursive call was successful, the original conclusion (= t1 t4) is + * justified with (= t1 t2) plus the chain built in the recursive call, + * i.e. (= t1 t2), (= t2 t4). + * + * Note that not all the premises were necessary to build a successful + * chain. Moreover, note that building a successful chain can depend on the + * order in which the equalities are chosen. When a recursive call fails to + * close a chain, the chosen equality is dismissed and another is searched for + * among the remaining ones. + * + * This method assumes that premises does not contain reflexivity premises. + * This is without loss of generality since such premises are spurious for a + * transitivity step. + * + * @param conclusion the conclusion of the transitivity chain of equalities + * @param premises the list of equalities to build a chain with + * @return whether premises successfully build a transitivity chain for the + * conclusion + */ + bool buildTransitivityChain(Node conclusion, + std::vector<Node>& premises) const; + + /** Reduce the a congruence EqProof into a transitivity matrix + * + * Given a congruence EqProof of (= (f a0 ... an-1) (f b0 ... bn-1)), reduce + * its justification into a matrix + * + * [0] -> p_{0,0} ... p_{m_0,0} + * ... + * [n-1] -> p_{0,n} ... p_{m_n-1,n-1} + * + * where f has arity n and each p_{0,i} ... p_{m_i, i} is a transitivity chain + * (modulo ordering) justifying (= ai bi). + * + * Congruence steps in EqProof are binary, representing reasoning over curried + * applications. In the simplest case the general shape of a congruence + * EqProof is: + * P0 + * ------- EQP::REFL ---------- + * P1 [] (= a0 b0) + * --------- ----------------------- EQP::CONG + * (= a1 b1) [] P2 + * ------------------------- EQP::CONG ----------- + * [] (= a2 b2) + * ------------------------------------------------ EQP::CONG + * (= (f a0 a1 a2) (f b0 b1 b2)) + * + * where [] stands for the null node, symbolizing "equality between partial + * applications". + * + * The reduction of such a proof is done by + * - converting the proof of the second CONG premise (via addToProof) and + * adding the resulting node to row i of the matrix + * - recursively reducing the first proof with i-1 + * + * In the above case the transitivity matrix would thus be + * [0] -> (= a0 b0) + * [1] -> (= a1 b1) + * [2] -> (= a2 b2) + * + * The more complex case of congruence proofs has transitivity steps as the + * first child of CONG steps. For example + * P0 + * ------- EQP::REFL ---------- + * P' [] (= a0 c) + * --------- ----------------------------- EQP::CONG + * (= b0 c) [] P1 + * ------------------------- EQP::TRANS ----------- + * [] (= a1 b1) + * ----------------------------------------------------- EQP::CONG + * (= (f a0 a1) (f b0 b1)) + * + * where when the first child of CONG is a transitivity step + * - the premises that are CONG steps are recursively reduced with *the same* + argument i + * - the other premises are processed with addToProof and added to the i row + * in the matrix + * + * In the above example the to which the transitivity matrix is + * [0] -> (= a0 c), (= b0 c) + * [1] -> (= a1 b1) + * + * The remaining complication is that when conclusion is an equality of n-ary + * applications of *different* arities, there is, necessarily, a transitivity + * step as a first child a CONG step whose conclusion is an equality of n-ary + * applications of different arities. For example + * P0 P1 + * -------------------------- EQP::TRANS ----------- + * (= (f a0 a1) (f b0)) (= a2 b1) + * -------------------------------------------------- EQP::CONG + * (= (f a0 a1 a2) (f b0 b1)) + * + * will be first reduced with i = 2 (maximal arity amorg the original + * conclusion's applications), adding (= a2 b1) to row 2 after processing + * P1. The first child is reduced with i = 1. Since it is a TRANS step whose + * conclusion is an equality of n-ary applications with mismatching arity, P0 + * is processed with addToProof and the result is added to row 1. Thus the + * transitivity matrix is + * [0] -> + * [1] -> (= (f a0 a1) (f b0)) + * [2] -> (= a2 b1) + * + * The empty row 0 is used by the original caller of reduceNestedCongruence to + * compute that the above congruence proof's conclusion is + * (= (f (f a0 a1) a2) (f (f b0) b1)) + * + * @param i the i-th argument of the congruent applications, initially being + * the maximal arity among conclusion's applications. + * @param conclusion the original congruence conclusion + * @param transitivityMatrix a matrix of equalities with each row justifying + * an equality between the congruent applications + * @param p a pointer to a CDProof to store the conversion of this EqProof + * @param visited a cache of the original EqProof conclusions and the + * resulting conclusion after conversion. + * @param assumptions the assumptions (and variants) of the original EqProof + * @param isNary whether conclusion is an equality of n-ary applications + */ + void reduceNestedCongruence( + unsigned i, + Node conclusion, + std::vector<std::vector<Node>>& transitivityMatrix, + CDProof* p, + std::unordered_map<Node, Node, NodeHashFunction>& visited, + std::unordered_set<Node, NodeHashFunction>& assumptions, + bool isNary) const; + +}; /* class EqProof */ + +} // Namespace eq +} // Namespace theory +} // Namespace CVC4 |