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/****************************************************************************** |
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* Top contributors (to current version): |
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* Gereon Kremer |
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* |
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* This file is part of the cvc5 project. |
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* |
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* Copyright (c) 2009-2021 by the authors listed in the file AUTHORS |
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* in the top-level source directory and their institutional affiliations. |
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* All rights reserved. See the file COPYING in the top-level source |
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* directory for licensing information. |
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* **************************************************************************** |
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* |
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* Implements the CDCAC approach as described in |
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* https://arxiv.org/pdf/2003.05633.pdf. |
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*/ |
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#include "cvc5_private.h" |
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#ifndef CVC5__THEORY__ARITH__NL__CAD__CDCAC_H |
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#define CVC5__THEORY__ARITH__NL__CAD__CDCAC_H |
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#ifdef CVC5_POLY_IMP |
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#include <poly/polyxx.h> |
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#include <vector> |
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#include "smt/env.h" |
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#include "theory/arith/nl/cad/cdcac_utils.h" |
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#include "theory/arith/nl/cad/constraints.h" |
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#include "theory/arith/nl/cad/proof_generator.h" |
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#include "theory/arith/nl/cad/variable_ordering.h" |
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namespace cvc5 { |
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namespace theory { |
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namespace arith { |
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namespace nl { |
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class NlModel; |
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namespace cad { |
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/** |
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* This class implements Cylindrical Algebraic Coverings as presented in |
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* https://arxiv.org/pdf/2003.05633.pdf |
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*/ |
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class CDCAC |
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{ |
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public: |
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/** Initialize this method with the given variable ordering. */ |
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CDCAC(Env& env, const std::vector<poly::Variable>& ordering = {}); |
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/** Reset this instance. */ |
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void reset(); |
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/** Collect variables from the constraints and compute a variable ordering. */ |
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void computeVariableOrdering(); |
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/** |
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* Extract an initial assignment from the given model. |
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* This initial assignment is used to guide sampling if possible. |
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* The ran_variable should be the variable used to encode real algebraic |
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* numbers in the model and is simply passed on to node_to_value. |
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*/ |
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void retrieveInitialAssignment(NlModel& model, const Node& ran_variable); |
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/** |
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* Returns the constraints as a non-const reference. Can be used to add new |
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* constraints. |
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*/ |
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Constraints& getConstraints(); |
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/** Returns the constraints as a const reference. */ |
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const Constraints& getConstraints() const; |
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/** |
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* Returns the current assignment. This is a satisfying model if |
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* get_unsat_cover() returned an empty vector. |
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*/ |
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const poly::Assignment& getModel() const; |
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/** Returns the current variable ordering. */ |
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const std::vector<poly::Variable>& getVariableOrdering() const; |
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/** |
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* Collect all unsatisfiable intervals for the given variable. |
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* Combines unsatisfiable regions from d_constraints evaluated over |
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* d_assignment. Implements Algorithm 2. |
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*/ |
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std::vector<CACInterval> getUnsatIntervals(std::size_t cur_variable); |
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/** |
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* Sample outside of the set of intervals. |
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* Uses a given initial value from mInitialAssignment if possible. |
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* Returns whether a sample was found (true) or the infeasible intervals cover |
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* the whole real line (false). |
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*/ |
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bool sampleOutsideWithInitial(const std::vector<CACInterval>& infeasible, |
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poly::Value& sample, |
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std::size_t cur_variable); |
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/** |
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* Collects the coefficients required for projection from the given |
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* polynomial. Implements Algorithm 6, depending on the command line |
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* arguments. Either directly implements Algorithm 6, or improved variants |
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* based on Lazard's projection. |
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*/ |
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PolyVector requiredCoefficients(const poly::Polynomial& p); |
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/** |
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* Constructs a characterization of the given covering. |
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* A characterization contains polynomials whose roots bound the region around |
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* the current assignment. Implements Algorithm 4. |
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*/ |
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PolyVector constructCharacterization(std::vector<CACInterval>& intervals); |
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/** |
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* Constructs an infeasible interval from a characterization. |
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* Implements Algorithm 5. |
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*/ |
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CACInterval intervalFromCharacterization(const PolyVector& characterization, |
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std::size_t cur_variable, |
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const poly::Value& sample); |
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/** |
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* Main method that checks for the satisfiability of the constraints. |
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* Recursively explores possible assignments and excludes regions based on the |
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* coverings. Returns either a covering for the lowest dimension or an empty |
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* vector. If the covering is empty, the result is SAT and an assignment can |
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* be obtained from d_assignment. If the covering is not empty, the result is |
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* UNSAT and an infeasible subset can be extracted from the returned covering. |
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* Implements Algorithm 2. |
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* @param curVariable The id of the variable (within d_variableOrdering) to |
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* be considered. This argument is used to manage the recursion internally and |
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* should always be zero if called externally. |
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* @param returnFirstInterval If true, the function returns after the first |
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* interval obtained from a recursive call. The result is not (necessarily) an |
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* unsat cover, but merely a list of infeasible intervals. |
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*/ |
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std::vector<CACInterval> getUnsatCover(std::size_t curVariable = 0, |
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bool returnFirstInterval = false); |
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void startNewProof(); |
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/** |
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* Finish the generated proof (if proofs are enabled) with a scope over the |
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* given assertions. |
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*/ |
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ProofGenerator* closeProof(const std::vector<Node>& assertions); |
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/** Get the proof generator */ |
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CADProofGenerator* getProof() { return d_proof.get(); } |
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private: |
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/** Check whether proofs are enabled */ |
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bool isProofEnabled() const { return d_proof != nullptr; } |
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/** |
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* Check whether the current sample satisfies the integrality condition of the |
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* current variable. Returns true if the variable is not integral or the |
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* sample is integral. |
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*/ |
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bool checkIntegrality(std::size_t cur_variable, const poly::Value& value); |
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/** |
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* Constructs an interval that excludes the non-integral region around the |
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* current sample. Assumes !check_integrality(cur_variable, value). |
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*/ |
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CACInterval buildIntegralityInterval(std::size_t cur_variable, |
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const poly::Value& value); |
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/** |
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* Check whether the polynomial has a real root above the given value (when |
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* evaluated over the current assignment). |
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*/ |
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bool hasRootAbove(const poly::Polynomial& p, const poly::Value& val) const; |
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/** |
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* Check whether the polynomial has a real root below the given value (when |
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* evaluated over the current assignment). |
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*/ |
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bool hasRootBelow(const poly::Polynomial& p, const poly::Value& val) const; |
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/** |
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* Sort intervals according to section 4.4.1. and removes fully redundant |
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* intervals as in 4.5. 1. by calling out to cleanIntervals. |
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* Additionally makes sure to prune proofs for removed intervals. |
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*/ |
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void pruneRedundantIntervals(std::vector<CACInterval>& intervals); |
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/** A reference to the environment */ |
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Env& d_env; |
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/** |
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* The current assignment. When the method terminates with SAT, it contains a |
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* model for the input constraints. |
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*/ |
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poly::Assignment d_assignment; |
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/** The set of input constraints to be checked for consistency. */ |
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Constraints d_constraints; |
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/** The computed variable ordering used for this method. */ |
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std::vector<poly::Variable> d_variableOrdering; |
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/** The object computing the variable ordering. */ |
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VariableOrdering d_varOrder; |
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/** The linear assignment used as an initial guess. */ |
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std::vector<poly::Value> d_initialAssignment; |
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/** The proof generator */ |
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std::unique_ptr<CADProofGenerator> d_proof; |
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}; |
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} // namespace cad |
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} // namespace nl |
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} // namespace arith |
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} // namespace theory |
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} // namespace cvc5 |
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#endif |
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#endif |