1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
|
/********************* */
/*! \file cdcac.h
** \verbatim
** Top contributors (to current version):
** Gereon Kremer
** 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 Implements the CDCAC approach.
**
** Implements the CDCAC approach as described in
** https://arxiv.org/pdf/2003.05633.pdf.
**/
#include "cvc4_private.h"
#ifndef CVC4__THEORY__ARITH__NL__CAD__CDCAC_H
#define CVC4__THEORY__ARITH__NL__CAD__CDCAC_H
#include "util/real_algebraic_number.h"
#ifdef CVC4_POLY_IMP
#include <poly/polyxx.h>
#include <vector>
#include "theory/arith/nl/cad/cdcac_utils.h"
#include "theory/arith/nl/cad/constraints.h"
#include "theory/arith/nl/cad/variable_ordering.h"
#include "theory/arith/nl/nl_model.h"
namespace CVC4 {
namespace theory {
namespace arith {
namespace nl {
namespace cad {
/**
* This class implements Cylindrical Algebraic Coverings as presented in
* https://arxiv.org/pdf/2003.05633.pdf
*/
class CDCAC
{
public:
/** Initialize without a variable ordering. */
CDCAC();
/** Initialize this method with the given variable ordering. */
CDCAC(const std::vector<poly::Variable>& ordering);
/** Reset this instance. */
void reset();
/** Collect variables from the constraints and compute a variable ordering. */
void computeVariableOrdering();
/**
* Extract an initial assignment from the given model.
* This initial assignment is used to guide sampling if possible.
* The ran_variable should be the variable used to encode real algebraic
* numbers in the model and is simply passed on to node_to_value.
*/
void retrieveInitialAssignment(NlModel& model, const Node& ran_variable);
/**
* Returns the constraints as a non-const reference. Can be used to add new
* constraints.
*/
Constraints& getConstraints();
/** Returns the constraints as a const reference. */
const Constraints& getConstraints() const;
/**
* Returns the current assignment. This is a satisfying model if
* get_unsat_cover() returned an empty vector.
*/
const poly::Assignment& getModel() const;
/** Returns the current variable ordering. */
const std::vector<poly::Variable>& getVariableOrdering() const;
/**
* Collect all unsatisfiable intervals for the given variable.
* Combines unsatisfiable regions from d_constraints evaluated over
* d_assignment. Implements Algorithm 2.
*/
std::vector<CACInterval> getUnsatIntervals(std::size_t cur_variable) const;
/**
* Sample outside of the set of intervals.
* Uses a given initial value from mInitialAssignment if possible.
* Returns whether a sample was found (true) or the infeasible intervals cover
* the whole real line (false).
*/
bool sampleOutsideWithInitial(const std::vector<CACInterval>& infeasible,
poly::Value& sample,
std::size_t cur_variable);
/**
* Collects the coefficients required for projection from the given
* polynomial. Implements Algorithm 6.
*/
std::vector<poly::Polynomial> requiredCoefficients(
const poly::Polynomial& p) const;
/**
* Constructs a characterization of the given covering.
* A characterization contains polynomials whose roots bound the region around
* the current assignment. Implements Algorithm 4.
*/
std::vector<poly::Polynomial> constructCharacterization(
std::vector<CACInterval>& intervals);
/**
* Constructs an infeasible interval from a characterization.
* Implements Algorithm 5.
*/
CACInterval intervalFromCharacterization(
const std::vector<poly::Polynomial>& characterization,
std::size_t cur_variable,
const poly::Value& sample);
/**
* Main method that checks for the satisfiability of the constraints.
* Recursively explores possible assignments and excludes regions based on the
* coverings. Returns either a covering for the lowest dimension or an empty
* vector. If the covering is empty, the result is SAT and an assignment can
* be obtained from d_assignment. If the covering is not empty, the result is
* UNSAT and an infeasible subset can be extracted from the returned covering.
* Implements Algorithm 2.
* @param curVariable The id of the variable (within d_variableOrdering) to
* be considered. This argument is used to manage the recursion internally and
* should always be zero if called externally.
* @param returnFirstInterval If true, the function returns after the first
* interval obtained from a recursive call. The result is not (necessarily) an
* unsat cover, but merely a list of infeasible intervals.
*/
std::vector<CACInterval> getUnsatCover(std::size_t curVariable = 0,
bool returnFirstInterval = false);
private:
/**
* Check whether the current sample satisfies the integrality condition of the
* current variable. Returns true if the variable is not integral or the
* sample is integral.
*/
bool checkIntegrality(std::size_t cur_variable, const poly::Value& value);
/**
* Constructs an interval that excludes the non-integral region around the
* current sample. Assumes !check_integrality(cur_variable, value).
*/
CACInterval buildIntegralityInterval(std::size_t cur_variable,
const poly::Value& value);
/**
* Check whether the polynomial has a real root above the given value (when
* evaluated over the current assignment).
*/
bool hasRootAbove(const poly::Polynomial& p, const poly::Value& val) const;
/**
* Check whether the polynomial has a real root below the given value (when
* evaluated over the current assignment).
*/
bool hasRootBelow(const poly::Polynomial& p, const poly::Value& val) const;
/**
* The current assignment. When the method terminates with SAT, it contains a
* model for the input constraints.
*/
poly::Assignment d_assignment;
/** The set of input constraints to be checked for consistency. */
Constraints d_constraints;
/** The computed variable ordering used for this method. */
std::vector<poly::Variable> d_variableOrdering;
/** The object computing the variable ordering. */
VariableOrdering d_varOrder;
/** The linear assignment used as an initial guess. */
std::vector<poly::Value> d_initialAssignment;
};
} // namespace cad
} // namespace nl
} // namespace arith
} // namespace theory
} // namespace CVC4
#endif
#endif
|
