"""Implements "A Fast Linear-Arithmetic Solver for DPLL(T)" The LRASolver class defined in this file can be used in conjunction with a SAT solver to check the satisfiability of formulas involving inequalities. Here's an example of how that would work: Suppose you want to check the satisfiability of the following formula: >>> from sympy.core.relational import Eq >>> from sympy.abc import x, y >>> f = ((x > 0) | (x < 0)) & (Eq(x, 0) | Eq(y, 1)) & (~Eq(y, 1) | Eq(1, 2)) First a preprocessing step should be done on f. During preprocessing, f should be checked for any predicates such as `Q.prime` that can't be handled. Also unequality like `~Eq(y, 1)` should be split. I should mention that the paper says to split both equalities and unequality, but this implementation only requires that unequality be split. >>> f = ((x > 0) | (x < 0)) & (Eq(x, 0) | Eq(y, 1)) & ((y < 1) | (y > 1) | Eq(1, 2)) Then an LRASolver instance needs to be initialized with this formula. >>> from sympy.assumptions.cnf import CNF, EncodedCNF >>> from sympy.assumptions.ask import Q >>> from sympy.logic.algorithms.lra_theory import LRASolver >>> cnf = CNF.from_prop(f) >>> enc = EncodedCNF() >>> enc.add_from_cnf(cnf) >>> lra, conflicts = LRASolver.from_encoded_cnf(enc) Any immediate one-lital conflicts clauses will be detected here. In this example, `~Eq(1, 2)` is one such conflict clause. We'll want to add it to `f` so that the SAT solver is forced to assign Eq(1, 2) to False. >>> f = f & ~Eq(1, 2) Now that the one-literal conflict clauses have been added and an lra object has been initialized, we can pass `f` to a SAT solver. The SAT solver will give us a satisfying assignment such as: (1 = 2): False (y = 1): True (y < 1): True (y > 1): True (x = 0): True (x < 0): True (x > 0): True Next you would pass this assignment to the LRASolver which will be able to determine that this particular assignment is satisfiable or not. Note that since EncodedCNF is inherently non-deterministic, the int each predicate is encoded as is not consistent. As a result, the code bellow likely does not reflect the assignment given above. >>> lra.assert_lit(-1) #doctest: +SKIP >>> lra.assert_lit(2) #doctest: +SKIP >>> lra.assert_lit(3) #doctest: +SKIP >>> lra.assert_lit(4) #doctest: +SKIP >>> lra.assert_lit(5) #doctest: +SKIP >>> lra.assert_lit(6) #doctest: +SKIP >>> lra.assert_lit(7) #doctest: +SKIP >>> is_sat, conflict_or_assignment = lra.check() As the particular assignment suggested is not satisfiable, the LRASolver will return unsat and a conflict clause when given that assignment. The conflict clause will always be minimal, but there can be multiple minimal conflict clauses. One possible conflict clause could be `~(x < 0) | ~(x > 0)`. We would then add whatever conflict clause is given to `f` to prevent the SAT solver from coming up with an assignment with the same conflicting literals. In this case, the conflict clause `~(x < 0) | ~(x > 0)` would prevent any assignment where both (x < 0) and (x > 0) were both true. The SAT solver would then find another assignment and we would check that assignment with the LRASolver and so on. Eventually either a satisfying assignment that the SAT solver and LRASolver agreed on would be found or enough conflict clauses would be added so that the boolean formula was unsatisfiable. This implementation is based on [1]_, which includes a detailed explanation of the algorithm and pseudocode for the most important functions. [1]_ also explains how backtracking and theory propagation could be implemented to speed up the current implementation, but these are not currently implemented. TODO: - Handle non-rational real numbers - Handle positive and negative infinity - Implement backtracking and theory proposition - Simplify matrix by removing unused variables using Gaussian elimination References ========== .. [1] Dutertre, B., de Moura, L.: A Fast Linear-Arithmetic Solver for DPLL(T) https://link.springer.com/chapter/10.1007/11817963_11 """ from sympy.solvers.solveset import linear_eq_to_matrix from sympy.matrices.dense import eye from sympy.assumptions import Predicate from sympy.assumptions.assume import AppliedPredicate from sympy.assumptions.ask import Q from sympy.core import Dummy from sympy.core.mul import Mul from sympy.core.add import Add from sympy.core.relational import Eq, Ne from sympy.core.sympify import sympify from sympy.core.singleton import S from sympy.core.numbers import Rational, oo from sympy.matrices.dense import Matrix class UnhandledInput(Exception): """ Raised while creating an LRASolver if non-linearity or non-rational numbers are present. """ # predicates that LRASolver understands and makes use of ALLOWED_PRED = {Q.eq, Q.gt, Q.lt, Q.le, Q.ge} # if true ~Q.gt(x, y) implies Q.le(x, y) HANDLE_NEGATION = True class LRASolver(): """ Linear Arithmetic Solver for DPLL(T) implemented with an algorithm based on the Dual Simplex method. Uses Bland's pivoting rule to avoid cycling. References ========== .. [1] Dutertre, B., de Moura, L.: A Fast Linear-Arithmetic Solver for DPLL(T) https://link.springer.com/chapter/10.1007/11817963_11 """ def __init__(self, A, slack_variables, nonslack_variables, enc_to_boundary, s_subs, testing_mode): """ Use the "from_encoded_cnf" method to create a new LRASolver. """ self.run_checks = testing_mode self.s_subs = s_subs # used only for test_lra_theory.test_random_problems if any(not isinstance(a, Rational) for a in A): raise UnhandledInput("Non-rational numbers are not handled") if any(not isinstance(b.bound, Rational) for b in enc_to_boundary.values()): raise UnhandledInput("Non-rational numbers are not handled") m, n = len(slack_variables), len(slack_variables)+len(nonslack_variables) if m != 0: assert A.shape == (m, n) if self.run_checks: assert A[:, n-m:] == -eye(m) self.enc_to_boundary = enc_to_boundary # mapping of int to Boundry objects self.boundary_to_enc = {value: key for key, value in enc_to_boundary.items()} self.A = A self.slack = slack_variables self.nonslack = nonslack_variables self.all_var = nonslack_variables + slack_variables self.slack_set = set(slack_variables) self.is_sat = True # While True, all constraints asserted so far are satisfiable self.result = None # always one of: (True, assignment), (False, conflict clause), None @staticmethod def from_encoded_cnf(encoded_cnf, testing_mode=False): """ Creates an LRASolver from an EncodedCNF object and a list of conflict clauses for propositions that can be simplified to True or False. Parameters ========== encoded_cnf : EncodedCNF testing_mode : bool Setting testing_mode to True enables some slow assert statements and sorting to reduce nonterministic behavior. Returns ======= (lra, conflicts) lra : LRASolver conflicts : list Contains a one-literal conflict clause for each proposition that can be simplified to True or False. Example ======= >>> from sympy.core.relational import Eq >>> from sympy.assumptions.cnf import CNF, EncodedCNF >>> from sympy.assumptions.ask import Q >>> from sympy.logic.algorithms.lra_theory import LRASolver >>> from sympy.abc import x, y, z >>> phi = (x >= 0) & ((x + y <= 2) | (x + 2 * y - z >= 6)) >>> phi = phi & (Eq(x + y, 2) | (x + 2 * y - z > 4)) >>> phi = phi & Q.gt(2, 1) >>> cnf = CNF.from_prop(phi) >>> enc = EncodedCNF() >>> enc.from_cnf(cnf) >>> lra, conflicts = LRASolver.from_encoded_cnf(enc, testing_mode=True) >>> lra #doctest: +SKIP >>> conflicts #doctest: +SKIP [[4]] """ # This function has three main jobs: # - raise errors if the input formula is not handled # - preprocesses the formula into a matirx and single variable constraints # - create one-literal conflict clauses from predicates that are always True # or always False such as Q.gt(3, 2) # # See the preprocessing section of "A Fast Linear-Arithmetic Solver for DPLL(T)" # for an explanation of how the formula is converted into a matrix # and a set of single variable constraints. encoding = {} # maps int to boundary A = [] basic = [] s_count = 0 s_subs = {} nonbasic = [] if testing_mode: # sort to reduce nondeterminism encoded_cnf_items = sorted(encoded_cnf.encoding.items(), key=lambda x: str(x)) else: encoded_cnf_items = encoded_cnf.encoding.items() empty_var = Dummy() var_to_lra_var = {} conflicts = [] for prop, enc in encoded_cnf_items: if isinstance(prop, Predicate): prop = prop(empty_var) if not isinstance(prop, AppliedPredicate): if prop == True: conflicts.append([enc]) continue if prop == False: conflicts.append([-enc]) continue raise ValueError(f"Unhandled Predicate: {prop}") assert prop.function in ALLOWED_PRED if prop.lhs == S.NaN or prop.rhs == S.NaN: raise ValueError(f"{prop} contains nan") if prop.lhs.is_imaginary or prop.rhs.is_imaginary: raise UnhandledInput(f"{prop} contains an imaginary component") if prop.lhs == oo or prop.rhs == oo: raise UnhandledInput(f"{prop} contains infinity") prop = _eval_binrel(prop) # simplify variable-less quantities to True / False if possible if prop == True: conflicts.append([enc]) continue elif prop == False: conflicts.append([-enc]) continue elif prop is None: raise UnhandledInput(f"{prop} could not be simplified") expr = prop.lhs - prop.rhs if prop.function in [Q.ge, Q.gt]: expr = -expr # expr should be less than (or equal to) 0 # otherwise prop is False if prop.function in [Q.le, Q.ge]: bool = (expr <= 0) elif prop.function in [Q.lt, Q.gt]: bool = (expr < 0) else: assert prop.function == Q.eq bool = Eq(expr, 0) if bool == True: conflicts.append([enc]) continue elif bool == False: conflicts.append([-enc]) continue vars, const = _sep_const_terms(expr) # example: (2x + 3y + 2) --> (2x + 3y), (2) vars, var_coeff = _sep_const_coeff(vars) # examples: (2x) --> (x, 2); (2x + 3y) --> (2x + 3y), (1) const = const / var_coeff terms = _list_terms(vars) # example: (2x + 3y) --> [2x, 3y] for term in terms: term, _ = _sep_const_coeff(term) assert len(term.free_symbols) > 0 if term not in var_to_lra_var: var_to_lra_var[term] = LRAVariable(term) nonbasic.append(term) if len(terms) > 1: if vars not in s_subs: s_count += 1 d = Dummy(f"s{s_count}") var_to_lra_var[d] = LRAVariable(d) basic.append(d) s_subs[vars] = d A.append(vars - d) var = s_subs[vars] else: var = terms[0] assert var_coeff != 0 equality = prop.function == Q.eq upper = var_coeff > 0 if not equality else None strict = prop.function in [Q.gt, Q.lt] b = Boundary(var_to_lra_var[var], -const, upper, equality, strict) encoding[enc] = b fs = [v.free_symbols for v in nonbasic + basic] assert all(len(syms) > 0 for syms in fs) fs_count = sum(len(syms) for syms in fs) if len(fs) > 0 and len(set.union(*fs)) < fs_count: raise UnhandledInput("Nonlinearity is not handled") A, _ = linear_eq_to_matrix(A, nonbasic + basic) nonbasic = [var_to_lra_var[nb] for nb in nonbasic] basic = [var_to_lra_var[b] for b in basic] for idx, var in enumerate(nonbasic + basic): var.col_idx = idx return LRASolver(A, basic, nonbasic, encoding, s_subs, testing_mode), conflicts def reset_bounds(self): """ Resets the state of the LRASolver to before anything was asserted. """ self.result = None for var in self.all_var: var.lower = LRARational(-float("inf"), 0) var.lower_from_eq = False var.lower_from_neg = False var.upper = LRARational(float("inf"), 0) var.upper_from_eq= False var.lower_from_neg = False var.assign = LRARational(0, 0) def assert_lit(self, enc_constraint): """ Assert a literal representing a constraint and update the internal state accordingly. Note that due to peculiarities of this implementation asserting ~(x > 0) will assert (x <= 0) but asserting ~Eq(x, 0) will not do anything. Parameters ========== enc_constraint : int A mapping of encodings to constraints can be found in `self.enc_to_boundary`. Returns ======= None or (False, explanation) explanation : set of ints A conflict clause that "explains" why the literals asserted so far are unsatisfiable. """ if abs(enc_constraint) not in self.enc_to_boundary: return None if not HANDLE_NEGATION and enc_constraint < 0: return None boundary = self.enc_to_boundary[abs(enc_constraint)] sym, c, negated = boundary.var, boundary.bound, enc_constraint < 0 if boundary.equality and negated: return None # negated equality is not handled and should only appear in conflict clauses upper = boundary.upper != negated if boundary.strict != negated: delta = -1 if upper else 1 c = LRARational(c, delta) else: c = LRARational(c, 0) if boundary.equality: res1 = self._assert_lower(sym, c, from_equality=True, from_neg=negated) if res1 and res1[0] == False: res = res1 else: res2 = self._assert_upper(sym, c, from_equality=True, from_neg=negated) res = res2 elif upper: res = self._assert_upper(sym, c, from_neg=negated) else: res = self._assert_lower(sym, c, from_neg=negated) if self.is_sat and sym not in self.slack_set: self.is_sat = res is None else: self.is_sat = False return res def _assert_upper(self, xi, ci, from_equality=False, from_neg=False): """ Adjusts the upper bound on variable xi if the new upper bound is more limiting. The assignment of variable xi is adjusted to be within the new bound if needed. Also calls `self._update` to update the assignment for slack variables to keep all equalities satisfied. """ if self.result: assert self.result[0] != False self.result = None if ci >= xi.upper: return None if ci < xi.lower: assert (xi.lower[1] >= 0) is True assert (ci[1] <= 0) is True lit1, neg1 = Boundary.from_lower(xi) lit2 = Boundary(var=xi, const=ci[0], strict=ci[1] != 0, upper=True, equality=from_equality) if from_neg: lit2 = lit2.get_negated() neg2 = -1 if from_neg else 1 conflict = [-neg1*self.boundary_to_enc[lit1], -neg2*self.boundary_to_enc[lit2]] self.result = False, conflict return self.result xi.upper = ci xi.upper_from_eq = from_equality xi.upper_from_neg = from_neg if xi in self.nonslack and xi.assign > ci: self._update(xi, ci) if self.run_checks and all(v.assign[0] != float("inf") and v.assign[0] != -float("inf") for v in self.all_var): M = self.A X = Matrix([v.assign[0] for v in self.all_var]) assert all(abs(val) < 10 ** (-10) for val in M * X) return None def _assert_lower(self, xi, ci, from_equality=False, from_neg=False): """ Adjusts the lower bound on variable xi if the new lower bound is more limiting. The assignment of variable xi is adjusted to be within the new bound if needed. Also calls `self._update` to update the assignment for slack variables to keep all equalities satisfied. """ if self.result: assert self.result[0] != False self.result = None if ci <= xi.lower: return None if ci > xi.upper: assert (xi.upper[1] <= 0) is True assert (ci[1] >= 0) is True lit1, neg1 = Boundary.from_upper(xi) lit2 = Boundary(var=xi, const=ci[0], strict=ci[1] != 0, upper=False, equality=from_equality) if from_neg: lit2 = lit2.get_negated() neg2 = -1 if from_neg else 1 conflict = [-neg1*self.boundary_to_enc[lit1],-neg2*self.boundary_to_enc[lit2]] self.result = False, conflict return self.result xi.lower = ci xi.lower_from_eq = from_equality xi.lower_from_neg = from_neg if xi in self.nonslack and xi.assign < ci: self._update(xi, ci) if self.run_checks and all(v.assign[0] != float("inf") and v.assign[0] != -float("inf") for v in self.all_var): M = self.A X = Matrix([v.assign[0] for v in self.all_var]) assert all(abs(val) < 10 ** (-10) for val in M * X) return None def _update(self, xi, v): """ Updates all slack variables that have equations that contain variable xi so that they stay satisfied given xi is equal to v. """ i = xi.col_idx for j, b in enumerate(self.slack): aji = self.A[j, i] b.assign = b.assign + (v - xi.assign)*aji xi.assign = v def check(self): """ Searches for an assignment that satisfies all constraints or determines that no such assignment exists and gives a minimal conflict clause that "explains" why the constraints are unsatisfiable. Returns ======= (True, assignment) or (False, explanation) assignment : dict of LRAVariables to values Assigned values are tuples that represent a rational number plus some infinatesimal delta. explanation : set of ints """ if self.is_sat: return True, {var: var.assign for var in self.all_var} if self.result: return self.result from sympy.matrices.dense import Matrix M = self.A.copy() basic = {s: i for i, s in enumerate(self.slack)} # contains the row index associated with each basic variable nonbasic = set(self.nonslack) iteration = 0 while True: iteration += 1 if self.run_checks: # nonbasic variables must always be within bounds assert all(((nb.assign >= nb.lower) == True) and ((nb.assign <= nb.upper) == True) for nb in nonbasic) # assignments for x must always satisfy Ax = 0 # probably have to turn this off when dealing with strict ineq if all(v.assign[0] != float("inf") and v.assign[0] != -float("inf") for v in self.all_var): X = Matrix([v.assign[0] for v in self.all_var]) assert all(abs(val) < 10**(-10) for val in M*X) # check upper and lower match this format: # x <= rat + delta iff x < rat # x >= rat - delta iff x > rat # this wouldn't make sense: # x <= rat - delta # x >= rat + delta assert all(x.upper[1] <= 0 for x in self.all_var) assert all(x.lower[1] >= 0 for x in self.all_var) cand = [b for b in basic if b.assign < b.lower or b.assign > b.upper] if len(cand) == 0: return True, {var: var.assign for var in self.all_var} xi = min(cand, key=lambda v: v.col_idx) # Bland's rule i = basic[xi] if xi.assign < xi.lower: cand = [nb for nb in nonbasic if (M[i, nb.col_idx] > 0 and nb.assign < nb.upper) or (M[i, nb.col_idx] < 0 and nb.assign > nb.lower)] if len(cand) == 0: N_plus = [nb for nb in nonbasic if M[i, nb.col_idx] > 0] N_minus = [nb for nb in nonbasic if M[i, nb.col_idx] < 0] conflict = [] conflict += [Boundary.from_upper(nb) for nb in N_plus] conflict += [Boundary.from_lower(nb) for nb in N_minus] conflict.append(Boundary.from_lower(xi)) conflict = [-neg*self.boundary_to_enc[c] for c, neg in conflict] return False, conflict xj = min(cand, key=str) M = self._pivot_and_update(M, basic, nonbasic, xi, xj, xi.lower) if xi.assign > xi.upper: cand = [nb for nb in nonbasic if (M[i, nb.col_idx] < 0 and nb.assign < nb.upper) or (M[i, nb.col_idx] > 0 and nb.assign > nb.lower)] if len(cand) == 0: N_plus = [nb for nb in nonbasic if M[i, nb.col_idx] > 0] N_minus = [nb for nb in nonbasic if M[i, nb.col_idx] < 0] conflict = [] conflict += [Boundary.from_upper(nb) for nb in N_minus] conflict += [Boundary.from_lower(nb) for nb in N_plus] conflict.append(Boundary.from_upper(xi)) conflict = [-neg*self.boundary_to_enc[c] for c, neg in conflict] return False, conflict xj = min(cand, key=lambda v: v.col_idx) M = self._pivot_and_update(M, basic, nonbasic, xi, xj, xi.upper) def _pivot_and_update(self, M, basic, nonbasic, xi, xj, v): """ Pivots basic variable xi with nonbasic variable xj, and sets value of xi to v and adjusts the values of all basic variables to keep equations satisfied. """ i, j = basic[xi], xj.col_idx assert M[i, j] != 0 theta = (v - xi.assign)*(1/M[i, j]) xi.assign = v xj.assign = xj.assign + theta for xk in basic: if xk != xi: k = basic[xk] akj = M[k, j] xk.assign = xk.assign + theta*akj # pivot basic[xj] = basic[xi] del basic[xi] nonbasic.add(xi) nonbasic.remove(xj) return self._pivot(M, i, j) @staticmethod def _pivot(M, i, j): """ Performs a pivot operation about entry i, j of M by performing a series of row operations on a copy of M and returing the result. The original M is left unmodified. Conceptually, M represents a system of equations and pivoting can be thought of as rearranging equation i to be in terms of variable j and then substituting in the rest of the equations to get rid of other occurances of variable j. Example ======= >>> from sympy.matrices.dense import Matrix >>> from sympy.logic.algorithms.lra_theory import LRASolver >>> from sympy import var >>> Matrix(3, 3, var('a:i')) Matrix([ [a, b, c], [d, e, f], [g, h, i]]) This matrix is equivalent to: 0 = a*x + b*y + c*z 0 = d*x + e*y + f*z 0 = g*x + h*y + i*z >>> LRASolver._pivot(_, 1, 0) Matrix([ [ 0, -a*e/d + b, -a*f/d + c], [-1, -e/d, -f/d], [ 0, h - e*g/d, i - f*g/d]]) We rearrange equation 1 in terms of variable 0 (x) and substitute to remove x from the other equations. 0 = 0 + (-a*e/d + b)*y + (-a*f/d + c)*z 0 = -x + (-e/d)*y + (-f/d)*z 0 = 0 + (h - e*g/d)*y + (i - f*g/d)*z """ _, _, Mij = M[i, :], M[:, j], M[i, j] if Mij == 0: raise ZeroDivisionError("Tried to pivot about zero-valued entry.") A = M.copy() A[i, :] = -A[i, :]/Mij for row in range(M.shape[0]): if row != i: A[row, :] = A[row, :] + A[row, j] * A[i, :] return A def _sep_const_coeff(expr): """ Example ======= >>> from sympy.logic.algorithms.lra_theory import _sep_const_coeff >>> from sympy.abc import x, y >>> _sep_const_coeff(2*x) (x, 2) >>> _sep_const_coeff(2*x + 3*y) (2*x + 3*y, 1) """ if isinstance(expr, Add): return expr, sympify(1) if isinstance(expr, Mul): coeffs = expr.args else: coeffs = [expr] var, const = [], [] for c in coeffs: c = sympify(c) if len(c.free_symbols)==0: const.append(c) else: var.append(c) return Mul(*var), Mul(*const) def _list_terms(expr): if not isinstance(expr, Add): return [expr] return expr.args def _sep_const_terms(expr): """ Example ======= >>> from sympy.logic.algorithms.lra_theory import _sep_const_terms >>> from sympy.abc import x, y >>> _sep_const_terms(2*x + 3*y + 2) (2*x + 3*y, 2) """ if isinstance(expr, Add): terms = expr.args else: terms = [expr] var, const = [], [] for t in terms: if len(t.free_symbols) == 0: const.append(t) else: var.append(t) return sum(var), sum(const) def _eval_binrel(binrel): """ Simplify binary relation to True / False if possible. """ if not (len(binrel.lhs.free_symbols) == 0 and len(binrel.rhs.free_symbols) == 0): return binrel if binrel.function == Q.lt: res = binrel.lhs < binrel.rhs elif binrel.function == Q.gt: res = binrel.lhs > binrel.rhs elif binrel.function == Q.le: res = binrel.lhs <= binrel.rhs elif binrel.function == Q.ge: res = binrel.lhs >= binrel.rhs elif binrel.function == Q.eq: res = Eq(binrel.lhs, binrel.rhs) elif binrel.function == Q.ne: res = Ne(binrel.lhs, binrel.rhs) if res == True or res == False: return res else: return None class Boundary: """ Represents an upper or lower bound or an equality between a symbol and some constant. """ def __init__(self, var, const, upper, equality, strict=None): if not equality in [True, False]: assert equality in [True, False] self.var = var if isinstance(const, tuple): s = const[1] != 0 if strict: assert s == strict self.bound = const[0] self.strict = s else: self.bound = const self.strict = strict self.upper = upper if not equality else None self.equality = equality self.strict = strict assert self.strict is not None @staticmethod def from_upper(var): neg = -1 if var.upper_from_neg else 1 b = Boundary(var, var.upper[0], True, var.upper_from_eq, var.upper[1] != 0) if neg < 0: b = b.get_negated() return b, neg @staticmethod def from_lower(var): neg = -1 if var.lower_from_neg else 1 b = Boundary(var, var.lower[0], False, var.lower_from_eq, var.lower[1] != 0) if neg < 0: b = b.get_negated() return b, neg def get_negated(self): return Boundary(self.var, self.bound, not self.upper, self.equality, not self.strict) def get_inequality(self): if self.equality: return Eq(self.var.var, self.bound) elif self.upper and self.strict: return self.var.var < self.bound elif not self.upper and self.strict: return self.var.var > self.bound elif self.upper: return self.var.var <= self.bound else: return self.var.var >= self.bound def __repr__(self): return repr("Boundry(" + repr(self.get_inequality()) + ")") def __eq__(self, other): other = (other.var, other.bound, other.strict, other.upper, other.equality) return (self.var, self.bound, self.strict, self.upper, self.equality) == other def __hash__(self): return hash((self.var, self.bound, self.strict, self.upper, self.equality)) class LRARational(): """ Represents a rational plus or minus some amount of arbitrary small deltas. """ def __init__(self, rational, delta): self.value = (rational, delta) def __lt__(self, other): return self.value < other.value def __le__(self, other): return self.value <= other.value def __eq__(self, other): return self.value == other.value def __add__(self, other): return LRARational(self.value[0] + other.value[0], self.value[1] + other.value[1]) def __sub__(self, other): return LRARational(self.value[0] - other.value[0], self.value[1] - other.value[1]) def __mul__(self, other): assert not isinstance(other, LRARational) return LRARational(self.value[0] * other, self.value[1] * other) def __getitem__(self, index): return self.value[index] def __repr__(self): return repr(self.value) class LRAVariable(): """ Object to keep track of upper and lower bounds on `self.var`. """ def __init__(self, var): self.upper = LRARational(float("inf"), 0) self.upper_from_eq = False self.upper_from_neg = False self.lower = LRARational(-float("inf"), 0) self.lower_from_eq = False self.lower_from_neg = False self.assign = LRARational(0,0) self.var = var self.col_idx = None def __repr__(self): return repr(self.var) def __eq__(self, other): if not isinstance(other, LRAVariable): return False return other.var == self.var def __hash__(self): return hash(self.var)