//===-- FastCexSolver.cpp -------------------------------------------------===// // // The KLEE Symbolic Virtual Machine // // This file is distributed under the University of Illinois Open Source // License. See LICENSE.TXT for details. // //===----------------------------------------------------------------------===// #include "klee/Solver.h" #include "klee/Constraints.h" #include "klee/Expr.h" #include "klee/IncompleteSolver.h" #include "klee/util/ExprEvaluator.h" #include "klee/util/ExprRangeEvaluator.h" #include "klee/util/ExprVisitor.h" // FIXME: Use APInt. #include "klee/Internal/Support/IntEvaluation.h" #include #include #include #include #include using namespace klee; /***/ // Hacker's Delight, pgs 58-63 static uint64_t minOR(uint64_t a, uint64_t b, uint64_t c, uint64_t d) { uint64_t temp, m = ((uint64_t) 1)<<63; while (m) { if (~a & c & m) { temp = (a | m) & -m; if (temp <= b) { a = temp; break; } } else if (a & ~c & m) { temp = (c | m) & -m; if (temp <= d) { c = temp; break; } } m >>= 1; } return a | c; } static uint64_t maxOR(uint64_t a, uint64_t b, uint64_t c, uint64_t d) { uint64_t temp, m = ((uint64_t) 1)<<63; while (m) { if (b & d & m) { temp = (b - m) | (m - 1); if (temp >= a) { b = temp; break; } temp = (d - m) | (m -1); if (temp >= c) { d = temp; break; } } m >>= 1; } return b | d; } static uint64_t minAND(uint64_t a, uint64_t b, uint64_t c, uint64_t d) { uint64_t temp, m = ((uint64_t) 1)<<63; while (m) { if (~a & ~c & m) { temp = (a | m) & -m; if (temp <= b) { a = temp; break; } temp = (c | m) & -m; if (temp <= d) { c = temp; break; } } m >>= 1; } return a & c; } static uint64_t maxAND(uint64_t a, uint64_t b, uint64_t c, uint64_t d) { uint64_t temp, m = ((uint64_t) 1)<<63; while (m) { if (b & ~d & m) { temp = (b & ~m) | (m - 1); if (temp >= a) { b = temp; break; } } else if (~b & d & m) { temp = (d & ~m) | (m - 1); if (temp >= c) { d = temp; break; } } m >>= 1; } return b & d; } /// class ValueRange { private: uint64_t m_min, m_max; public: ValueRange() : m_min(1),m_max(0) {} ValueRange(uint64_t value) : m_min(value), m_max(value) {} ValueRange(uint64_t _min, uint64_t _max) : m_min(_min), m_max(_max) {} ValueRange(const ValueRange &b) : m_min(b.m_min), m_max(b.m_max) {} void print(std::ostream &os) const { if (isFixed()) { os << m_min; } else { os << "[" << m_min << "," << m_max << "]"; } } bool isEmpty() const { return m_min>m_max; } bool contains(uint64_t value) const { return this->intersects(ValueRange(value)); } bool intersects(const ValueRange &b) const { return !this->set_intersection(b).isEmpty(); } bool isFullRange(unsigned bits) { return m_min==0 && m_max==bits64::maxValueOfNBits(bits); } ValueRange set_intersection(const ValueRange &b) const { return ValueRange(std::max(m_min,b.m_min), std::min(m_max,b.m_max)); } ValueRange set_union(const ValueRange &b) const { return ValueRange(std::min(m_min,b.m_min), std::max(m_max,b.m_max)); } ValueRange set_difference(const ValueRange &b) const { if (b.isEmpty() || b.m_min > m_max || b.m_max < m_min) { // no intersection return *this; } else if (b.m_min <= m_min && b.m_max >= m_max) { // empty return ValueRange(1,0); } else if (b.m_min <= m_min) { // one range out // cannot overflow because b.m_max < m_max return ValueRange(b.m_max+1, m_max); } else if (b.m_max >= m_max) { // cannot overflow because b.min > m_min return ValueRange(m_min, b.m_min-1); } else { // two ranges, take bottom return ValueRange(m_min, b.m_min-1); } } ValueRange binaryAnd(const ValueRange &b) const { // XXX assert(!isEmpty() && !b.isEmpty() && "XXX"); if (isFixed() && b.isFixed()) { return ValueRange(m_min & b.m_min); } else { return ValueRange(minAND(m_min, m_max, b.m_min, b.m_max), maxAND(m_min, m_max, b.m_min, b.m_max)); } } ValueRange binaryAnd(uint64_t b) const { return binaryAnd(ValueRange(b)); } ValueRange binaryOr(ValueRange b) const { // XXX assert(!isEmpty() && !b.isEmpty() && "XXX"); if (isFixed() && b.isFixed()) { return ValueRange(m_min | b.m_min); } else { return ValueRange(minOR(m_min, m_max, b.m_min, b.m_max), maxOR(m_min, m_max, b.m_min, b.m_max)); } } ValueRange binaryOr(uint64_t b) const { return binaryOr(ValueRange(b)); } ValueRange binaryXor(ValueRange b) const { if (isFixed() && b.isFixed()) { return ValueRange(m_min ^ b.m_min); } else { uint64_t t = m_max | b.m_max; while (!bits64::isPowerOfTwo(t)) t = bits64::withoutRightmostBit(t); return ValueRange(0, (t<<1)-1); } } ValueRange binaryShiftLeft(unsigned bits) const { return ValueRange(m_min<>bits, m_max>>bits); } ValueRange concat(const ValueRange &b, unsigned bits) const { return binaryShiftLeft(bits).binaryOr(b); } ValueRange extract(uint64_t lowBit, uint64_t maxBit) const { return binaryShiftRight(lowBit).binaryAnd(bits64::maxValueOfNBits(maxBit-lowBit)); } ValueRange add(const ValueRange &b, unsigned width) const { return ValueRange(0, bits64::maxValueOfNBits(width)); } ValueRange sub(const ValueRange &b, unsigned width) const { return ValueRange(0, bits64::maxValueOfNBits(width)); } ValueRange mul(const ValueRange &b, unsigned width) const { return ValueRange(0, bits64::maxValueOfNBits(width)); } ValueRange udiv(const ValueRange &b, unsigned width) const { return ValueRange(0, bits64::maxValueOfNBits(width)); } ValueRange sdiv(const ValueRange &b, unsigned width) const { return ValueRange(0, bits64::maxValueOfNBits(width)); } ValueRange urem(const ValueRange &b, unsigned width) const { return ValueRange(0, bits64::maxValueOfNBits(width)); } ValueRange srem(const ValueRange &b, unsigned width) const { return ValueRange(0, bits64::maxValueOfNBits(width)); } // use min() to get value if true (XXX should we add a method to // make code clearer?) bool isFixed() const { return m_min==m_max; } bool operator==(const ValueRange &b) const { return m_min==b.m_min && m_max==b.m_max; } bool operator!=(const ValueRange &b) const { return !(*this==b); } bool mustEqual(const uint64_t b) const { return m_min==m_max && m_min==b; } bool mayEqual(const uint64_t b) const { return m_min<=b && m_max>=b; } bool mustEqual(const ValueRange &b) const { return isFixed() && b.isFixed() && m_min==b.m_min; } bool mayEqual(const ValueRange &b) const { return this->intersects(b); } uint64_t min() const { assert(!isEmpty() && "cannot get minimum of empty range"); return m_min; } uint64_t max() const { assert(!isEmpty() && "cannot get maximum of empty range"); return m_max; } int64_t minSigned(unsigned bits) const { assert((m_min>>bits)==0 && (m_max>>bits)==0 && "range is outside given number of bits"); // if max allows sign bit to be set then it can be smallest value, // otherwise since the range is not empty, min cannot have a sign // bit uint64_t smallest = ((uint64_t) 1 << (bits-1)); if (m_max >= smallest) { return ints::sext(smallest, 64, bits); } else { return m_min; } } int64_t maxSigned(unsigned bits) const { assert((m_min>>bits)==0 && (m_max>>bits)==0 && "range is outside given number of bits"); uint64_t smallest = ((uint64_t) 1 << (bits-1)); // if max and min have sign bit then max is max, otherwise if only // max has sign bit then max is largest signed integer, otherwise // max is max if (m_min < smallest && m_max >= smallest) { return smallest - 1; } else { return ints::sext(m_max, 64, bits); } } }; inline std::ostream &operator<<(std::ostream &os, const ValueRange &vr) { vr.print(os); return os; } // XXX waste of space, rather have ByteValueRange typedef ValueRange CexValueData; class CexObjectData { /// possibleContents - An array of "possible" values for the object. /// /// The possible values is an inexact approximation for the set of values for /// each array location. std::vector possibleContents; /// exactContents - An array of exact values for the object. /// /// The exact values are a conservative approximation for the set of values /// for each array location. std::vector exactContents; CexObjectData(const CexObjectData&); // DO NOT IMPLEMENT void operator=(const CexObjectData&); // DO NOT IMPLEMENT public: CexObjectData(uint64_t size) : possibleContents(size), exactContents(size) { for (uint64_t i = 0; i != size; ++i) { possibleContents[i] = ValueRange(0, 255); exactContents[i] = ValueRange(0, 255); } } const CexValueData getPossibleValues(size_t index) const { return possibleContents[index]; } void setPossibleValues(size_t index, CexValueData values) { possibleContents[index] = values; } void setPossibleValue(size_t index, unsigned char value) { possibleContents[index] = CexValueData(value); } const CexValueData getExactValues(size_t index) const { return exactContents[index]; } void setExactValues(size_t index, CexValueData values) { exactContents[index] = values; } /// getPossibleValue - Return some possible value. unsigned char getPossibleValue(size_t index) const { const CexValueData &cvd = possibleContents[index]; return cvd.min() + (cvd.max() - cvd.min()) / 2; } }; class CexRangeEvaluator : public ExprRangeEvaluator { public: std::map &objects; CexRangeEvaluator(std::map &_objects) : objects(_objects) {} ValueRange getInitialReadRange(const Array &os, ValueRange index) { return ValueRange(0, 255); } }; class CexPossibleEvaluator : public ExprEvaluator { protected: ref getInitialValue(const Array& array, unsigned index) { // If the index is out of range, we cannot assign it a value, since that // value cannot be part of the assignment. if (index >= array.size) return ReadExpr::create(UpdateList(&array, true, 0), ConstantExpr::alloc(index, Expr::Int32)); std::map::iterator it = objects.find(array.id); return ConstantExpr::alloc((it == objects.end() ? 127 : it->second->getPossibleValue(index)), Expr::Int8); } public: std::map &objects; CexPossibleEvaluator(std::map &_objects) : objects(_objects) {} }; class CexExactEvaluator : public ExprEvaluator { protected: ref getInitialValue(const Array& array, unsigned index) { // If the index is out of range, we cannot assign it a value, since that // value cannot be part of the assignment. if (index >= array.size) return ReadExpr::create(UpdateList(&array, true, 0), ConstantExpr::alloc(index, Expr::Int32)); std::map::iterator it = objects.find(array.id); if (it == objects.end()) return ReadExpr::create(UpdateList(&array, true, 0), ConstantExpr::alloc(index, Expr::Int32)); CexValueData cvd = it->second->getExactValues(index); if (!cvd.isFixed()) return ReadExpr::create(UpdateList(&array, true, 0), ConstantExpr::alloc(index, Expr::Int32)); return ConstantExpr::alloc(cvd.min(), Expr::Int8); } public: std::map &objects; CexExactEvaluator(std::map &_objects) : objects(_objects) {} }; class CexData { public: std::map objects; CexData(const CexData&); // DO NOT IMPLEMENT void operator=(const CexData&); // DO NOT IMPLEMENT public: CexData() {} ~CexData() { for (std::map::iterator it = objects.begin(), ie = objects.end(); it != ie; ++it) delete it->second; } CexObjectData &getObjectData(const Array *A) { CexObjectData *&Entry = objects[A->id]; if (!Entry) Entry = new CexObjectData(A->size); return *Entry; } void propogatePossibleValue(ref e, uint64_t value) { propogatePossibleValues(e, CexValueData(value,value)); } void propogateExactValue(ref e, uint64_t value) { propogateExactValues(e, CexValueData(value,value)); } void propogatePossibleValues(ref e, CexValueData range) { switch (e->getKind()) { case Expr::Constant: { // rather a pity if the constant isn't in the range, but how can // we use this? break; } // Special case Expr::NotOptimized: break; case Expr::Read: { ReadExpr *re = cast(e); const Array *array = re->updates.root; CexObjectData &cod = getObjectData(array); // FIXME: This is imprecise, we need to look through the existing writes // to see if this is an initial read or not. if (ConstantExpr *CE = dyn_cast(re->index)) { if (CE->getConstantValue() < array->size) { // If the range is fixed, just set that; even if it conflicts with the // previous range it should be a better guess. if (range.isFixed()) { cod.setPossibleValue(CE->getConstantValue(), range.min()); } else { CexValueData cvd = cod.getPossibleValues(CE->getConstantValue()); CexValueData tmp = cvd.set_intersection(range); if (!tmp.isEmpty()) cod.setPossibleValues(CE->getConstantValue(), tmp); } } } else { // XXX fatal("XXX not implemented"); } break; } case Expr::Select: { SelectExpr *se = cast(e); ValueRange cond = evalRangeForExpr(se->cond); if (cond.isFixed()) { if (cond.min()) { propogatePossibleValues(se->trueExpr, range); } else { propogatePossibleValues(se->falseExpr, range); } } else { // XXX imprecise... we have a choice here. One method is to // simply force both sides into the specified range (since the // condition is indetermined). This may lose in two ways, the // first is that the condition chosen may limit further // restrict the range in each of the children, however this is // less of a problem as the range will be a superset of legal // values. The other is if the condition ends up being forced // by some other constraints, then we needlessly forced one // side into the given range. // // The other method would be to force the condition to one // side and force that side into the given range. This loses // when we force the condition to an unsatisfiable value // (either because the condition cannot be that, or the // resulting range given that condition is not in the required // range). // // Currently we just force both into the range. A hybrid would // be to evaluate the ranges for each of the children... if // one of the ranges happens to already be a subset of the // required range then it may be preferable to force the // condition to that side. propogatePossibleValues(se->trueExpr, range); propogatePossibleValues(se->falseExpr, range); } break; } // XXX imprecise... the problem here is that extracting bits // loses information about what bits are connected across the // bytes. if a value can be 1 or 256 then either the top or // lower byte is 0, but just extraction loses this information // and will allow neither,one,or both to be 1. // // we can protect against this in a limited fashion by writing // the extraction a byte at a time, then checking the evaluated // value, isolating for that range, and continuing. case Expr::Concat: { ConcatExpr *ce = cast(e); Expr::Width LSBWidth = ce->getKid(1)->getWidth(); Expr::Width MSBWidth = ce->getKid(1)->getWidth(); propogatePossibleValues(ce->getKid(0), range.extract(LSBWidth, MSBWidth)); propogatePossibleValues(ce->getKid(1), range.extract(0, LSBWidth)); break; } case Expr::Extract: { // XXX break; } // Casting // Simply intersect the output range with the range of all possible // outputs and then truncate to the desired number of bits. // For ZExt this simplifies to just intersection with the possible input // range. case Expr::ZExt: { CastExpr *ce = cast(e); unsigned inBits = ce->src->getWidth(); ValueRange input = range.set_intersection(ValueRange(0, bits64::maxValueOfNBits(inBits))); propogatePossibleValues(ce->src, input); break; } // For SExt instead of doing the intersection we just take the output // range minus the impossible values. This is nicer since it is a single // interval. case Expr::SExt: { CastExpr *ce = cast(e); unsigned inBits = ce->src->getWidth(); unsigned outBits = ce->width; ValueRange output = range.set_difference(ValueRange(1<<(inBits-1), (bits64::maxValueOfNBits(outBits) - bits64::maxValueOfNBits(inBits-1)-1))); ValueRange input = output.binaryAnd(bits64::maxValueOfNBits(inBits)); propogatePossibleValues(ce->src, input); break; } // Binary case Expr::And: { BinaryExpr *be = cast(e); if (be->getWidth()==Expr::Bool) { if (range.isFixed()) { ValueRange left = evalRangeForExpr(be->left); ValueRange right = evalRangeForExpr(be->right); if (!range.min()) { if (left.mustEqual(0) || right.mustEqual(0)) { // all is well } else { // XXX heuristic, which order propogatePossibleValue(be->left, 0); left = evalRangeForExpr(be->left); // see if that worked if (!left.mustEqual(1)) propogatePossibleValue(be->right, 0); } } else { if (!left.mustEqual(1)) propogatePossibleValue(be->left, 1); if (!right.mustEqual(1)) propogatePossibleValue(be->right, 1); } } } else { // XXX } break; } case Expr::Or: { BinaryExpr *be = cast(e); if (be->getWidth()==Expr::Bool) { if (range.isFixed()) { ValueRange left = evalRangeForExpr(be->left); ValueRange right = evalRangeForExpr(be->right); if (range.min()) { if (left.mustEqual(1) || right.mustEqual(1)) { // all is well } else { // XXX heuristic, which order? // force left to value we need propogatePossibleValue(be->left, 1); left = evalRangeForExpr(be->left); // see if that worked if (!left.mustEqual(1)) propogatePossibleValue(be->right, 1); } } else { if (!left.mustEqual(0)) propogatePossibleValue(be->left, 0); if (!right.mustEqual(0)) propogatePossibleValue(be->right, 0); } } } else { // XXX } break; } case Expr::Xor: break; // Comparison case Expr::Eq: { BinaryExpr *be = cast(e); if (range.isFixed()) { if (ConstantExpr *CE = dyn_cast(be->left)) { uint64_t value = CE->getConstantValue(); if (range.min()) { propogatePossibleValue(be->right, value); } else { if (value==0) { propogatePossibleValues(be->right, CexValueData(1, ints::sext(1, be->right->getWidth(), 1))); } else { // XXX heuristic / lossy, could be better to pick larger range? propogatePossibleValues(be->right, CexValueData(0, value-1)); } } } else { // XXX what now } } break; } case Expr::Ult: { BinaryExpr *be = cast(e); // XXX heuristic / lossy, what order if conflict if (range.isFixed()) { ValueRange left = evalRangeForExpr(be->left); ValueRange right = evalRangeForExpr(be->right); uint64_t maxValue = bits64::maxValueOfNBits(be->right->getWidth()); // XXX should deal with overflow (can lead to empty range) if (left.isFixed()) { if (range.min()) { propogatePossibleValues(be->right, CexValueData(left.min()+1, maxValue)); } else { propogatePossibleValues(be->right, CexValueData(0, left.min())); } } else if (right.isFixed()) { if (range.min()) { propogatePossibleValues(be->left, CexValueData(0, right.min()-1)); } else { propogatePossibleValues(be->left, CexValueData(right.min(), maxValue)); } } else { // XXX ??? } } break; } case Expr::Ule: { BinaryExpr *be = cast(e); // XXX heuristic / lossy, what order if conflict if (range.isFixed()) { ValueRange left = evalRangeForExpr(be->left); ValueRange right = evalRangeForExpr(be->right); // XXX should deal with overflow (can lead to empty range) uint64_t maxValue = bits64::maxValueOfNBits(be->right->getWidth()); if (left.isFixed()) { if (range.min()) { propogatePossibleValues(be->right, CexValueData(left.min(), maxValue)); } else { propogatePossibleValues(be->right, CexValueData(0, left.min()-1)); } } else if (right.isFixed()) { if (range.min()) { propogatePossibleValues(be->left, CexValueData(0, right.min())); } else { propogatePossibleValues(be->left, CexValueData(right.min()+1, maxValue)); } } else { // XXX ??? } } break; } case Expr::Ne: case Expr::Ugt: case Expr::Uge: case Expr::Sgt: case Expr::Sge: assert(0 && "invalid expressions (uncanonicalized"); default: break; } } void propogateExactValues(ref e, CexValueData range) { switch (e->getKind()) { case Expr::Constant: { // FIXME: Assert that range contains this constant. break; } // Special case Expr::NotOptimized: break; case Expr::Read: { break; } case Expr::Select: { break; } case Expr::Concat: { break; } case Expr::Extract: { break; } // Casting case Expr::ZExt: { break; } case Expr::SExt: { break; } // Binary case Expr::And: { break; } case Expr::Or: { break; } case Expr::Xor: { break; } // Comparison case Expr::Eq: { break; } case Expr::Ult: { break; } case Expr::Ule: { break; } case Expr::Ne: case Expr::Ugt: case Expr::Uge: case Expr::Sgt: case Expr::Sge: assert(0 && "invalid expressions (uncanonicalized"); default: break; } } ValueRange evalRangeForExpr(ref &e) { CexRangeEvaluator ce(objects); return ce.evaluate(e); } /// evaluate - Try to evaluate the given expression using a consistent fixed /// value for the current set of possible ranges. ref evaluatePossible(ref e) { return CexPossibleEvaluator(objects).visit(e); } ref evaluateExact(ref e) { return CexExactEvaluator(objects).visit(e); } }; /* *** */ class FastCexSolver : public IncompleteSolver { public: FastCexSolver(); ~FastCexSolver(); IncompleteSolver::PartialValidity computeTruth(const Query&); bool computeValue(const Query&, ref &result); bool computeInitialValues(const Query&, const std::vector &objects, std::vector< std::vector > &values, bool &hasSolution); }; FastCexSolver::FastCexSolver() { } FastCexSolver::~FastCexSolver() { } /// propogateValues - Propogate value ranges for the given query and return the /// propogation results. /// /// \param query - The query to propogate values for. /// /// \param cd - The initial object values resulting from the propogation. /// /// \param checkExpr - Include the query expression in the constraints to /// propogate. /// /// \param isValid - If the propogation succeeds (returns true), whether the /// constraints were proven valid or invalid. /// /// \return - True if the propogation was able to prove validity or invalidity. static bool propogateValues(const Query& query, CexData &cd, bool checkExpr, bool &isValid) { for (ConstraintManager::const_iterator it = query.constraints.begin(), ie = query.constraints.end(); it != ie; ++it) { cd.propogatePossibleValue(*it, 1); cd.propogateExactValue(*it, 1); } if (checkExpr) { cd.propogatePossibleValue(query.expr, 0); cd.propogateExactValue(query.expr, 0); } // Check the result. if (checkExpr) { if (!cd.evaluatePossible(query.expr)->isFalse()) return false; // If the query is known to be true, then we have proved validity. if (cd.evaluateExact(query.expr)->isTrue()) { isValid = true; return true; } } for (ConstraintManager::const_iterator it = query.constraints.begin(), ie = query.constraints.end(); it != ie; ++it) { if (!cd.evaluatePossible(*it)->isTrue()) return false; // If this constraint is known to be false, then we can prove anything, so // the query is valid. if (cd.evaluateExact(*it)->isFalse()) { isValid = true; return true; } } isValid = false; return true; } IncompleteSolver::PartialValidity FastCexSolver::computeTruth(const Query& query) { CexData cd; bool isValid; bool success = propogateValues(query, cd, true, isValid); if (!success) return IncompleteSolver::None; return isValid ? IncompleteSolver::MustBeTrue : IncompleteSolver::MayBeFalse; } bool FastCexSolver::computeValue(const Query& query, ref &result) { CexData cd; bool isValid; bool success = propogateValues(query, cd, false, isValid); // Check if propogation wasn't able to determine anything. if (!success) return false; // FIXME: We don't have a way to communicate valid constraints back. if (isValid) return false; // Propogation found a satisfying assignment, evaluate the expression. ref value = cd.evaluatePossible(query.expr); if (isa(value)) { // FIXME: We should be able to make sure this never fails? result = value; return true; } else { return false; } } bool FastCexSolver::computeInitialValues(const Query& query, const std::vector &objects, std::vector< std::vector > &values, bool &hasSolution) { CexData cd; bool isValid; bool success = propogateValues(query, cd, true, isValid); // Check if propogation wasn't able to determine anything. if (!success) return false; hasSolution = !isValid; if (!hasSolution) return true; // Propogation found a satisfying assignment, compute the initial values. for (unsigned i = 0; i != objects.size(); ++i) { const Array *array = objects[i]; std::vector data; data.reserve(array->size); for (unsigned i=0; i < array->size; i++) { ref read = ReadExpr::create(UpdateList(array, true, 0), ConstantExpr::create(i, kMachinePointerType)); ref value = cd.evaluatePossible(read); if (ConstantExpr *CE = dyn_cast(value)) { data.push_back((unsigned char) CE->getConstantValue()); } else { // FIXME: When does this happen? return false; } } values.push_back(data); } return true; } Solver *klee::createFastCexSolver(Solver *s) { return new Solver(new StagedSolverImpl(new FastCexSolver(), s)); }