From 3d486c4c98b2a6d07ca397f4fd2d2d5257d19e95 Mon Sep 17 00:00:00 2001 From: Nikolaj Bjorner Date: Tue, 2 Apr 2013 15:28:45 -0700 Subject: [PATCH] add abstraction and instantiation Signed-off-by: Nikolaj Bjorner --- src/ast/rewriter/expr_safe_replace.cpp | 6 + src/ast/rewriter/expr_safe_replace.h | 2 + src/muz_qe/dl_context.cpp | 4 + src/muz_qe/dl_mk_quantifier_abstraction.cpp | 206 ++++++++++++ src/muz_qe/dl_mk_quantifier_abstraction.h | 61 ++++ src/muz_qe/dl_mk_quantifier_instantiation.cpp | 295 ++++++++++++++++++ src/muz_qe/dl_mk_quantifier_instantiation.h | 120 +++++++ src/muz_qe/fixedpoint_params.pyg | 2 + 8 files changed, 696 insertions(+) create mode 100644 src/muz_qe/dl_mk_quantifier_abstraction.cpp create mode 100644 src/muz_qe/dl_mk_quantifier_abstraction.h create mode 100644 src/muz_qe/dl_mk_quantifier_instantiation.cpp create mode 100644 src/muz_qe/dl_mk_quantifier_instantiation.h diff --git a/src/ast/rewriter/expr_safe_replace.cpp b/src/ast/rewriter/expr_safe_replace.cpp index b3b4d5138..1ec810414 100644 --- a/src/ast/rewriter/expr_safe_replace.cpp +++ b/src/ast/rewriter/expr_safe_replace.cpp @@ -100,3 +100,9 @@ void expr_safe_replace::operator()(expr* e, expr_ref& res) { } res = cache.find(e); } + +void expr_safe_replace::reset() { + m_src.reset(); + m_dst.reset(); + m_subst.reset(); +} diff --git a/src/ast/rewriter/expr_safe_replace.h b/src/ast/rewriter/expr_safe_replace.h index 6af819596..b6131906a 100644 --- a/src/ast/rewriter/expr_safe_replace.h +++ b/src/ast/rewriter/expr_safe_replace.h @@ -38,6 +38,8 @@ public: void operator()(expr_ref& e) { (*this)(e.get(), e); } void operator()(expr* src, expr_ref& e); + + void reset(); }; #endif /* __EXPR_SAFE_REPLACE_H__ */ diff --git a/src/muz_qe/dl_context.cpp b/src/muz_qe/dl_context.cpp index c667c7775..9207fbe61 100644 --- a/src/muz_qe/dl_context.cpp +++ b/src/muz_qe/dl_context.cpp @@ -46,6 +46,7 @@ Revision History: #include"dl_mk_bit_blast.h" #include"dl_mk_array_blast.h" #include"dl_mk_karr_invariants.h" +#include"dl_mk_quantifier_abstraction.h" #include"datatype_decl_plugin.h" #include"expr_abstract.h" @@ -905,6 +906,9 @@ namespace datalog { m_transf.register_plugin(alloc(datalog::mk_rule_inliner, *this, 34890)); m_transf.register_plugin(alloc(datalog::mk_subsumption_checker, *this, 34880)); + + m_transf.register_plugin(alloc(datalog::mk_quantifier_abstraction, *this, 33000)); + m_transf.register_plugin(alloc(datalog::mk_bit_blast, *this, 35000)); m_transf.register_plugin(alloc(datalog::mk_array_blast, *this, 36000)); m_transf.register_plugin(alloc(datalog::mk_karr_invariants, *this, 36010)); diff --git a/src/muz_qe/dl_mk_quantifier_abstraction.cpp b/src/muz_qe/dl_mk_quantifier_abstraction.cpp new file mode 100644 index 000000000..511ab7b9a --- /dev/null +++ b/src/muz_qe/dl_mk_quantifier_abstraction.cpp @@ -0,0 +1,206 @@ +/*++ +Copyright (c) 2013 Microsoft Corporation + +Module Name: + + dl_mk_quantifier_abstraction.cpp + +Abstract: + + Create quantified Horn clauses from benchmarks with arrays. + +Author: + + Ken McMillan + Andrey Rybalchenko + Nikolaj Bjorner (nbjorner) 2013-04-02 + +Revision History: + +--*/ + +#include "dl_mk_quantifier_abstraction.h" +#include "dl_context.h" + +namespace datalog { + + mk_quantifier_abstraction::mk_quantifier_abstraction( + context & ctx, unsigned priority): + plugin(priority), + m(ctx.get_manager()), + m_ctx(ctx), + a(m), + m_refs(m) { + } + + mk_quantifier_abstraction::~mk_quantifier_abstraction() { + + } + + func_decl* mk_quantifier_abstraction::declare_pred(func_decl* old_p) { + + unsigned sz = old_p->get_arity(); + unsigned num_arrays = 0; + for (unsigned i = 0; i < sz; ++i) { + if (a.is_array(old_p->get_domain(i))) { + num_arrays++; + } + } + if (num_arrays == 0) { + return 0; + } + + func_decl* new_p = 0; + if (!m_old2new.find(old_p, new_p)) { + sort_ref_vector domain(m); + for (unsigned i = 0; i < sz; ++i) { + sort* s = old_p->get_domain(i); + while (a.is_array(s)) { + unsigned arity = get_array_arity(s); + for (unsigned j = 0; j < arity; ++j) { + domain.push_back(get_array_domain(s, j)); + } + s = get_array_range(s); + } + domain.push_back(s); + } + SASSERT(old_p->get_range() == m.mk_bool_sort()); + new_p = m.mk_func_decl(old_p->get_name(), domain.size(), domain.c_ptr(), old_p->get_range()); + m_refs.push_back(new_p); + m_ctx.register_predicate(new_p); + } + return new_p; + } + + app_ref mk_quantifier_abstraction::mk_head(app* p, unsigned idx) { + func_decl* new_p = declare_pred(p->get_decl()); + if (!new_p) { + return app_ref(p, m); + } + expr_ref_vector args(m); + expr_ref arg(m); + unsigned sz = p->get_num_args(); + for (unsigned i = 0; i < sz; ++i) { + arg = p->get_arg(i); + sort* s = m.get_sort(arg); + while (a.is_array(s)) { + unsigned arity = get_array_arity(s); + for (unsigned j = 0; j < arity; ++j) { + args.push_back(m.mk_var(idx++, get_array_domain(s, j))); + } + ptr_vector args2; + args2.push_back(arg); + args2.append(arity, args.c_ptr()-arity); + arg = a.mk_select(args2.size(), args2.c_ptr()); + s = get_array_range(s); + } + args.push_back(arg); + } + return app_ref(m.mk_app(new_p, args.size(), args.c_ptr()), m); + } + + app_ref mk_quantifier_abstraction::mk_tail(app* p) { + func_decl* old_p = p->get_decl(); + func_decl* new_p = declare_pred(old_p); + if (!new_p) { + return app_ref(p, m); + } + SASSERT(new_p->get_arity() > old_p->get_arity()); + unsigned num_extra_args = new_p->get_arity() - old_p->get_arity(); + var_shifter shift(m); + expr_ref p_shifted(m); + shift(p, num_extra_args, p_shifted); + app* ps = to_app(p_shifted); + expr_ref_vector args(m); + app_ref_vector pats(m); + sort_ref_vector vars(m); + svector names; + expr_ref arg(m); + unsigned idx = 0; + unsigned sz = p->get_num_args(); + for (unsigned i = 0; i < sz; ++i) { + arg = ps->get_arg(i); + sort* s = m.get_sort(arg); + bool is_pattern = false; + while (a.is_array(s)) { + is_pattern = true; + unsigned arity = get_array_arity(s); + for (unsigned j = 0; j < arity; ++j) { + vars.push_back(get_array_domain(s, j)); + names.push_back(symbol(idx)); + args.push_back(m.mk_var(idx++, vars.back())); + } + ptr_vector args2; + args2.push_back(arg); + args2.append(arity, args.c_ptr()-arity); + arg = a.mk_select(args2.size(), args2.c_ptr()); + s = get_array_range(s); + } + if (is_pattern) { + pats.push_back(to_app(arg)); + } + args.push_back(arg); + } + expr* pat = 0; + expr_ref pattern(m); + pattern = m.mk_pattern(pats.size(), pats.c_ptr()); + pat = pattern.get(); + app_ref result(m); + symbol qid, skid; + result = m.mk_app(new_p, args.size(), args.c_ptr()); + result = m.mk_eq(m.mk_forall(vars.size(), vars.c_ptr(), names.c_ptr(), result, 1, qid, skid, 1, &pat), m.mk_true()); + return result; + } + + rule_set * mk_quantifier_abstraction::operator()(rule_set const & source) { + if (!m_ctx.get_params().quantify_arrays()) { + return 0; + } + m_refs.reset(); + m_old2new.reset(); + m_new2old.reset(); + rule_manager& rm = source.get_rule_manager(); + rule_set * result = alloc(rule_set, m_ctx); + unsigned sz = source.get_num_rules(); + rule_ref new_rule(rm); + app_ref_vector tail(m); + app_ref head(m); + svector neg; + rule_counter& vc = rm.get_counter(); + + for (unsigned i = 0; i < sz; ++i) { + tail.reset(); + neg.reset(); + rule & r = *source.get_rule(i); + unsigned cnt = vc.get_max_rule_var(r)+1; + unsigned utsz = r.get_uninterpreted_tail_size(); + unsigned tsz = r.get_tail_size(); + for (unsigned j = 0; j < utsz; ++j) { + tail.push_back(mk_tail(r.get_tail(j))); + neg.push_back(r.is_neg_tail(j)); + } + for (unsigned j = utsz; j < tsz; ++j) { + tail.push_back(r.get_tail(j)); + neg.push_back(false); + } + head = mk_head(r.get_head(), cnt); + + new_rule = rm.mk(head, tail.size(), tail.c_ptr(), neg.c_ptr(), r.name(), true); + result->add_rule(new_rule); + + } + + // model converter: TBD. + // proof converter: TBD. + + if (m_old2new.empty()) { + dealloc(result); + result = 0; + } + return result; + } + + +}; + + diff --git a/src/muz_qe/dl_mk_quantifier_abstraction.h b/src/muz_qe/dl_mk_quantifier_abstraction.h new file mode 100644 index 000000000..8c26e277a --- /dev/null +++ b/src/muz_qe/dl_mk_quantifier_abstraction.h @@ -0,0 +1,61 @@ +/*++ +Copyright (c) 2013 Microsoft Corporation + +Module Name: + + dl_mk_quantifier_abstraction.h + +Abstract: + + Convert clauses with array arguments to predicates + into Quantified Horn clauses. + +Author: + + Ken McMillan + Andrey Rybalchenko + Nikolaj Bjorner (nbjorner) 2013-04-02 + +Revision History: + + Based on approach suggested in SAS 2013 paper + "On Solving Universally Quantified Horn Clauses" + +--*/ +#ifndef _DL_MK_QUANTIFIER_ABSTRACTION_H_ +#define _DL_MK_QUANTIFIER_ABSTRACTION_H_ + + +#include"dl_rule_transformer.h" +#include"array_decl_plugin.h" + +namespace datalog { + + class context; + + class mk_quantifier_abstraction : public rule_transformer::plugin { + ast_manager& m; + context& m_ctx; + array_util a; + func_decl_ref_vector m_refs; + obj_map m_new2old; + obj_map m_old2new; + + func_decl* declare_pred(func_decl* old_p); + app_ref mk_head(app* p, unsigned idx); + app_ref mk_tail(app* p); + + public: + mk_quantifier_abstraction(context & ctx, unsigned priority); + + virtual ~mk_quantifier_abstraction(); + + rule_set * operator()(rule_set const & source); + }; + + + +}; + +#endif /* _DL_MK_QUANTIFIER_ABSTRACTION_H_ */ + diff --git a/src/muz_qe/dl_mk_quantifier_instantiation.cpp b/src/muz_qe/dl_mk_quantifier_instantiation.cpp new file mode 100644 index 000000000..a5b80af4f --- /dev/null +++ b/src/muz_qe/dl_mk_quantifier_instantiation.cpp @@ -0,0 +1,295 @@ +/*++ +Copyright (c) 2013 Microsoft Corporation + +Module Name: + + dl_mk_quantifier_instantiation.cpp + +Abstract: + + Convert Quantified Horn clauses into non-quantified clauses using + instantiation. + +Author: + + Ken McMillan + Andrey Rybalchenko + Nikolaj Bjorner (nbjorner) 2013-04-02 + +Revision History: + + Based on approach suggested in SAS 2013 paper + "On Solving Universally Quantified Horn Clauses" + +--*/ + +#include "dl_mk_quantifier_instantiation.h" +#include "dl_context.h" + +namespace datalog { + + mk_quantifier_instantiation::mk_quantifier_instantiation( + context & ctx, unsigned priority): + plugin(priority), + m(ctx.get_manager()), + m_ctx(ctx), + m_var2cnst(m), + m_cnst2var(m), + a(m) { + } + + mk_quantifier_instantiation::~mk_quantifier_instantiation() { + + } + + void mk_quantifier_instantiation::extract_quantifiers(rule& r, expr_ref_vector& conjs, quantifier_ref_vector& qs) { + conjs.reset(); + qs.reset(); + unsigned tsz = r.get_tail_size(); + for (unsigned j = 0; j < tsz; ++j) { + conjs.push_back(r.get_tail(j)); + + } + datalog::flatten_and(conjs); + for (unsigned j = 0; j < conjs.size(); ++j) { + expr* e = conjs[j].get(); + quantifier* q; + if (rule_manager::is_forall(m, e, q)) { + qs.push_back(q); + conjs[j] = conjs.back(); + conjs.pop_back(); + --j; + } + } + } + + void mk_quantifier_instantiation::instantiate_quantifier(quantifier* q, expr_ref_vector & conjs) { + expr_ref qe(m); + qe = q; + m_var2cnst(qe); + q = to_quantifier(qe); + unsigned num_patterns = q->get_num_patterns(); + for (unsigned i = 0; i < num_patterns; ++i) { + expr * pat = q->get_pattern(i); + SASSERT(m.is_pattern(pat)); + instantiate_quantifier(q, to_app(pat), conjs); + } + } + + + void mk_quantifier_instantiation::instantiate_quantifier(quantifier* q, app* pat, expr_ref_vector & conjs) { + unsigned sz = pat->get_num_args(); + m_binding.reset(); + m_binding.resize(q->get_num_decls()); + term_pairs todo; + match(0, pat, 0, todo, q, conjs); + } + + void mk_quantifier_instantiation::match(unsigned i, app* pat, unsigned j, term_pairs& todo, quantifier* q, expr_ref_vector& conjs) { + while (j < todo.size()) { + expr* p = todo[j].first; + expr* t = todo[j].second; + if (is_var(p)) { + unsigned idx = to_var(p)->get_idx(); + expr* t2 = m_binding[idx]; + if (!t2) { + m_binding[idx] = t; + match(i, pat, j + 1, todo, q, conjs); + m_binding[idx] = 0; + return; + } + else if (m_uf.find(t2->get_id()) != m_uf.find(t->get_id())) { + // matching failed. + return; + } + j += 1; + continue; + } + if (!is_app(p)) { + return; + } + app* a1 = to_app(p); + unsigned id = t->get_id(); + unsigned next_id = id; + unsigned sz = todo.size(); + do { + expr* t2 = m_terms[next_id]; + if (is_app(t2)) { + app* a2 = to_app(t2); + if (a1->get_decl() == a2->get_decl() && + a1->get_num_args() == a2->get_num_args()) { + for (unsigned k = 0; k < a1->get_num_args(); ++k) { + todo.push_back(std::make_pair(a1->get_arg(k), a2->get_arg(k))); + } + match(i, pat, j + 1, todo, q, conjs); + todo.resize(sz); + } + } + next_id = m_uf.next(id); + } + while (next_id != id); + return; + } + + if (i == pat->get_num_args()) { + yield_binding(q, conjs); + return; + } + expr* arg = pat->get_arg(i); + ptr_vector* terms = 0; + + if (m_funs.find(to_app(arg)->get_decl(), terms)) { + for (unsigned k = 0; k < terms->size(); ++k) { + todo.push_back(std::make_pair(arg, (*terms)[k])); + match(i + 1, pat, j, todo, q, conjs); + todo.pop_back(); + } + } + } + + void mk_quantifier_instantiation::yield_binding(quantifier* q, expr_ref_vector& conjs) { + DEBUG_CODE( + for (unsigned i = 0; i < m_binding.size(); ++i) { + SASSERT(m_binding[i]); + }); + m_binding.reverse(); + expr_ref res(m); + instantiate(m, q, m_binding.c_ptr(), res); + m_binding.reverse(); + conjs.push_back(res); + } + + void mk_quantifier_instantiation::merge(expr* e1, expr* e2) { + unsigned i1 = e1->get_id(); + unsigned i2 = e2->get_id(); + unsigned n = std::max(i1, i2); + while (n >= m_uf.get_num_vars()) { + m_uf.mk_var(); + } + m_uf.merge(i1, i2); + } + + void mk_quantifier_instantiation::collect_egraph(expr* e) { + expr* e1, *e2; + m_todo.push_back(e); + expr_fast_mark1 visited; + while (!m_todo.empty()) { + e = m_todo.back(); + m_todo.pop_back(); + if (visited.is_marked(e)) { + continue; + } + if (e->get_id() >= m_terms.size()) { + m_terms.resize(e->get_id()+1); + } + m_terms[e->get_id()] = e; + visited.mark(e); + if (m.is_eq(e, e1, e2) || m.is_iff(e, e1, e2)) { + merge(e1, e2); + } + if (is_app(e)) { + app* ap = to_app(e); + ptr_vector* terms = 0; + if (m_funs.find(ap->get_decl(), terms)) { + terms = alloc(ptr_vector); + m_funs.insert(ap->get_decl(), terms); + } + terms->push_back(e); + m_todo.append(ap->get_num_args(), ap->get_args()); + } + } + } + + void mk_quantifier_instantiation::instantiate_rule(rule& r, expr_ref_vector& conjs, quantifier_ref_vector& qs, rule_set& rules) { + rule_manager& rm = m_ctx.get_rule_manager(); + expr_ref fml(m), cnst(m); + var_ref var(m); + ptr_vector sorts; + r.get_vars(sorts); + m_uf.reset(); + m_terms.reset(); + m_var2cnst.reset(); + m_cnst2var.reset(); + fml = m.mk_and(conjs.size(), conjs.c_ptr()); + + for (unsigned i = 0; i < sorts.size(); ++i) { + if (!sorts[i]) { + sorts[i] = m.mk_bool_sort(); + } + var = m.mk_var(i, sorts[i]); + cnst = m.mk_fresh_const("C", sorts[i]); + m_var2cnst.insert(var, cnst); + m_cnst2var.insert(cnst, var); + } + + fml = m.mk_and(conjs.size(), conjs.c_ptr()); + m_var2cnst(fml); + collect_egraph(fml); + + for (unsigned i = 0; i < qs.size(); ++i) { + instantiate_quantifier(qs[i].get(), conjs); + } + obj_map*>::iterator it = m_funs.begin(), end = m_funs.end(); + for (; it != end; ++it) { + dealloc(it->m_value); + } + m_funs.reset(); + + fml = m.mk_and(conjs.size(), conjs.c_ptr()); + fml = m.mk_implies(fml, r.get_head()); + + rule_ref_vector added_rules(rm); + proof_ref pr(m); // proofs are TBD. + rm.mk_rule(fml, pr, added_rules); + rules.add_rules(added_rules.size(), added_rules.c_ptr()); + } + + rule_set * mk_quantifier_instantiation::operator()(rule_set const & source) { + if (!m_ctx.get_params().instantiate_quantifiers()) { + return 0; + } + bool has_quantifiers = false; + unsigned sz = source.get_num_rules(); + for (unsigned i = 0; !has_quantifiers && i < sz; ++i) { + rule& r = *source.get_rule(i); + has_quantifiers = has_quantifiers || r.has_quantifiers(); + if (r.has_negation()) { + return 0; + } + } + if (!has_quantifiers) { + return 0; + } + + expr_ref_vector conjs(m); + quantifier_ref_vector qs(m); + rule_set * result = alloc(rule_set, m_ctx); + + bool instantiated = false; + + for (unsigned i = 0; i < sz; ++i) { + rule * r = source.get_rule(i); + extract_quantifiers(*r, conjs, qs); + if (qs.empty()) { + result->add_rule(r); + } + else { + instantiate_rule(*r, conjs, qs, *result); + instantiated = true; + } + } + + // model converter: TBD. + // proof converter: TBD. + + if (!instantiated) { + dealloc(result); + result = 0; + } + return result; + } + + +}; + + diff --git a/src/muz_qe/dl_mk_quantifier_instantiation.h b/src/muz_qe/dl_mk_quantifier_instantiation.h new file mode 100644 index 000000000..81a77e86a --- /dev/null +++ b/src/muz_qe/dl_mk_quantifier_instantiation.h @@ -0,0 +1,120 @@ +/*++ +Copyright (c) 2013 Microsoft Corporation + +Module Name: + + dl_mk_quantifier_instantiation.h + +Abstract: + + Convert Quantified Horn clauses into non-quantified clauses using + instantiation. + +Author: + + Ken McMillan + Andrey Rybalchenko + Nikolaj Bjorner (nbjorner) 2013-04-02 + +Revision History: + + Based on approach suggested in SAS 2013 paper + "On Solving Universally Quantified Horn Clauses" + +--*/ +#ifndef _DL_MK_QUANTIFIER_INSTANTIATION_H_ +#define _DL_MK_QUANTIFIER_INSTANTIATION_H_ + + +#include"dl_rule_transformer.h" +#include"array_decl_plugin.h" +#include"expr_safe_replace.h" + + +namespace datalog { + + class context; + + class mk_quantifier_instantiation : public rule_transformer::plugin { + typedef svector > term_pairs; + + class union_find { + unsigned_vector m_find; + unsigned_vector m_size; + unsigned_vector m_next; + public: + unsigned mk_var() { + unsigned r = m_find.size(); + m_find.push_back(r); + m_size.push_back(1); + m_next.push_back(r); + return r; + } + unsigned get_num_vars() const { return m_find.size(); } + + unsigned find(unsigned v) const { + while (true) { + unsigned new_v = m_find[v]; + if (new_v == v) + return v; + v = new_v; + } + } + + unsigned next(unsigned v) const { return m_next[v]; } + + bool is_root(unsigned v) const { return m_find[v] == v; } + + void merge(unsigned v1, unsigned v2) { + unsigned r1 = find(v1); + unsigned r2 = find(v2); + if (r1 == r2) + return; + if (m_size[r1] > m_size[r2]) + std::swap(r1, r2); + m_find[r1] = r2; + m_size[r2] += m_size[r1]; + std::swap(m_next[r1], m_next[r2]); + } + + void reset() { + m_find.reset(); + m_next.reset(); + m_size.reset(); + } + }; + ast_manager& m; + context& m_ctx; + expr_safe_replace m_var2cnst; + expr_safe_replace m_cnst2var; + array_util a; + union_find m_uf; + ptr_vector m_todo; + ptr_vector m_terms; + ptr_vector m_binding; + obj_map*> m_funs; + + + void merge(expr* e1, expr* e2); + void extract_quantifiers(rule& r, expr_ref_vector& conjs, quantifier_ref_vector& qs); + void collect_egraph(expr* e); + void instantiate_rule(rule& r, expr_ref_vector& conjs, quantifier_ref_vector& qs, rule_set& rules); + void instantiate_quantifier(quantifier* q, expr_ref_vector & conjs); + void mk_quantifier_instantiation::instantiate_quantifier(quantifier* q, app* pat, expr_ref_vector & conjs); + void mk_quantifier_instantiation::match(unsigned i, app* pat, unsigned j, term_pairs& todo, quantifier* q, expr_ref_vector& conjs); + void mk_quantifier_instantiation::yield_binding(quantifier* q, expr_ref_vector& conjs); + + public: + mk_quantifier_instantiation(context & ctx, unsigned priority); + + virtual ~mk_quantifier_instantiation(); + + rule_set * operator()(rule_set const & source); + }; + + + +}; + +#endif /* _DL_MK_QUANTIFIER_INSTANTIATION_H_ */ + diff --git a/src/muz_qe/fixedpoint_params.pyg b/src/muz_qe/fixedpoint_params.pyg index 7a3316c56..774559cdb 100644 --- a/src/muz_qe/fixedpoint_params.pyg +++ b/src/muz_qe/fixedpoint_params.pyg @@ -42,6 +42,8 @@ def_module_params('fixedpoint', ('simplify_formulas_post', BOOL, False, "PDR: simplify derived formulas after inductive propagation"), ('slice', BOOL, True, "PDR: simplify clause set using slicing"), ('karr', BOOL, False, "Add linear invariants to clauses using Karr's method"), + ('quantify_arrays', BOOL, False, "create quantified Horn clauses from clauses with arrays"), + ('instantiate_quantifiers', BOOL, False, "instantiate quantified Horn clauses using E-matching heuristic"), ('coalesce_rules', BOOL, False, "BMC: coalesce rules"), ('use_multicore_generalizer', BOOL, False, "PDR: extract multiple cores for blocking states"), ('use_inductive_generalizer', BOOL, True, "PDR: generalize lemmas using induction strengthening"),