use structured proof hints

Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>

examples/python/tutorial/jupyter/guide.ipynb

@@ -1395,7 +1395,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.7.1"
+   "version": "3.8.8"
   }
  },
  "nbformat": 4,

src/sat/sat_drat.cpp

@@ -19,8 +19,10 @@ Author:
 Notes:
 
 --*/
-#include "sat_solver.h"
-#include "sat_drat.h"
+
+#include "util/rational.h"
+#include "sat/sat_solver.h"
+#include "sat/sat_drat.h"
 
 
 namespace sat {
@@ -137,13 +139,13 @@ namespace sat {
             }
         }
         buffer[len++] = '0';
-        if (st.get_pragma()) {
+        if (st.get_hint()) {
             buffer[len++] = ' ';
             buffer[len++] = 'p';
             buffer[len++] = ' ';
-            char const* ps = st.get_pragma();
-            while (*ps) 
-                buffer[len++] = *ps++;
+            auto* ps = st.get_hint();
+            for (auto ch : ps->to_string())
+                buffer[len++] = ch;
         }
         buffer[len++] = '\n';
         m_out->write(buffer, len);
@@ -905,6 +907,106 @@ namespace sat {
         if (!st.is_sat())
             out << " " << p.th(st.get_th());
         return out;
-    }    
+    }
+
+
+    std::string proof_hint::to_string() const {
+        std::ostringstream ous;
+        switch (m_ty) {
+        case hint_type::null_h:
+            return std::string();
+        case hint_type::farkas_h:
+            ous << "farkas ";
+            break;
+        case hint_type::bound_h:
+            ous << "bound ";
+            break;
+        case hint_type::cut_h:
+            ous << "cut ";
+            break;
+        }
+        for (auto const& [q, l] : m_literals)
+            ous << rational(q) << " * " << l << " ";
+        for (auto const& [q, a, b] : m_eqs)
+            ous << rational(q) << " = " << a << " " << b << " ";
+        return ous.str();
+    }
+
+    proof_hint proof_hint::from_string(char const* s) {
+        proof_hint h;
+        h.m_ty = hint_type::null_h;
+        if (!s)
+            return h;
+        auto ws = [&]() {
+            while (*s == ' ' || *s == '\n' || *s == '\t')
+                ++s;
+        };
+
+        auto parse_type = [&]() {
+            if (0 == strncmp(s, "farkas", 6)) {
+                h.m_ty = hint_type::farkas_h;
+                s += 6;
+                return true;
+            }
+            if (0 == strncmp(s, "bound", 5)) {
+                h.m_ty = hint_type::bound_h;
+                s += 5;
+                return true;
+            }
+            return false;
+        };
+
+        sbuffer<char> buff;
+        auto parse_coeff = [&]() {
+            buff.reset();
+            while (*s && *s != ' ') {
+                buff.push_back(*s);
+                ++s;
+            }
+            buff.push_back(0);
+            return rational(buff.data());
+        };
+
+        auto parse_literal = [&]() {
+            rational r = parse_coeff();
+            if (!r.is_int())
+                return sat::null_literal;
+            if (r < 0)
+                return sat::literal((-r).get_unsigned(), true);
+            return sat::literal(r.get_unsigned(), false);
+        };
+        auto parse_coeff_literal = [&]() {
+            rational coeff = parse_coeff();
+            ws();
+            if (*s == '*') {
+                ++s;
+                ws();
+                sat::literal lit = parse_literal();
+                h.m_literals.push_back(std::make_pair(coeff, lit));
+                return true;
+            }
+            if (*s == '=') {
+                ++s;
+                ws();
+                unsigned a = parse_coeff().get_unsigned();
+                ws();
+                unsigned b = parse_coeff().get_unsigned();
+                h.m_eqs.push_back(std::make_tuple(coeff, a, b));
+                return true;
+            }
+            return false;
+        };
+
+        ws();
+        if (!parse_type())
+            return h;
+        ws();
+        while (*s) {
+            if (!parse_coeff_literal())
+                return h;
+            ws();            
+        }
+        return h;
+    }
 
 }

src/sat/sat_types.h

@@ -28,6 +28,7 @@ Revision History:
 #include "util/stopwatch.h"
 #include "util/symbol.h"
 #include "util/sat_literal.h"
+#include "util/rational.h"
 
 class params_ref;
 class reslimit;
@@ -93,29 +94,44 @@ namespace sat {
 
     };
 
+    enum class hint_type {
+        null_h,
+        farkas_h,
+        bound_h,
+        cut_h
+    };
+
+    struct proof_hint {
+        hint_type                                   m_ty;
+        vector<std::pair<rational, literal>>             m_literals;
+        vector<std::tuple<rational, unsigned, unsigned>> m_eqs;
+        std::string to_string() const;
+        static proof_hint from_string(char const* s);
+    };
+
     class status {
     public:
         enum class st { input, asserted, redundant, deleted };
         st m_st;
         int m_orig;
-        char const* m_pragma;
+        const proof_hint* m_hint;
     public:
-        status(st s, int o, char const* ps = nullptr) : m_st(s), m_orig(o), m_pragma(ps) {};
-        status(status const& s) : m_st(s.m_st), m_orig(s.m_orig), m_pragma(s.m_pragma) {}
-        status(status&& s) noexcept { m_st = st::asserted; m_orig = -1; std::swap(m_st, s.m_st); std::swap(m_orig, s.m_orig); std::swap(m_pragma, s.m_pragma); }
+        status(st s, int o, proof_hint const* ps = nullptr) : m_st(s), m_orig(o), m_hint(ps) {};
+        status(status const& s) : m_st(s.m_st), m_orig(s.m_orig), m_hint(s.m_hint) {}
+        status(status&& s) noexcept { m_st = st::asserted; m_orig = -1; std::swap(m_st, s.m_st); std::swap(m_orig, s.m_orig); std::swap(m_hint, s.m_hint); }
         status& operator=(status const& other) { m_st = other.m_st; m_orig = other.m_orig; return *this; }
         static status redundant() { return status(status::st::redundant, -1); }
         static status asserted() { return status(status::st::asserted, -1); }
         static status deleted() { return status(status::st::deleted, -1); }
         static status input() { return status(status::st::input, -1); }
 
-        static status th(bool redundant, int id, char const* ps = nullptr) { return status(redundant ? st::redundant : st::asserted, id, ps); }
+        static status th(bool redundant, int id, proof_hint const* ps = nullptr) { return status(redundant ? st::redundant : st::asserted, id, ps); }
 
         bool is_input() const { return st::input == m_st; }
         bool is_redundant() const { return st::redundant == m_st; }
         bool is_asserted() const { return st::asserted == m_st; }
         bool is_deleted() const { return st::deleted == m_st; }
-        char const* get_pragma() const { return m_pragma; }
+        proof_hint const* get_hint() const { return m_hint; }
 
         bool is_sat() const { return -1 == m_orig; }
         int  get_th() const { return m_orig;  }

src/sat/smt/arith_axioms.cpp

@@ -233,46 +233,55 @@ namespace arith {
         SASSERT(b1.get_var() == b2.get_var());
         if (k1 == k2 && kind1 == kind2) return;
         SASSERT(k1 != k2 || kind1 != kind2);
-        char const* bound_params = "farkas 1 1";
 
+        auto bin_clause = [&](sat::literal l1, sat::literal l2) {
+            sat::proof_hint* bound_params = nullptr;
+            if (ctx.use_drat()) {
+                bound_params = &m_farkas2;
+                m_farkas2.m_literals[0] = std::make_pair(rational(1), l1);
+                m_farkas2.m_literals[1] = std::make_pair(rational(1), l2);
+            }
+            add_clause(l1, l2, bound_params);            
+        };
+        
         if (kind1 == lp_api::lower_t) {
             if (kind2 == lp_api::lower_t) {
                 if (k2 <= k1)
-                    add_clause(~l1, l2, bound_params);
+                    bin_clause(~l1, l2);
                 else
-                    add_clause(l1, ~l2, bound_params);
+                    bin_clause(l1, ~l2);
             }
             else if (k1 <= k2)
                 // k1 <= k2, k1 <= x or x <= k2
-                add_clause(l1, l2);
+                bin_clause(l1, l2);
             else {
                 // k1 > hi_inf, k1 <= x => ~(x <= hi_inf)
-                add_clause(~l1, ~l2, bound_params);
+                bin_clause(~l1, ~l2);
                 if (v_is_int && k1 == k2 + rational(1))
                     // k1 <= x or x <= k1-1
-                    add_clause(l1, l2, bound_params);
+                    bin_clause(l1, l2);
             }
         }
         else if (kind2 == lp_api::lower_t) {
             if (k1 >= k2)
                 // k1 >= lo_inf, k1 >= x or lo_inf <= x
-                add_clause(l1, l2, bound_params);
+                bin_clause(l1, l2);
             else {
                 // k1 < k2, k2 <= x => ~(x <= k1)
-                add_clause(~l1, ~l2, bound_params);
+                bin_clause(~l1, ~l2);
                 if (v_is_int && k1 == k2 - rational(1))
                     // x <= k1 or k1+l <= x
-                    add_clause(l1, l2, bound_params);
+                    bin_clause(l1, l2);
             }
         }
         else {
             // kind1 == A_UPPER, kind2 == A_UPPER
             if (k1 >= k2)
                 // k1 >= k2, x <= k2 => x <= k1
-                add_clause(l1, ~l2, bound_params);
+                bin_clause(l1, ~l2);
             else
                 // k1 <= hi_sup , x <= k1 =>  x <= hi_sup
-                add_clause(~l1, l2, bound_params);
+                bin_clause(~l1, l2);
         }
     }
 

src/sat/smt/arith_diagnostics.cpp

@@ -80,16 +80,38 @@ namespace arith {
         if (m_nla) m_nla->collect_statistics(st);
     }
 
-    char const* solver::bounds_pragma() {
+    /**
+     * Assumption:
+     * A bound literal ax <= b is explained by a set of weighted literals
+     * r1*(a1*x <= b1) + .... + r_k*(a_k*x <= b_k), where r_i > 0
+     * such that there is a r >= 1
+     * (r1*a1+..+r_k*a_k) = r*a, (r1*b1+..+r_k*b_k) <= r*b
+     */
+    sat::proof_hint const* solver::explain(sat::hint_type ty) {
         if (!ctx.use_drat())
             return nullptr;
-        m_bounds_pragma.clear();
-        m_bounds_pragma += "bounds ";
-        for (sat::literal c : m_core) {
-            if (c.sign()) m_bounds_pragma += "-";
-            m_bounds_pragma += std::to_string(c.var());
-            m_bounds_pragma += " ";
+        m_bounds_pragma.m_ty = ty;
+        m_bounds_pragma.m_literals.reset();
+        m_bounds_pragma.m_eqs.reset();
+        for (auto ev : m_explanation) {
+            auto idx = ev.ci();
+            if (UINT_MAX == idx)
+                continue;
+            switch (m_constraint_sources[idx]) {
+            case inequality_source: {
+                literal lit = m_inequalities[idx];
+                m_bounds_pragma.m_literals.push_back({ev.coeff(), lit});
+                break;                
+            }
+            case equality_source: {
+                auto [u, v] = m_equalities[idx];
+                m_bounds_pragma.m_eqs.push_back({ev.coeff(), u->get_expr_id(), v->get_expr_id()});
+                break;
+            }
+            default:
+                break;
+            }
         }
-        return m_bounds_pragma.c_str();
+        return &m_bounds_pragma;
     }
 }

src/sat/smt/arith_solver.cpp

@@ -39,6 +39,8 @@ namespace arith {
         lp().settings().set_random_seed(get_config().m_random_seed);
         
         m_lia = alloc(lp::int_solver, *m_solver.get());
+        m_farkas2.m_ty = sat::hint_type::farkas_h;
+        m_farkas2.m_literals.resize(2);
     }
 
     solver::~solver() {
@@ -195,7 +197,13 @@ namespace arith {
         reset_evidence();
         m_core.push_back(lit1);
         TRACE("arith", tout << lit2 << " <- " << m_core << "\n";);
-        assign(lit2, m_core, m_eqs, "farkas 1 1");
+        sat::proof_hint* ph = nullptr;
+        if (ctx.use_drat()) {
+            ph = &m_farkas2;
+            m_farkas2.m_literals[0] = std::make_pair(rational(1), lit1);
+            m_farkas2.m_literals[1] = std::make_pair(rational(1), lit2);
+        }
+        assign(lit2, m_core, m_eqs, ph); 
         ++m_stats.m_bounds_propagations;
     }
 
@@ -255,7 +263,7 @@ namespace arith {
             TRACE("arith", for (auto lit : m_core) tout << lit << ": " << s().value(lit) << "\n";);
             DEBUG_CODE(for (auto lit : m_core) { VERIFY(s().value(lit) == l_true); });
             ++m_stats.m_bound_propagations1;
-            assign(lit, m_core, m_eqs, bounds_pragma()); 
+            assign(lit, m_core, m_eqs, explain(sat::hint_type::bound_h));
         }
 
         if (should_refine_bounds() && first)
@@ -370,7 +378,7 @@ namespace arith {
         reset_evidence();
         m_explanation.clear();
         lp().explain_implied_bound(be, m_bp);
-        assign(bound, m_core, m_eqs, nullptr);
+        assign(bound, m_core, m_eqs, explain(sat::hint_type::bound_h));
     }
 
 
@@ -1169,7 +1177,7 @@ namespace arith {
             app_ref b = mk_bound(m_lia->get_term(), m_lia->get_offset(), !m_lia->is_upper());
             IF_VERBOSE(4, verbose_stream() << "cut " << b << "\n");
             literal lit = expr2literal(b);
-            assign(lit, m_core, m_eqs, nullptr);
+            assign(lit, m_core, m_eqs, explain(sat::hint_type::cut_h));
             lia_check = l_false;
             break;
         }
@@ -1191,7 +1199,7 @@ namespace arith {
         return lia_check;
     }
 
-    void solver::assign(literal lit, literal_vector const& core, svector<enode_pair> const& eqs, char const* pma) {        
+    void solver::assign(literal lit, literal_vector const& core, svector<enode_pair> const& eqs, sat::proof_hint const* pma) {        
         if (core.size() < small_lemma_size() && eqs.empty()) {
             m_core2.reset();
             for (auto const& c : core)
@@ -1238,7 +1246,7 @@ namespace arith {
         for (literal& c : m_core)
             c.neg();
 
-        add_clause(m_core);
+        add_clause(m_core, explain(sat::hint_type::farkas_h));
     }
 
     bool solver::is_infeasible() const {

src/sat/smt/arith_solver.h

@@ -414,13 +414,14 @@ namespace arith {
         void set_conflict();
         void set_conflict_or_lemma(literal_vector const& core, bool is_conflict);
         void set_evidence(lp::constraint_index idx);
-        void assign(literal lit, literal_vector const& core, svector<enode_pair> const& eqs, char const* pma);
+        void assign(literal lit, literal_vector const& core, svector<enode_pair> const& eqs, sat::proof_hint const* pma);
 
         void false_case_of_check_nla(const nla::lemma& l);        
         void dbg_finalize_model(model& mdl);
 
-        std::string m_bounds_pragma;
-        char const* bounds_pragma();
+        sat::proof_hint m_bounds_pragma;
+        sat::proof_hint m_farkas2;
+        sat::proof_hint const* explain(sat::hint_type ty);
 
 
     public:

src/sat/smt/sat_th.cpp

@@ -125,7 +125,7 @@ namespace euf {
             pop_core(n);        
     }
 
-    sat::status th_euf_solver::mk_status(char const* ps) {
+    sat::status th_euf_solver::mk_status(sat::proof_hint const* ps) {
         return sat::status::th(m_is_redundant, get_id(), ps);
     }
 
@@ -149,7 +149,7 @@ namespace euf {
         return add_clause(2, lits);
     }
 
-    bool th_euf_solver::add_clause(sat::literal a, sat::literal b, char const* ps) {      
+    bool th_euf_solver::add_clause(sat::literal a, sat::literal b, sat::proof_hint const* ps) {      
         sat::literal lits[2] = { a, b };
         return add_clause(2, lits, ps);
     }
@@ -164,7 +164,7 @@ namespace euf {
         return add_clause(4, lits);
     }
 
-    bool th_euf_solver::add_clause(unsigned n, sat::literal* lits, char const* ps) {
+    bool th_euf_solver::add_clause(unsigned n, sat::literal* lits, sat::proof_hint const* ps) {
         bool was_true = false;
         for (unsigned i = 0; i < n; ++i)       
             was_true |= is_true(lits[i]);
@@ -226,11 +226,11 @@ namespace euf {
         return ctx.s().rand()();
     }
 
-    size_t th_explain::get_obj_size(unsigned num_lits, unsigned num_eqs, char const* pma) {
-        return sat::constraint_base::obj_size(sizeof(th_explain) + sizeof(sat::literal) * num_lits + sizeof(enode_pair) * num_eqs + (pma?strlen(pma)+1:1));
+    size_t th_explain::get_obj_size(unsigned num_lits, unsigned num_eqs, sat::proof_hint const* pma) {
+        return sat::constraint_base::obj_size(sizeof(th_explain) + sizeof(sat::literal) * num_lits + sizeof(enode_pair) * num_eqs + (pma?pma->to_string().length()+1:1));
     }
 
-    th_explain::th_explain(unsigned n_lits, sat::literal const* lits, unsigned n_eqs, enode_pair const* eqs, sat::literal c, enode_pair const& p, char const* pma) {
+    th_explain::th_explain(unsigned n_lits, sat::literal const* lits, unsigned n_eqs, enode_pair const* eqs, sat::literal c, enode_pair const& p, sat::proof_hint const* pma) {
         m_consequent = c;
         m_eq = p;
         m_num_literals = n_lits;
@@ -246,23 +246,26 @@ namespace euf {
             m_eqs[i] = eqs[i];
         base_ptr += sizeof(enode_pair) * n_eqs;            
         m_pragma = reinterpret_cast<char*>(base_ptr);
-        for (i = 0; pma && pma[i]; ++i)
-            m_pragma[i] = pma[i];
+        if (pma) {
+            std::string s = pma->to_string();
+            for (i = 0; s[i]; ++i)
+                m_pragma[i] = s[i];
+        }
         m_pragma[i] = 0;
     }
 
-    th_explain* th_explain::mk(th_euf_solver& th, unsigned n_lits, sat::literal const* lits, unsigned n_eqs, enode_pair const* eqs, sat::literal c, enode* x, enode* y, char const* pma) {
+    th_explain* th_explain::mk(th_euf_solver& th, unsigned n_lits, sat::literal const* lits, unsigned n_eqs, enode_pair const* eqs, sat::literal c, enode* x, enode* y, sat::proof_hint const* pma) {
         region& r = th.ctx.get_region();
         void* mem = r.allocate(get_obj_size(n_lits, n_eqs, pma));
         sat::constraint_base::initialize(mem, &th);
         return new (sat::constraint_base::ptr2mem(mem)) th_explain(n_lits, lits, n_eqs, eqs, c, enode_pair(x, y));
     }
 
-    th_explain* th_explain::propagate(th_euf_solver& th, sat::literal_vector const& lits, enode_pair_vector const& eqs, sat::literal consequent, char const* pma) {
+    th_explain* th_explain::propagate(th_euf_solver& th, sat::literal_vector const& lits, enode_pair_vector const& eqs, sat::literal consequent, sat::proof_hint const* pma) {
         return mk(th, lits.size(), lits.data(), eqs.size(), eqs.data(), consequent, nullptr, nullptr, pma);
     }
 
-    th_explain* th_explain::propagate(th_euf_solver& th, sat::literal_vector const& lits, enode_pair_vector const& eqs, euf::enode* x, euf::enode* y, char const* pma) {
+    th_explain* th_explain::propagate(th_euf_solver& th, sat::literal_vector const& lits, enode_pair_vector const& eqs, euf::enode* x, euf::enode* y, sat::proof_hint const* pma) {
         return mk(th, lits.size(), lits.data(), eqs.size(), eqs.data(), sat::null_literal, x, y, pma);
     }
 

src/sat/smt/sat_th.h

@@ -143,16 +143,16 @@ namespace euf {
         region& get_region();
 
 
-        sat::status mk_status(char const* ps = nullptr);
+        sat::status mk_status(sat::proof_hint const* ps = nullptr);
         bool add_unit(sat::literal lit);
         bool add_units(sat::literal_vector const& lits);
         bool add_clause(sat::literal lit) { return add_unit(lit); }
         bool add_clause(sat::literal a, sat::literal b);
-        bool add_clause(sat::literal a, sat::literal b, char const* ps);
+        bool add_clause(sat::literal a, sat::literal b, sat::proof_hint const* ps);
         bool add_clause(sat::literal a, sat::literal b, sat::literal c);
         bool add_clause(sat::literal a, sat::literal b, sat::literal c, sat::literal d);
-        bool add_clause(sat::literal_vector const& lits, char const* ps = nullptr) { return add_clause(lits.size(), lits.data(), ps); }
-        bool add_clause(unsigned n, sat::literal* lits, char const* ps = nullptr);
+        bool add_clause(sat::literal_vector const& lits, sat::proof_hint const* ps = nullptr) { return add_clause(lits.size(), lits.data(), ps); }
+        bool add_clause(unsigned n, sat::literal* lits, sat::proof_hint const* ps = nullptr);
         void add_equiv(sat::literal a, sat::literal b);
         void add_equiv_and(sat::literal a, sat::literal_vector const& bs);
 
@@ -221,9 +221,9 @@ namespace euf {
         sat::literal*  m_literals;
         enode_pair*    m_eqs;
         char*          m_pragma;
-        static size_t get_obj_size(unsigned num_lits, unsigned num_eqs, char const* pma);
-        th_explain(unsigned n_lits, sat::literal const* lits, unsigned n_eqs, enode_pair const* eqs, sat::literal c, enode_pair const& eq, char const* pma = nullptr);
-        static th_explain* mk(th_euf_solver& th, unsigned n_lits, sat::literal const* lits, unsigned n_eqs, enode_pair const* eqs, sat::literal c, enode* x, enode* y, char const* pma = nullptr);
+        static size_t get_obj_size(unsigned num_lits, unsigned num_eqs, sat::proof_hint const* pma);
+        th_explain(unsigned n_lits, sat::literal const* lits, unsigned n_eqs, enode_pair const* eqs, sat::literal c, enode_pair const& eq, sat::proof_hint const* pma = nullptr);
+        static th_explain* mk(th_euf_solver& th, unsigned n_lits, sat::literal const* lits, unsigned n_eqs, enode_pair const* eqs, sat::literal c, enode* x, enode* y, sat::proof_hint const* pma = nullptr);
 
     public:
         static th_explain* conflict(th_euf_solver& th, sat::literal_vector const& lits, enode_pair_vector const& eqs);
@@ -234,8 +234,8 @@ namespace euf {
         static th_explain* conflict(th_euf_solver& th, sat::literal lit, euf::enode* x, euf::enode* y);
         static th_explain* conflict(th_euf_solver& th, euf::enode* x, euf::enode* y);
         static th_explain* propagate(th_euf_solver& th, sat::literal lit, euf::enode* x, euf::enode* y);
-        static th_explain* propagate(th_euf_solver& th, sat::literal_vector const& lits, enode_pair_vector const& eqs, sat::literal consequent, char const* pma = nullptr);
-        static th_explain* propagate(th_euf_solver& th, sat::literal_vector const& lits, enode_pair_vector const& eqs, euf::enode* x, euf::enode* y, char const* pma = nullptr);
+        static th_explain* propagate(th_euf_solver& th, sat::literal_vector const& lits, enode_pair_vector const& eqs, sat::literal consequent, sat::proof_hint const* pma = nullptr);
+        static th_explain* propagate(th_euf_solver& th, sat::literal_vector const& lits, enode_pair_vector const& eqs, euf::enode* x, euf::enode* y, sat::proof_hint const* pma = nullptr);
 
         sat::ext_constraint_idx to_index() const {
             return sat::constraint_base::mem2base(this);
@@ -270,7 +270,7 @@ namespace euf {
 
         enode_pair eq_consequent() const { return m_eq; }
 
-        char const* get_pragma() const { return *m_pragma ? m_pragma : nullptr; }
+        sat::proof_hint const* get_pragma() const { return nullptr; } //*m_pragma ? m_pragma : nullptr; }
 
     };
 

src/util/sat_literal.h

@@ -95,6 +95,7 @@ namespace sat {
     inline bool operator!=(literal const & l1, literal const & l2) { return l1.m_val != l2.m_val; }
 
     inline std::ostream & operator<<(std::ostream & out, sat::literal l) { if (l == sat::null_literal) out << "null"; else out << (l.sign() ? "-" : "") << l.var(); return out; }
+    
 
 
     typedef svector<literal> literal_vector;
@@ -192,3 +193,11 @@ namespace sat {
     }
 
 };
+
+namespace std {
+
+    inline std::string to_string(sat::literal l) {
+        if (l.sign()) return "-" + to_string(l.var());
+        return to_string(l.var());
+    }
+};