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Add seq::derive class for symbolic regex derivatives

Browse source 
Implement a new seq::derive class (seq_derive.h/cpp) that computes
symbolic derivatives of regular expressions using ITE-trees, based on
the RE# approach (Varatalu, Veanes, Ernits - POPL 2025).

Key features:
- Two-argument operator()(ele, r): computes derivative of regex r w.r.t.
  element ele (concrete character or de Bruijn variable for symbolic mode)
- ACI canonicalization (flatten, stable_sort, dedup) for union/intersection
- ITE-tree combinators for binary/unary operations
- Info-based nullability with recursive fallback
- Complement absorption rules
- Depth-bounded recursion to prevent stack overflow

Integration with seq_rewriter:
- mk_derivative(ele, r) and mk_derivative(r) now delegate to m_derive
- Removed dead mk_derivative_rec function
- Added ITE hoisting in mk_re_star, mk_re_concat, mk_re_union0,
  mk_re_inter0, mk_re_complement
- Added depth limiting in Antimirov derivative helpers

Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>
This commit is contained in:
Nikolaj Bjorner 2026-06-03 10:36:19 -07:00
parent 80facee023
commit 52c7e89c31
6 changed files with 928 additions and 245 deletions
Download patch file Download diff file Expand all files Collapse all files

346
src/ast/rewriter/seq_rewriter.cpp
View file

@ -2947,11 +2947,7 @@ bool seq_rewriter::check_deriv_normal_form(expr* r, int level) {
#endif
expr_ref seq_rewriter::mk_derivative(expr* r) {
sort* seq_sort = nullptr, * ele_sort = nullptr;
VERIFY(m_util.is_re(r, seq_sort));
VERIFY(m_util.is_seq(seq_sort, ele_sort));
expr_ref v(m().mk_var(0, ele_sort), m());
return mk_antimirov_deriv(v, r, m().mk_true());
return m_derive(r);
}
expr_ref seq_rewriter::mk_brz_derivative(expr* r) {
@ -2963,7 +2959,7 @@ expr_ref seq_rewriter::mk_brz_derivative(expr* r) {
}
expr_ref seq_rewriter::mk_derivative(expr* ele, expr* r) {
return mk_antimirov_deriv(ele, r, m().mk_true());
return m_derive(ele, r);
}
expr_ref seq_rewriter::mk_antimirov_deriv(expr* e, expr* r, expr* path) {
@ -3183,13 +3179,18 @@ expr_ref seq_rewriter::mk_antimirov_deriv_intersection(expr* e, expr* d1, expr*
VERIFY(m_util.is_seq(seq_sort, ele_sort));
expr_ref result(m());
expr* c, * a, * b;
if (re().is_empty(d1))
if (m_re_deriv_depth >= m_max_re_deriv_depth) {
// Depth limit reached: construct intersection without further decomposition
result = mk_regex_inter_normalize(d1, d2);
}
else if (re().is_empty(d1))
result = d1;
else if (re().is_empty(d2))
result = d2;
else if (m().is_ite(d1, c, a, b)) {
expr_ref path_and_c(simplify_path(e, m().mk_and(path, c)), m());
expr_ref path_and_notc(simplify_path(e, m().mk_and(path, m().mk_not(c))), m());
++m_re_deriv_depth;
if (m().is_false(path_and_c))
result = mk_antimirov_deriv_intersection(e, b, d2, path);
else if (m().is_false(path_and_notc))
@ -3197,22 +3198,32 @@ expr_ref seq_rewriter::mk_antimirov_deriv_intersection(expr* e, expr* d1, expr*
else
result = m().mk_ite(c, mk_antimirov_deriv_intersection(e, a, d2, path_and_c),
mk_antimirov_deriv_intersection(e, b, d2, path_and_notc));
--m_re_deriv_depth;
}
else if (m().is_ite(d2))
else if (m().is_ite(d2)) {
// swap d1 and d2
++m_re_deriv_depth;
result = mk_antimirov_deriv_intersection(e, d2, d1, path);
--m_re_deriv_depth;
}
else if (d1 == d2 || re().is_full_seq(d2))
result = mk_antimirov_deriv_restrict(e, d1, path);
else if (re().is_full_seq(d1))
result = mk_antimirov_deriv_restrict(e, d2, path);
else if (re().is_union(d1, a, b))
else if (re().is_union(d1, a, b)) {
// distribute intersection over the union in d1
++m_re_deriv_depth;
result = mk_antimirov_deriv_union(mk_antimirov_deriv_intersection(e, a, d2, path),
mk_antimirov_deriv_intersection(e, b, d2, path));
else if (re().is_union(d2, a, b))
--m_re_deriv_depth;
}
else if (re().is_union(d2, a, b)) {
// distribute intersection over the union in d2
++m_re_deriv_depth;
result = mk_antimirov_deriv_union(mk_antimirov_deriv_intersection(e, d1, a, path),
mk_antimirov_deriv_intersection(e, d1, b, path));
--m_re_deriv_depth;
}
else
result = mk_regex_inter_normalize(d1, d2);
return result;
@ -3222,13 +3233,22 @@ expr_ref seq_rewriter::mk_antimirov_deriv_concat(expr* d, expr* r) {
expr_ref result(m());
expr_ref _r(r, m()), _d(d, m());
expr* c, * t, * e;
if (m().is_ite(d, c, t, e)) {
if (m_re_deriv_depth >= m_max_re_deriv_depth) {
// Depth limit reached: construct concat without further decomposition
result = mk_re_append(d, r);
}
else if (m().is_ite(d, c, t, e)) {
++m_re_deriv_depth;
auto r2 = mk_antimirov_deriv_concat(e, r);
auto r1 = mk_antimirov_deriv_concat(t, r);
--m_re_deriv_depth;
result = m().mk_ite(c, r1, r2);
}
else if (re().is_union(d, t, e))
else if (re().is_union(d, t, e)) {
++m_re_deriv_depth;
result = mk_antimirov_deriv_union(mk_antimirov_deriv_concat(t, r), mk_antimirov_deriv_concat(e, r));
--m_re_deriv_depth;
}
else
result = mk_re_append(d, r);
SASSERT(result.get());
@ -3244,7 +3264,11 @@ expr_ref seq_rewriter::mk_antimirov_deriv_negate(expr* elem, expr* d) {
auto dotplus = [&]() { return expr_ref(re().mk_plus(re().mk_full_char(d->get_sort())), m()); };
expr_ref result(m());
expr* c, * t, * e;
if (re().is_empty(d))
if (m_re_deriv_depth >= m_max_re_deriv_depth) {
// Depth limit reached: construct complement without further decomposition
result = re().mk_complement(d);
}
else if (re().is_empty(d))
result = dotstar();
else if (re().is_epsilon(d))
result = dotplus();
@ -3252,12 +3276,21 @@ expr_ref seq_rewriter::mk_antimirov_deriv_negate(expr* elem, expr* d) {
result = nothing();
else if (re().is_dot_plus(d))
result = epsilon();
else if (m().is_ite(d, c, t, e))
else if (m().is_ite(d, c, t, e)) {
++m_re_deriv_depth;
result = m().mk_ite(c, mk_antimirov_deriv_negate(elem, t), mk_antimirov_deriv_negate(elem, e));
else if (re().is_union(d, t, e))
--m_re_deriv_depth;
}
else if (re().is_union(d, t, e)) {
++m_re_deriv_depth;
result = mk_antimirov_deriv_intersection(elem, mk_antimirov_deriv_negate(elem, t), mk_antimirov_deriv_negate(elem, e), m().mk_true());
else if (re().is_intersection(d, t, e))
--m_re_deriv_depth;
}
else if (re().is_intersection(d, t, e)) {
++m_re_deriv_depth;
result = mk_antimirov_deriv_union(mk_antimirov_deriv_negate(elem, t), mk_antimirov_deriv_negate(elem, e));
--m_re_deriv_depth;
}
else if (re().is_complement(d, t))
result = t;
else
@ -3298,15 +3331,21 @@ expr_ref seq_rewriter::mk_antimirov_deriv_restrict(expr* e, expr* d, expr* cond)
result = re().mk_empty(d->get_sort());
else if (re().is_empty(d) || m().is_true(cond))
result = d;
else if (m_re_deriv_depth >= m_max_re_deriv_depth)
result = d;
else if (m().is_ite(d, c, a, b)) {
expr_ref path_and_c(simplify_path(e, m().mk_and(cond, c)), m());
expr_ref path_and_notc(simplify_path(e, m().mk_and(cond, m().mk_not(c))), m());
++m_re_deriv_depth;
result = re().mk_ite_simplify(c, mk_antimirov_deriv_restrict(e, a, path_and_c),
mk_antimirov_deriv_restrict(e, b, path_and_notc));
--m_re_deriv_depth;
}
else if (re().is_union(d, a, b)) {
++m_re_deriv_depth;
expr_ref a1(mk_antimirov_deriv_restrict(e, a, cond), m());
expr_ref b1(mk_antimirov_deriv_restrict(e, b, cond), m());
--m_re_deriv_depth;
result = mk_antimirov_deriv_union(a1, b1);
}
return result;
@ -3970,230 +4009,7 @@ expr_ref seq_rewriter::mk_der_cond(expr* cond, expr* ele, sort* seq_sort) {
return result;
}
/*
Classical Brzozowski derivative used by the regex_bisim equivalence
procedure. Unlike `mk_antimirov_deriv`, this variant never creates
_OP_RE_ANTIMIROV_UNION nodes — it stays in a classical (single regex
tree) form. The bisimulation algorithm relies on this so that each
leaf of D(p XOR q) is a coherent XOR pair (D_v p) XOR (D_v q).
*/
expr_ref seq_rewriter::mk_derivative_rec(expr* ele, expr* r) {
expr_ref result(m());
sort* seq_sort = nullptr, *ele_sort = nullptr;
VERIFY(m_util.is_re(r, seq_sort));
VERIFY(m_util.is_seq(seq_sort, ele_sort));
SASSERT(ele_sort == ele->get_sort());
expr* r1 = nullptr, *r2 = nullptr, *p = nullptr;
auto mk_empty = [&]() { return expr_ref(re().mk_empty(r->get_sort()), m()); };
unsigned lo = 0, hi = 0;
if (re().is_concat(r, r1, r2)) {
expr_ref is_n = is_nullable(r1);
expr_ref dr1 = mk_derivative_rec(ele, r1);
result = mk_der_concat(dr1, r2);
if (m().is_false(is_n)) {
return result;
}
expr_ref dr2 = mk_derivative_rec(ele, r2);
is_n = re_predicate(is_n, seq_sort);
if (re().is_empty(dr2)) {
//do not concatenate [], it is a deade-end
return result;
}
else {
// Classical Brzozowski union: keep the derivative tree free of
// antimirov-union nodes so the bisimulation procedure sees a
// single regex tree whose leaves are XOR pairs.
return mk_der_union(result, mk_der_concat(is_n, dr2));
}
}
else if (re().is_star(r, r1)) {
return mk_der_concat(mk_derivative_rec(ele, r1), r);
}
else if (re().is_plus(r, r1)) {
expr_ref star(re().mk_star(r1), m());
return mk_derivative_rec(ele, star);
}
else if (re().is_union(r, r1, r2)) {
return mk_der_union(mk_derivative_rec(ele, r1), mk_derivative_rec(ele, r2));
}
else if (re().is_intersection(r, r1, r2)) {
return mk_der_inter(mk_derivative_rec(ele, r1), mk_derivative_rec(ele, r2));
}
else if (re().is_diff(r, r1, r2)) {
return mk_der_inter(mk_derivative_rec(ele, r1), mk_der_compl(mk_derivative_rec(ele, r2)));
}
else if (re().is_xor(r, r1, r2)) {
return mk_der_xor(mk_derivative_rec(ele, r1), mk_derivative_rec(ele, r2));
}
else if (m().is_ite(r, p, r1, r2)) {
// there is no BDD normalization here
result = m().mk_ite(p, mk_derivative_rec(ele, r1), mk_derivative_rec(ele, r2));
return result;
}
else if (re().is_opt(r, r1)) {
return mk_derivative_rec(ele, r1);
}
else if (re().is_complement(r, r1)) {
return mk_der_compl(mk_derivative_rec(ele, r1));
}
else if (re().is_loop(r, r1, lo)) {
if (lo > 0) {
lo--;
}
result = mk_derivative_rec(ele, r1);
//do not concatenate with [] (emptyset)
if (re().is_empty(result)) {
return result;
}
else {
//do not create loop r1{0,}, instead create r1*
return mk_der_concat(result, (lo == 0 ? re().mk_star(r1) : re().mk_loop(r1, lo)));
}
}
else if (re().is_loop(r, r1, lo, hi)) {
if (hi == 0) {
return mk_empty();
}
hi--;
if (lo > 0) {
lo--;
}
result = mk_derivative_rec(ele, r1);
//do not concatenate with [] (emptyset) or handle the rest of the loop if no more iterations remain
if (re().is_empty(result) || hi == 0) {
return result;
}
else {
return mk_der_concat(result, re().mk_loop_proper(r1, lo, hi));
}
}
else if (re().is_full_seq(r) ||
re().is_empty(r)) {
return expr_ref(r, m());
}
else if (re().is_to_re(r, r1)) {
// r1 is a string here (not a regexp)
expr_ref hd(m()), tl(m());
if (get_head_tail(r1, hd, tl)) {
// head must be equal; if so, derivative is tail
// Use mk_der_cond to normalize
STRACE(seq_verbose, tout << "deriv to_re" << std::endl;);
result = m().mk_eq(ele, hd);
result = mk_der_cond(result, ele, seq_sort);
expr_ref r1(re().mk_to_re(tl), m());
result = mk_der_concat(result, r1);
return result;
}
else if (str().is_empty(r1)) {
//observe: str().is_empty(r1) checks that r = () = epsilon
//while mk_empty() = [], because deriv(epsilon) = [] = nothing
return mk_empty();
}
else if (str().is_itos(r1)) {
//
// here r1 = (str.from_int r2) and r2 is non-ground
// or else the expression would have been simplified earlier
// so r1 must be nonempty and must consists of decimal digits
// '0' <= elem <= '9'
// if ((isdigit ele) and (ele = (hd r1))) then (to_re (tl r1)) else []
//
hd = mk_seq_first(r1);
// isolate nested conjunction for deterministic evaluation
auto a0 = u().mk_le(m_util.mk_char('0'), ele);
auto a1 = u().mk_le(ele, m_util.mk_char('9'));
auto a2 = m().mk_not(m().mk_eq(r1, str().mk_empty(seq_sort)));
auto a3 = m().mk_eq(hd, ele);
auto inner = m().mk_and(a2, a3);
m_br.mk_and(a0, a1, inner, result);
tl = re().mk_to_re(mk_seq_rest(r1));
return re_and(result, tl);
}
else {
// recall: [] denotes the empty language (nothing) regex, () denotes epsilon or empty sequence
// construct the term (if (r1 != () and (ele = (first r1)) then (to_re (rest r1)) else []))
hd = mk_seq_first(r1);
m_br.mk_and(m().mk_not(m().mk_eq(r1, str().mk_empty(seq_sort))), m().mk_eq(hd, ele), result);
tl = re().mk_to_re(mk_seq_rest(r1));
return re_and(result, tl);
}
}
else if (re().is_reverse(r, r1)) {
if (re().is_to_re(r1, r2)) {
// First try to extract hd and tl such that r = hd ++ tl and |tl|=1
expr_ref hd(m()), tl(m());
if (get_head_tail_reversed(r2, hd, tl)) {
// Use mk_der_cond to normalize
STRACE(seq_verbose, tout << "deriv reverse to_re" << std::endl;);
result = m().mk_eq(ele, tl);
result = mk_der_cond(result, ele, seq_sort);
result = mk_der_concat(result, re().mk_reverse(re().mk_to_re(hd)));
return result;
}
else if (str().is_empty(r2)) {
return mk_empty();
}
else {
// construct the term (if (r2 != () and (ele = (last r2)) then reverse(to_re (butlast r2)) else []))
// hd = first of reverse(r2) i.e. last of r2
// tl = rest of reverse(r2) i.e. butlast of r2
//hd = str().mk_nth_i(r2, m_autil.mk_sub(str().mk_length(r2), one()));
hd = mk_seq_last(r2);
// factor nested constructor calls to enforce deterministic argument evaluation order
auto a_non_empty = m().mk_not(m().mk_eq(r2, str().mk_empty(seq_sort)));
auto a_eq = m().mk_eq(hd, ele);
m_br.mk_and(a_non_empty, a_eq, result);
tl = re().mk_to_re(mk_seq_butlast(r2));
return re_and(result, re().mk_reverse(tl));
}
}
}
else if (re().is_range(r, r1, r2)) {
// r1, r2 are sequences.
zstring s1, s2;
if (str().is_string(r1, s1) && str().is_string(r2, s2)) {
if (s1.length() == 1 && s2.length() == 1) {
expr_ref ch1(m_util.mk_char(s1[0]), m());
expr_ref ch2(m_util.mk_char(s2[0]), m());
// Use mk_der_cond to normalize
STRACE(seq_verbose, tout << "deriv range zstring" << std::endl;);
expr_ref p1(u().mk_le(ch1, ele), m());
p1 = mk_der_cond(p1, ele, seq_sort);
expr_ref p2(u().mk_le(ele, ch2), m());
p2 = mk_der_cond(p2, ele, seq_sort);
result = mk_der_inter(p1, p2);
return result;
}
else {
return mk_empty();
}
}
expr* e1 = nullptr, * e2 = nullptr;
if (str().is_unit(r1, e1) && str().is_unit(r2, e2)) {
SASSERT(u().is_char(e1));
// Use mk_der_cond to normalize
STRACE(seq_verbose, tout << "deriv range str" << std::endl;);
expr_ref p1(u().mk_le(e1, ele), m());
p1 = mk_der_cond(p1, ele, seq_sort);
expr_ref p2(u().mk_le(ele, e2), m());
p2 = mk_der_cond(p2, ele, seq_sort);
result = mk_der_inter(p1, p2);
return result;
}
}
else if (re().is_full_char(r)) {
return expr_ref(re().mk_to_re(str().mk_empty(seq_sort)), m());
}
else if (re().is_of_pred(r, p)) {
array_util array(m());
expr* args[2] = { p, ele };
result = array.mk_select(2, args);
// Use mk_der_cond to normalize
STRACE(seq_verbose, tout << "deriv of_pred" << std::endl;);
return mk_der_cond(result, ele, seq_sort);
}
// stuck cases: re.derivative, re variable,
return expr_ref(re().mk_derivative(ele, r), m());
}
/*************************************************
***** End Derivative Code *****
@ -4697,6 +4513,16 @@ br_status seq_rewriter::mk_re_concat(expr* a, expr* b, expr_ref& result) {
}
std::swap(a, b);
}
// Hoist ite out of concat: concat(ite(c, r1, r2), b) → ite(c, concat(r1, b), concat(r2, b))
expr* c = nullptr;
if (m().is_ite(a, c, a1, b1)) {
result = m().mk_ite(c, re().mk_concat(a1, b), re().mk_concat(b1, b));
return BR_REWRITE3;
}
if (m().is_ite(b, c, a1, b1)) {
result = m().mk_ite(c, re().mk_concat(a, a1), re().mk_concat(a, b1));
return BR_REWRITE3;
}
return BR_FAILED;
}
@ -4745,6 +4571,21 @@ br_status seq_rewriter::mk_re_union0(expr* a, expr* b, expr_ref& result) {
result = b;
return BR_DONE;
}
// r ∪ ~r → Σ* (complement absorption)
if (are_complements(a, b)) {
result = re().mk_full_seq(a->get_sort());
return BR_DONE;
}
// Hoist ite out of union: union(ite(c, r1, r2), b) → ite(c, union(r1, b), union(r2, b))
expr *c = nullptr, *r1 = nullptr, *r2 = nullptr;
if (m().is_ite(a, c, r1, r2)) {
result = m().mk_ite(c, re().mk_union(r1, b), re().mk_union(r2, b));
return BR_REWRITE3;
}
if (m().is_ite(b, c, r1, r2)) {
result = m().mk_ite(c, re().mk_union(a, r1), re().mk_union(a, r2));
return BR_REWRITE3;
}
return BR_FAILED;
}
@ -4787,6 +4628,12 @@ br_status seq_rewriter::mk_re_complement(expr* a, expr_ref& result) {
result = re().mk_plus(re().mk_full_char(a->get_sort()));
return BR_DONE;
}
// Hoist ite out of complement: ~(ite(c, r1, r2)) → ite(c, ~r1, ~r2)
expr* c = nullptr;
if (m().is_ite(a, c, e1, e2)) {
result = m().mk_ite(c, re().mk_complement(e1), re().mk_complement(e2));
return BR_REWRITE3;
}
return BR_FAILED;
}
@ -4812,6 +4659,21 @@ br_status seq_rewriter::mk_re_inter0(expr* a, expr* b, expr_ref& result) {
result = a;
return BR_DONE;
}
// r ∩ ~r → ∅ (complement absorption)
if (are_complements(a, b)) {
result = re().mk_empty(a->get_sort());
return BR_DONE;
}
// Hoist ite out of intersection: inter(ite(c, r1, r2), b) → ite(c, inter(r1, b), inter(r2, b))
expr *c = nullptr, *r1 = nullptr, *r2 = nullptr;
if (m().is_ite(a, c, r1, r2)) {
result = m().mk_ite(c, re().mk_inter(r1, b), re().mk_inter(r2, b));
return BR_REWRITE3;
}
if (m().is_ite(b, c, r1, r2)) {
result = m().mk_ite(c, re().mk_inter(a, r1), re().mk_inter(a, r2));
return BR_REWRITE3;
}
return BR_FAILED;
}
@ -5051,7 +4913,9 @@ br_status seq_rewriter::mk_re_star(expr* a, expr_ref& result) {
result = re().mk_full_seq(b1->get_sort());
return BR_REWRITE2;
}
// Hoist ite out of star: (ite c r1 r2)* → ite(c, r1*, r2*)
result = m().mk_ite(c, re().mk_star(b1), re().mk_star(c1));
return BR_REWRITE3;
}
return BR_FAILED;
}