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refactor(cc): fix bugs, use list of nodes in equiv class

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This commit is contained in:
Simon Cruanes 2018-08-18 18:06:16 -05:00
parent c2d79b2e6a
commit ca531d73a6
8 changed files with 186 additions and 131 deletions
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3
src/sat/Internal.ml
View file

@ -910,10 +910,11 @@ module Make (Th : Theory_intf.S) = struct
(* A literal is unassigned, we nedd to add it back to
the heap of potentially assignable literals, unless it has
a level lower than [lvl], in which case we just move it back. *)
assert (a.var.v_level > lvl);
if a.var.v_level <= lvl then (
(* It is a late propagation, which has a level
lower than where we backtrack, so we just move it to the head
of the queue, to be propagated again. *)
of the queue, to be theory-propagated again (BCP will be fine). *)
Vec.set st.trail !head a;
head := !head + 1
) else (

278
src/smt/Congruence_closure.ml
View file

@ -34,9 +34,7 @@ end
type actions = (module ACTIONS)
type task =
| T_pending of node
| T_merge of node * node * explanation
type explanation_thunk = explanation lazy_t
type t = {
tst: Term.state;
@ -51,8 +49,8 @@ type t = {
The critical property is that all members of an equivalence class
that have the same "shape" (including head symbol)
have the same signature *)
tasks: task Vec.t;
(* tasks to perform *)
pending: node Vec.t;
combine: (node * node * explanation_thunk) Vec.t;
on_backtrack:(unit->unit)->unit;
mutable ps_lits: Lit.Set.t;
(* proof state *)
@ -69,10 +67,7 @@ type t = {
let[@inline] on_backtrack cc f : unit = cc.on_backtrack f
let[@inline] is_root_ (n:node) : bool = n.n_root == n
let[@inline] size_ (r:repr) =
assert (is_root_ r);
Bag.size r.n_parents
let[@inline] size_ (r:repr) = r.n_size
(* check if [t] is in the congruence closure.
Invariant: [in_cc t ∧ do_cc t => forall u subterm t, in_cc u] *)
@ -87,6 +82,15 @@ let rec find_rec cc (n:node) : repr =
root
)
(* traverse the equivalence class of [n] *)
let iter_class_ (n:node) : node Sequence.t =
fun yield ->
let rec aux u =
yield u;
if u.n_next != n then aux u.n_next
in
aux n
let[@inline] true_ cc = cc.true_
let[@inline] false_ cc = cc.false_
@ -106,6 +110,27 @@ let[@inline] find_tn cc (t:term) : repr = get_ cc t |> find cc
let[@inline] same_class cc (n1:node)(n2:node): bool =
N.equal (find cc n1) (find cc n2)
(* print full state *)
let pp_full out (cc:t) : unit =
let pp_next out n =
Fmt.fprintf out "@ :next %a" N.pp n.n_next in
let pp_root out n =
if is_root_ n then Fmt.string out " :is-root" else Fmt.fprintf out "@ :root %a" N.pp n.n_root in
let pp_expl out n = match n.n_expl with
| E_none -> ()
| E_some e ->
Fmt.fprintf out " (@[:forest %a :expl %a@])" N.pp e.next Explanation.pp e.expl
in
let pp_n out n =
Fmt.fprintf out "(@[%a%a%a%a@])" Term.pp n.n_term pp_root n pp_next n pp_expl n
and pp_sig_e out (s,n) =
Fmt.fprintf out "(@[<1>%a@ <--> %a%a@])" Signature.pp s N.pp n pp_root n
in
Fmt.fprintf out
"(@[@{<yellow>cc.state@}@ (@[<hv>:nodes@ %a@])@ (@[<hv>:sig@ %a@])@])"
(Util.pp_seq ~sep:" " pp_n) (Term.Tbl.values cc.tbl)
(Util.pp_seq ~sep:" " pp_sig_e) (Sig_tbl.to_seq cc.signatures_tbl)
(* compute signature *)
let signature cc (t:term): node Term.view option =
let find = find_tn cc in
@ -124,13 +149,15 @@ let find_by_signature cc (t:term) : repr option =
| None -> None
| Some s -> Sig_tbl.get cc.signatures_tbl s
let add_signature cc (t:term) (r:node): unit =
match signature cc t with
let add_signature cc (r:node): unit =
match signature cc r.n_term with
| None -> ()
| Some s ->
(* add, but only if not present already *)
begin match Sig_tbl.find cc.signatures_tbl s with
| exception Not_found ->
Log.debugf 15
(fun k->k "(@[cc.add_sig@ %a@ <--> %a@])" Signature.pp s N.pp r);
on_backtrack cc (fun () -> Sig_tbl.remove cc.signatures_tbl s);
Sig_tbl.add cc.signatures_tbl s r;
| r' ->
@ -141,21 +168,20 @@ let push_pending cc t : unit =
if not @@ N.get_field N.field_is_pending t then (
Log.debugf 5 (fun k->k "(@[<hv1>cc.push_pending@ %a@])" N.pp t);
N.set_field N.field_is_pending true t;
Vec.push cc.tasks (T_pending t)
Vec.push cc.pending t
)
let push_combine cc t u e : unit =
Log.debugf 5
(fun k->k "(@[<hv1>cc.push_combine@ :t1 %a@ :t2 %a@ :expl %a@])"
N.pp t N.pp u Explanation.pp e);
Vec.push cc.tasks @@ T_merge (t,u,e)
N.pp t N.pp u Explanation.pp (Lazy.force e));
Vec.push cc.combine (t,u,e)
(* re-root the explanation tree of the equivalence class of [n]
so that it points to [n].
postcondition: [n.n_expl = None] *)
let rec reroot_expl (cc:t) (n:node): unit =
let old_expl = n.n_expl in
on_backtrack cc (fun () -> n.n_expl <- old_expl);
begin match old_expl with
| E_none -> () (* already root *)
| E_some {next=u; expl=e_n_u} ->
@ -167,12 +193,9 @@ let rec reroot_expl (cc:t) (n:node): unit =
let raise_conflict (cc:t) (e:conflict): _ =
let (module A) = cc.acts in
(* clear tasks queue *)
Vec.iter
(function
| T_pending n -> N.set_field N.field_is_pending false n
| T_merge _ -> ())
cc.tasks;
Vec.clear cc.tasks;
Vec.iter (N.set_field N.field_is_pending false) cc.pending;
Vec.clear cc.pending;
Vec.clear cc.combine;
A.raise_conflict e
let[@inline] all_classes cc : repr Sequence.t =
@ -251,7 +274,7 @@ let explain_loop (cc : t) : Lit.Set.t =
while not (Vec.is_empty cc.ps_queue) do
let a, b = Vec.pop_last cc.ps_queue in
Log.debugf 5
(fun k->k "(@[cc.explain_loop at@ %a@ %a@])" N.pp a N.pp b);
(fun k->k "(@[cc.explain_loop.at@ %a@ =?= %a@])" N.pp a N.pp b);
assert (N.equal (find cc a) (find cc b));
let c = find_common_ancestor a b in
explain_along_path cc a c;
@ -288,6 +311,9 @@ let add_tag_n cc (n:node) (tag:int) (expl:explanation) : unit =
n.n_tags <- Util.Int_map.add tag (n,expl) n.n_tags;
)
(* TODO: payload for set of tags *)
(* TODO: payload for mapping an equiv class to a set of literals, for bool prop *)
let relevant_subterms (t:Term.t) : Term.t Sequence.t =
fun yield ->
match t.term_view with
@ -302,13 +328,11 @@ let relevant_subterms (t:Term.t) : Term.t Sequence.t =
(* main CC algo: add terms from [pending] to the signature table,
check for collisions *)
let rec update_tasks (cc:t): unit =
(* step 2 deal with pending (parent) terms whose equiv class
might have changed *)
while not (Vec.is_empty cc.tasks) do
let task = Vec.pop_last cc.tasks in
match task with
| T_pending n -> task_pending_ cc n
| T_merge (t,u,expl) -> task_merge_ cc t u expl
while not (Vec.is_empty cc.pending && Vec.is_empty cc.combine) do
Vec.iter (task_pending_ cc) cc.pending;
Vec.clear cc.pending;
Vec.iter (task_combine_ cc) cc.combine;
Vec.clear cc.combine;
done
and task_pending_ cc n =
@ -317,25 +341,24 @@ and task_pending_ cc n =
begin match find_by_signature cc n.n_term with
| None ->
(* add to the signature table [sig(n) --> n] *)
add_signature cc n.n_term n
add_signature cc n
| Some u when n == u -> ()
| Some u ->
(* must combine [t] with [r] *)
if not @@ same_class cc n u then (
(* [t1] and [t2] must be applications of the same symbol to
(* [t1] and [t2] must be applications of the same symbol to
arguments that are pairwise equal *)
assert (n != u);
let expl = match n.n_term.term_view, u.n_term.term_view with
| App_cst (f1, a1), App_cst (f2, a2) ->
assert (Cst.equal f1 f2);
assert (IArray.length a1 = IArray.length a2);
Explanation.mk_merges @@ IArray.map2 (fun u1 u2 -> add_ cc u1, add_ cc u2) a1 a2
| If _, _ | App_cst _, _ | Bool _, _
-> assert false
in
push_combine cc n u expl
)
assert (n != u);
let expl = lazy (
match n.n_term.term_view, u.n_term.term_view with
| App_cst (f1, a1), App_cst (f2, a2) ->
assert (Cst.equal f1 f2);
assert (IArray.length a1 = IArray.length a2);
Explanation.mk_merges @@ IArray.map2 (fun u1 u2 -> add_ cc u1, add_ cc u2) a1 a2
| If _, _ | App_cst _, _ | Bool _, _
-> assert false
) in
push_combine cc n u expl
end;
(* TODO: evaluate [(= t u) := true] when [find t==find u] *)
(* FIXME: when to actually evaluate?
eval_pending cc;
*)
@ -343,15 +366,34 @@ and task_pending_ cc n =
(* main CC algo: merge equivalence classes in [st.combine].
@raise Exn_unsat if merge fails *)
and task_merge_ cc a b e_ab : unit =
and task_combine_ cc (a,b,e_ab) : unit =
let ra = find cc a in
let rb = find cc b in
if not (N.equal ra rb) then (
assert (is_root_ ra);
assert (is_root_ rb);
let lazy e_ab = e_ab in
(* We will merge [r_from] into [r_into].
we try to ensure that [size ra <= size rb] in general *)
let r_from, r_into = if size_ ra > size_ rb then rb, ra else ra, rb in
we try to ensure that [size ra <= size rb] in general, but always
keep values as representative *)
let r_from, r_into =
if Term.is_value ra.n_term then rb, ra
else if Term.is_value rb.n_term then ra, rb
else if size_ ra > size_ rb then rb, ra
else ra, rb
in
(* check we're not merging [true] and [false] *)
if (N.equal ra cc.true_ && N.equal rb cc.false_) ||
(N.equal rb cc.true_ && N.equal ra cc.false_) then (
Log.debugf 5
(fun k->k "(@[<hv>cc.merge.true_false_conflict@ @[:r1 %a@]@ @[:r2 %a@]@ :e_ab %a@])"
N.pp ra N.pp rb Explanation.pp e_ab);
let lits = explain_unfold cc e_ab in
let lits = explain_eq_n ~init:lits cc a ra in
let lits = explain_eq_n ~init:lits cc b rb in
raise_conflict cc @@ Lit.Set.elements lits
);
(* update set of tags the new node cannot be equal to *)
let new_tags =
Util.Int_map.union
(fun _i (n1,e1) (n2,e2) ->
@ -370,33 +412,48 @@ and task_merge_ cc a b e_ab : unit =
raise_conflict cc @@ Lit.Set.elements lits)
ra.n_tags rb.n_tags
in
(* remove [ra.parents] from signature, put them into [st.pending] *)
begin
Bag.to_seq r_from.n_parents
(fun parent -> push_pending cc parent)
end;
(* perform [union ra rb] *)
(* perform [union r_from r_into] *)
Log.debugf 15 (fun k->k "(@[cc.merge@ :from %a@ :into %a@])" N.pp r_from N.pp r_into);
begin
let r_into_old_parents = r_into.n_parents in
(* for each node in [r_from]'s class:
- make it point to [r_into]
- push it into [st.pending] *)
iter_class_ r_from
(fun u ->
assert (u.n_root == r_from);
on_backtrack cc (fun () -> u.n_root <- r_from);
u.n_root <- r_into;
Bag.to_seq u.n_parents
(fun parent -> push_pending cc parent));
(* now merge the classes *)
let r_into_old_tags = r_into.n_tags in
let r_into_old_next = r_into.n_next in
let r_from_old_next = r_from.n_next in
on_backtrack cc
(fun () ->
Log.debugf 15
(fun k->k "(@[cc.undo_merge@ :of %a :into %a@])"
(fun k->k "(@[cc.undo_merge@ :from %a :into %a@])"
Term.pp r_from.n_term Term.pp r_into.n_term);
r_from.n_root <- r_from;
r_into.n_tags <- r_into_old_tags;
r_into.n_parents <- r_into_old_parents);
r_from.n_root <- r_into;
r_into.n_next <- r_into_old_next;
r_from.n_next <- r_from_old_next;
r_into.n_tags <- r_into_old_tags);
r_into.n_tags <- new_tags;
r_from.n_parents <- Bag.append r_into_old_parents r_from.n_parents;
r_into.n_next <- r_from_old_next;
r_from.n_next <- r_into_old_next;
end;
(* update explanations (a -> b), arbitrarily *)
(* update explanations (a -> b), arbitrarily.
Note that here we merge the classes by adding a bridge between [a]
and [b], not their roots. *)
begin
reroot_expl cc a;
assert (a.n_expl = E_none);
on_backtrack cc (fun () -> a.n_expl <- E_none);
(* on backtracking, link may be inverted, but we delete the one
that bridges between [a] and [b] *)
on_backtrack cc
(fun () -> match a.n_expl, b.n_expl with
| E_some e, _ when N.equal e.next b -> a.n_expl <- E_none
| _, E_some e when N.equal e.next a -> b.n_expl <- E_none
| _ -> assert false);
a.n_expl <- E_some {next=b; expl=e_ab};
end;
(* notify listeners of the merge *)
@ -421,7 +478,6 @@ and add_new_term_ cc (t:term) : node =
let old_parents = sub.n_parents in
on_backtrack cc (fun () -> sub.n_parents <- old_parents);
sub.n_parents <- Bag.cons n sub.n_parents;
push_pending cc sub
in
(* add sub-term to [cc], and register [n] to its parents *)
let add_sub_t (u:term) : unit =
@ -446,6 +502,43 @@ and[@inline] add_ cc t : node =
try Term.Tbl.find cc.tbl t
with Not_found -> add_new_term_ cc t
let check_invariants_ (cc:t) =
Log.debug 5 "(cc.check-invariants)";
Log.debugf 15 (fun k-> k "%a" pp_full cc);
assert (Term.equal (Term.true_ cc.tst) cc.true_.n_term);
assert (Term.equal (Term.false_ cc.tst) cc.false_.n_term);
assert (not @@ same_class cc cc.true_ cc.false_);
assert (Vec.is_empty cc.combine);
assert (Vec.is_empty cc.pending);
(* check that subterms are internalized *)
Term.Tbl.iter
(fun t n ->
assert (Term.equal t n.n_term);
assert (not @@ N.get_field N.field_is_pending n);
relevant_subterms t
(fun u -> assert (Term.Tbl.mem cc.tbl u));
assert (N.equal n.n_root n.n_next.n_root);
(* check proper signature.
note that some signatures in the sig table can be obsolete (they
were not removed) but there must be a valid, up-to-date signature for
each term *)
begin match signature cc t with
| None -> ()
| Some s ->
Log.debugf 15 (fun k->k "(@[cc.check-sig@ %a@ :sig %a@])" Term.pp t Signature.pp s);
(* add, but only if not present already *)
begin match Sig_tbl.find cc.signatures_tbl s with
| exception Not_found -> assert false
| repr_s -> assert (same_class cc n repr_s)
end
end;
)
cc.tbl;
()
let[@inline] check_invariants (cc:t) : unit =
if Util._CHECK_INVARIANTS then check_invariants_ cc
let add cc t : node =
let n = add_ cc t in
update_tasks cc;
@ -468,14 +561,14 @@ let assert_lit cc lit : unit =
basically, just have [n] point to true/false and thus acquire
the corresponding value, so its superterms (like [ite]) can evaluate
properly *)
push_combine cc n rhs (E_lit lit);
push_combine cc n rhs (Lazy.from_val @@ E_lit lit);
update_tasks cc
let assert_eq cc (t:term) (u:term) e : unit =
let n1 = add_ cc t in
let n2 = add_ cc u in
if not (same_class cc n1 n2) then (
let e = Explanation.E_lits e in
let e = Lazy.from_val @@ Explanation.E_lits e in
push_combine cc n1 n2 e;
);
update_tasks cc
@ -510,37 +603,23 @@ let create ?(size=2048) ~actions (tst:Term.state) : t =
acts=actions;
tbl = Term.Tbl.create size;
signatures_tbl = Sig_tbl.create size;
tasks=Vec.make_empty (T_pending N.dummy);
pending=Vec.make_empty N.dummy;
combine=Vec.make_empty (N.dummy,N.dummy,Lazy.from_val Explanation.dummy);
ps_lits=Lit.Set.empty;
on_backtrack=A.on_backtrack;
ps_queue=Vec.make_empty (nd,nd);
true_ = N.dummy;
false_ = N.dummy;
} in
cc.true_ <- add cc (Term.true_ tst);
cc.false_ <- add cc (Term.false_ tst);
cc.true_ <- add_ cc (Term.true_ tst);
cc.false_ <- add_ cc (Term.false_ tst);
update_tasks cc;
cc
let final_check cc : unit =
Log.debug 5 "(CC.final_check)";
update_tasks cc
let pp_full out (cc:t) : unit =
let pp_next out n =
if n==n.n_root then Fmt.string out " :is-root"
else Fmt.fprintf out "@ :next %a" N.pp n.n_root in
let pp_root out n =
let u = find cc n in if n==u||n.n_root==u then () else Fmt.fprintf out "@ :root %a" N.pp u in
let pp_n out n =
Fmt.fprintf out "(@[%a%a%a@])" Term.pp n.n_term pp_next n pp_root n
and pp_sig_e out (s,n) =
Fmt.fprintf out "(@[<1>%a@ -> %a%a%a@])" Signature.pp s N.pp n pp_next n pp_root n
in
Fmt.fprintf out
"(@[@{<yellow>cc.state@}@ (@[<hv>:nodes@ %a@])@ (@[<hv>:sig@ %a@])@])"
(Util.pp_seq ~sep:" " pp_n) (Term.Tbl.values cc.tbl)
(Util.pp_seq ~sep:" " pp_sig_e) (Sig_tbl.to_seq cc.signatures_tbl)
(* model: map each uninterpreted equiv class to some ID *)
let mk_model (cc:t) (m:Model.t) : Model.t =
Log.debugf 15 (fun k->k "(@[cc.mk_model@ %a@])" pp_full cc);
@ -621,34 +700,3 @@ let mk_model (cc:t) (m:Model.t) : Model.t =
funs
in
Model.add_funs funs m
let check_invariants_ (cc:t) =
Log.debug 5 "(cc.check-invariants)";
Log.debugf 15 (fun k-> k "%a" pp_full cc);
assert (Term.equal (Term.true_ cc.tst) cc.true_.n_term);
assert (Term.equal (Term.false_ cc.tst) cc.false_.n_term);
assert (Vec.is_empty cc.tasks);
(* check that subterms are internalized *)
Term.Tbl.iter
(fun t n ->
assert (Term.equal t n.n_term);
assert (not @@ N.get_field N.field_is_pending n);
relevant_subterms t
(fun u -> assert (Term.Tbl.mem cc.tbl u));
(* check proper signature *)
begin match signature cc t with
| None -> ()
| Some s ->
Log.debugf 15 (fun k->k "(@[cc.check-sig@ %a@ :sig %a@])" Term.pp t Signature.pp s);
(* add, but only if not present already *)
begin match Sig_tbl.find cc.signatures_tbl s with
| exception Not_found -> assert false
| repr_s -> assert (same_class cc n repr_s)
end
end;
)
cc.tbl;
()
let[@inline] check_invariants (cc:t) : unit =
if Util._CHECK_INVARIANTS then check_invariants_ cc

1
src/smt/Congruence_closure.mli
View file

@ -70,4 +70,5 @@ val mk_model : t -> Model.t -> Model.t
(**/**)
val check_invariants : t -> unit
val pp_full : t Fmt.printer
(**/**)

8
src/smt/Equiv_class.ml
View file

@ -1,8 +1,8 @@
open Solver_types
type t = cc_node
type payload = cc_node_payload = ..
type t = equiv_class
type payload = equiv_class_payload = ..
let field_is_active = Node_bits.mk_field()
let field_is_pending = Node_bits.mk_field()
@ -22,6 +22,8 @@ let make (t:term) : t =
n_root=n;
n_expl=E_none;
n_payload=[];
n_next=n;
n_size=1;
n_tags=Util.Int_map.empty;
} in
n
@ -64,7 +66,7 @@ let[@inline] get_field f t = Node_bits.get f t.n_bits
let[@inline] set_field f b t = t.n_bits <- Node_bits.set f b t.n_bits
module Tbl = CCHashtbl.Make(struct
type t = cc_node
type t = equiv_class
let equal = equal
let hash = hash
end)

4
src/smt/Equiv_class.mli
View file

@ -20,8 +20,8 @@ open Solver_types
merged, to detect conflicts and solve equations à la Shostak.
*)
type t = cc_node
type payload = cc_node_payload = ..
type t = equiv_class
type payload = equiv_class_payload = ..
val field_is_active : Node_bits.field
(** The term is needed for evaluation. We must try to evaluate it

3
src/smt/Explanation.ml
View file

@ -3,7 +3,7 @@ open Solver_types
type t = explanation =
| E_reduction (* by pure reduction, tautologically equal *)
| E_merges of (cc_node * cc_node) IArray.t (* caused by these merges *)
| E_merges of (equiv_class * equiv_class) IArray.t (* caused by these merges *)
| E_lit of lit (* because of this literal *)
| E_lits of lit list (* because of this (true) conjunction *)
@ -17,6 +17,7 @@ let mk_lit l : t = E_lit l
let mk_lits = function [x] -> mk_lit x | l -> E_lits l
let mk_reduction : t = E_reduction
let dummy = E_lit Lit.dummy
let[@inline] lit l : t = E_lit l
module Set = CCSet.Make(struct

18
src/smt/Solver_types.ml
View file

@ -26,30 +26,32 @@ and 'a term_view =
If there is a normal form in the congruence class, then the
representative is a normal form *)
and cc_node = {
and equiv_class = {
n_term: term;
mutable n_bits: Node_bits.t; (* bitfield for various properties *)
mutable n_parents: cc_node Bag.t; (* parent terms of the whole equiv class *)
mutable n_root: cc_node; (* representative of congruence class (itself if a representative) *)
mutable n_parents: equiv_class Bag.t; (* parent terms of this node *)
mutable n_root: equiv_class; (* representative of congruence class (itself if a representative) *)
mutable n_next: equiv_class; (* pointer to next element of congruence class *)
mutable n_size: int; (* size of the class *)
mutable n_expl: explanation_forest_link; (* the rooted forest for explanations *)
mutable n_payload: cc_node_payload list; (* list of theory payloads *)
mutable n_tags: (cc_node * explanation) Util.Int_map.t; (* "distinct" tags (i.e. set of `(distinct t1…tn)` terms this belongs to *)
mutable n_payload: equiv_class_payload list; (* list of theory payloads *)
mutable n_tags: (equiv_class * explanation) Util.Int_map.t; (* "distinct" tags (i.e. set of `(distinct t1…tn)` terms this belongs to *)
}
(** Theory-extensible payloads *)
and cc_node_payload = ..
and equiv_class_payload = ..
and explanation_forest_link =
| E_none
| E_some of {
next: cc_node;
next: equiv_class;
expl: explanation;
}
(* atomic explanation in the congruence closure *)
and explanation =
| E_reduction (* by pure reduction, tautologically equal *)
| E_merges of (cc_node * cc_node) IArray.t (* caused by these merges *)
| E_merges of (equiv_class * equiv_class) IArray.t (* caused by these merges *)
| E_lit of lit (* because of this literal *)
| E_lits of lit list (* because of this (true) conjunction *)

2
src/smt/Theory_combine.ml
View file

@ -91,7 +91,7 @@ let assume_real (self:t) (slice:Lit.t Sat_solver.slice_actions) =
let add_formula (self:t) (lit:Lit.t) =
let t = Lit.view lit in
let lazy cc = self.cc in
ignore (C_clos.add cc t : cc_node)
ignore (C_clos.add cc t : Equiv_class.t)
(* propagation from the bool solver *)
let assume (self:t) (slice:_ Sat_solver.slice_actions) =