(** Comparison and arithmetic relations over the natural domain. *)
(* TODO: all relations here are hardcoded to naturals. The make_comparison
factory accepts a domain_name string but membership_criteria always uses
Stdlib.compare on Obj.t, which only works correctly for unboxed integers.
Extending to other domains requires a proper typed comparison dispatch. *)
let mk v = { Attribute.value = Obj.repr v }
(** Build a comparison relation over a single domain. [pred] receives the result
of [Stdlib.compare left right]. *)
let make_comparison ~name ~domain_name ~pred ~cardinality ~generator =
let schema =
Schema.empty
|> Schema.add "left" domain_name
|> Schema.add "right" domain_name
in
let membership_criteria : (string -> Merkle.t option) -> Tuple.t -> bool =
fun _tree_of -> function
| Tuple.Materialized m -> (
match
( Tuple.AttributeMap.find_opt "left" m.attributes,
Tuple.AttributeMap.find_opt "right" m.attributes )
with
| Some l, Some r ->
pred (Stdlib.compare l.Attribute.value r.Attribute.value)
| _ -> false)
| Tuple.NonMaterialized _ -> false
in
Relation.make ~hash:None ~name ~schema ~tree:None ~constraints:None
~cardinality ~generator:(Some generator) ~membership_criteria
~provenance:(Relation.Provenance.Base name)
~lineage:(Relation.Lineage.Base name)
(** Enumerate pairs (a, b) where a < b using triangular indexing. *)
let pair_of_nat_lt n =
let nf = float_of_int n in
let r = int_of_float (floor ((1. +. sqrt (1. +. (8. *. nf))) /. 2.)) in
let t_prev = r * (r - 1) / 2 in
let left = n - t_prev in
(left, r)
(** Enumerate all pairs (a, b) via Cantor pairing. *)
let cantor_pair_of_nat n =
let w =
int_of_float (floor ((sqrt (float_of_int ((8 * n) + 1)) -. 1.) /. 2.))
in
let t = w * (w + 1) / 2 in
let b = n - t in
let a = w - b in
(a, b)
let less_than_natural : Relation.t =
let rec generator (position : int option) : Generator.result =
match position with
| Some position ->
let left, right = pair_of_nat_lt position in
let attributes : Attribute.materialized Tuple.AttributeMap.t =
Tuple.AttributeMap.of_list [ ("left", mk left); ("right", mk right) ]
in
Generator.Value
(Tuple.make_materialized ~relation:"less_than" ~attributes, generator)
| None ->
Error
"Cannot produce a randomly enumerated value for less_than over \
naturals."
in
make_comparison ~name:"natural_natural_less_than" ~domain_name:"natural"
~pred:(fun c -> c < 0)
~cardinality:Conventions.Cardinality.AlephZero ~generator
let less_than_or_equal_natural : Relation.t =
let rec generator (position : int option) : Generator.result =
match position with
| Some n ->
let a, b = cantor_pair_of_nat n in
(* a <= b is always true when a <= b; we enumerate all (a, b) with a <= b *)
let left = min a b in
let right = max a b in
let attributes =
Tuple.AttributeMap.of_list [ ("left", mk left); ("right", mk right) ]
in
Generator.Value
( Tuple.make_materialized ~relation:"less_than_or_equal" ~attributes,
generator )
| None -> Error "Cannot enumerate less_than_or_equal randomly."
in
make_comparison ~name:"natural_natural_less_than_or_equal"
~domain_name:"natural"
~pred:(fun c -> c <= 0)
~cardinality:Conventions.Cardinality.AlephZero ~generator
let greater_than_natural : Relation.t =
let rec generator (position : int option) : Generator.result =
match position with
| Some n ->
let left, right = pair_of_nat_lt n in
(* Swap: right > left *)
let attributes =
Tuple.AttributeMap.of_list [ ("left", mk right); ("right", mk left) ]
in
Generator.Value
( Tuple.make_materialized ~relation:"greater_than" ~attributes,
generator )
| None -> Error "Cannot enumerate greater_than randomly."
in
make_comparison ~name:"natural_natural_greater_than" ~domain_name:"natural"
~pred:(fun c -> c > 0)
~cardinality:Conventions.Cardinality.AlephZero ~generator
let greater_than_or_equal_natural : Relation.t =
let rec generator (position : int option) : Generator.result =
match position with
| Some n ->
let a, b = cantor_pair_of_nat n in
let left = max a b in
let right = min a b in
let attributes =
Tuple.AttributeMap.of_list [ ("left", mk left); ("right", mk right) ]
in
Generator.Value
( Tuple.make_materialized ~relation:"greater_than_or_equal" ~attributes,
generator )
| None -> Error "Cannot enumerate greater_than_or_equal randomly."
in
make_comparison ~name:"natural_natural_greater_than_or_equal"
~domain_name:"natural"
~pred:(fun c -> c >= 0)
~cardinality:Conventions.Cardinality.AlephZero ~generator
let equal_natural : Relation.t =
let rec generator (position : int option) : Generator.result =
match position with
| Some n ->
let attributes =
Tuple.AttributeMap.of_list [ ("left", mk n); ("right", mk n) ]
in
Generator.Value
(Tuple.make_materialized ~relation:"equal" ~attributes, generator)
| None -> Error "Cannot enumerate equal randomly."
in
make_comparison ~name:"natural_natural_equal" ~domain_name:"natural"
~pred:(fun c -> c = 0)
~cardinality:Conventions.Cardinality.AlephZero ~generator
let not_equal_natural : Relation.t =
let rec generator (position : int option) : Generator.result =
match position with
| Some n ->
let a, b = cantor_pair_of_nat n in
let left, right = if a = b then (a, b + 1) else (a, b) in
let attributes =
Tuple.AttributeMap.of_list [ ("left", mk left); ("right", mk right) ]
in
Generator.Value
(Tuple.make_materialized ~relation:"not_equal" ~attributes, generator)
| None -> Error "Cannot enumerate not_equal randomly."
in
make_comparison ~name:"natural_natural_not_equal" ~domain_name:"natural"
~pred:(fun c -> c <> 0)
~cardinality:Conventions.Cardinality.AlephZero ~generator
let plus_natural : Relation.t =
let schema =
Schema.empty |> Schema.add "a" "natural" |> Schema.add "b" "natural"
|> Schema.add "sum" "natural"
in
let rec generator (position : int option) : Generator.result =
match position with
| Some n ->
let a, b = cantor_pair_of_nat n in
let s = a + b in
let attributes =
Tuple.AttributeMap.of_list [ ("a", mk a); ("b", mk b); ("sum", mk s) ]
in
Generator.Value
(Tuple.make_materialized ~relation:"plus" ~attributes, generator)
| None -> Error "Cannot enumerate plus randomly."
in
let membership_criteria : (string -> Merkle.t option) -> Tuple.t -> bool =
fun _tree_of -> function
| Tuple.Materialized m -> (
match
( Tuple.AttributeMap.find_opt "a" m.attributes,
Tuple.AttributeMap.find_opt "b" m.attributes,
Tuple.AttributeMap.find_opt "sum" m.attributes )
with
| Some a, Some b, Some s ->
(Obj.magic a.Attribute.value : int)
+ (Obj.magic b.Attribute.value : int)
= (Obj.magic s.Attribute.value : int)
| _ -> false)
| Tuple.NonMaterialized _ -> false
in
Relation.make ~hash:None ~name:"natural_plus" ~schema ~tree:None
~constraints:None ~cardinality:Conventions.Cardinality.AlephZero
~generator:(Some generator) ~membership_criteria
~provenance:(Relation.Provenance.Base "plus")
~lineage:(Relation.Lineage.Base "plus")
let times_natural : Relation.t =
let schema =
Schema.empty |> Schema.add "a" "natural" |> Schema.add "b" "natural"
|> Schema.add "product" "natural"
in
let rec generator (position : int option) : Generator.result =
match position with
| Some n ->
let a, b = cantor_pair_of_nat n in
let p = a * b in
let attributes =
Tuple.AttributeMap.of_list
[ ("a", mk a); ("b", mk b); ("product", mk p) ]
in
Generator.Value
(Tuple.make_materialized ~relation:"times" ~attributes, generator)
| None -> Error "Cannot enumerate times randomly."
in
let membership_criteria : (string -> Merkle.t option) -> Tuple.t -> bool =
fun _tree_of -> function
| Tuple.Materialized m -> (
match
( Tuple.AttributeMap.find_opt "a" m.attributes,
Tuple.AttributeMap.find_opt "b" m.attributes,
Tuple.AttributeMap.find_opt "product" m.attributes )
with
| Some a, Some b, Some p ->
(Obj.magic a.Attribute.value : int)
* (Obj.magic b.Attribute.value : int)
= (Obj.magic p.Attribute.value : int)
| _ -> false)
| Tuple.NonMaterialized _ -> false
in
Relation.make ~hash:None ~name:"natural_times" ~schema ~tree:None
~constraints:None ~cardinality:Conventions.Cardinality.AlephZero
~generator:(Some generator) ~membership_criteria
~provenance:(Relation.Provenance.Base "times")
~lineage:(Relation.Lineage.Base "times")
let minus_natural : Relation.t =
let schema =
Schema.empty |> Schema.add "a" "natural" |> Schema.add "b" "natural"
|> Schema.add "difference" "natural"
in
let rec generator (position : int option) : Generator.result =
match position with
| Some n ->
(* Enumerate pairs where a >= b, so difference >= 0 *)
let b, diff = cantor_pair_of_nat n in
let a = b + diff in
let attributes =
Tuple.AttributeMap.of_list
[ ("a", mk a); ("b", mk b); ("difference", mk diff) ]
in
Generator.Value
(Tuple.make_materialized ~relation:"minus" ~attributes, generator)
| None -> Error "Cannot enumerate minus randomly."
in
let membership_criteria : (string -> Merkle.t option) -> Tuple.t -> bool =
fun _tree_of -> function
| Tuple.Materialized m -> (
match
( Tuple.AttributeMap.find_opt "a" m.attributes,
Tuple.AttributeMap.find_opt "b" m.attributes,
Tuple.AttributeMap.find_opt "difference" m.attributes )
with
| Some a, Some b, Some d ->
let av = (Obj.magic a.Attribute.value : int) in
let bv = (Obj.magic b.Attribute.value : int) in
let dv = (Obj.magic d.Attribute.value : int) in
av - bv = dv && dv >= 0
| _ -> false)
| Tuple.NonMaterialized _ -> false
in
Relation.make ~hash:None ~name:"natural_minus" ~schema ~tree:None
~constraints:None ~cardinality:Conventions.Cardinality.AlephZero
~generator:(Some generator) ~membership_criteria
~provenance:(Relation.Provenance.Base "minus")
~lineage:(Relation.Lineage.Base "minus")
let divide_natural : Relation.t =
let schema =
Schema.empty |> Schema.add "a" "natural" |> Schema.add "b" "natural"
|> Schema.add "quotient" "natural"
|> Schema.add "remainder" "natural"
in
let rec generator (position : int option) : Generator.result =
match position with
| Some n ->
(* Enumerate (b, q, r) triples where b > 0, then a = b*q + r, 0 <= r < b *)
(* We use a nested Cantor pairing: n -> (b_idx, qr_idx) -> (q, r) *)
let b_idx, qr_idx = cantor_pair_of_nat n in
let b = b_idx + 1 in
(* b > 0 *)
let q, r_raw = cantor_pair_of_nat qr_idx in
let r = r_raw mod b in
let a = (b * q) + r in
let attributes =
Tuple.AttributeMap.of_list
[
("a", mk a); ("b", mk b); ("quotient", mk q); ("remainder", mk r);
]
in
Generator.Value
(Tuple.make_materialized ~relation:"divide" ~attributes, generator)
| None -> Error "Cannot enumerate divide randomly."
in
let membership_criteria : (string -> Merkle.t option) -> Tuple.t -> bool =
fun _tree_of -> function
| Tuple.Materialized m -> (
match
( Tuple.AttributeMap.find_opt "a" m.attributes,
Tuple.AttributeMap.find_opt "b" m.attributes,
Tuple.AttributeMap.find_opt "quotient" m.attributes,
Tuple.AttributeMap.find_opt "remainder" m.attributes )
with
| Some a, Some b, Some q, Some r ->
let av = (Obj.magic a.Attribute.value : int) in
let bv = (Obj.magic b.Attribute.value : int) in
let qv = (Obj.magic q.Attribute.value : int) in
let rv = (Obj.magic r.Attribute.value : int) in
bv > 0 && av = (bv * qv) + rv && rv >= 0 && rv < bv
| _ -> false)
| Tuple.NonMaterialized _ -> false
in
Relation.make ~hash:None ~name:"natural_divide" ~schema ~tree:None
~constraints:None ~cardinality:Conventions.Cardinality.AlephZero
~generator:(Some generator) ~membership_criteria
~provenance:(Relation.Provenance.Base "divide")
~lineage:(Relation.Lineage.Base "divide")
let prelude_relations =
[ less_than_natural
; less_than_or_equal_natural
; greater_than_natural
; greater_than_or_equal_natural
; equal_natural
; not_equal_natural
; plus_natural
; times_natural
; minus_natural
; divide_natural ]
