(** Predefined fundamental domains. Each domain carries a generator (for
enumeration) and membership criteria (for validation). Values are stored as
{!Conventions.AbstractValue.t} via [Obj.magic], following the same
convention as {!Standard}. *)
(** Bijection from natural number to integer: 0->0, 1->1, 2->-1, 3->2, 4->-2,
... *)
let nat_to_int n = if n = 0 then 0 else if n mod 2 = 1 then (n + 1) / 2 else -(n / 2)
(** Cantor pairing inverse: given position [n], returns a pair [(a, b)] of
natural numbers such that [a + b] lies on the [n]-th diagonal. *)
let cantor_unpair 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
(** Integer domain: all integers, enumerated via a bijection from the naturals.
Schema: [value: integer]. *)
let integer : Relation.domain =
let rec generator (position : int option) : Generator.result =
match position with
| Some n ->
let v = nat_to_int n in
let attributes = Tuple.AttributeMap.of_list ["value", Obj.magic v] in
Generator.Value (Tuple.make_materialized ~relation:"integer" ~attributes, generator)
| None -> Generator.Error "Cannot produce a randomly enumerated value for the integer domain."
in
let membership_criteria : Tuple.t -> bool = function
| Tuple.Materialized _ -> true
| Tuple.NonMaterialized _ -> false
in
let schema = Schema.empty |> Schema.add "value" "integer" in
new Relation.domain
~name:"integer" ~generator ~membership_criteria ~schema ~constraints:None
~cardinality:Conventions.Cardinality.AlephZero ~lineage:None ~provenance:None
(** Natural number domain: non-negative integers (0, 1, 2, …). Schema:
[value: natural]. *)
let natural : Relation.domain =
let rec generator (position : int option) : Generator.result =
match position with
| Some n ->
let attributes = Tuple.AttributeMap.of_list ["value", Obj.magic n] in
Generator.Value (Tuple.make_materialized ~relation:"natural" ~attributes, generator)
| None -> Generator.Error "Cannot produce a randomly enumerated value for the natural domain."
in
let membership_criteria : Tuple.t -> bool = function
| Tuple.Materialized m -> (
match Tuple.AttributeMap.find_opt "value" m.attributes with
| Some attr -> (Obj.magic attr : int) >= 0
| None -> false )
| Tuple.NonMaterialized _ -> false
in
let schema = Schema.empty |> Schema.add "value" "natural" in
new Relation.domain
~name:"natural" ~generator ~membership_criteria ~schema ~constraints:None
~cardinality:Conventions.Cardinality.AlephZero ~lineage:None ~provenance:None
(** Rational number domain: pairs [(numerator, denominator)] where the
denominator is never zero. The generator enumerates all such pairs via
Cantor pairing on the integers. Schema:
[numerator: integer, denominator: integer]. *)
let rational : Relation.domain =
let rec generator (position : int option) : Generator.result =
match position with
| Some n ->
let a, b = cantor_unpair n in
let num = nat_to_int a in
(* Map b -> non-zero integer: 0->1, 1->-1, 2->2, 3->-2, … *)
let den = if b mod 2 = 0 then (b / 2) + 1 else -((b + 1) / 2) in
let attributes =
Tuple.AttributeMap.of_list ["numerator", Obj.magic num; "denominator", Obj.magic den]
in
Generator.Value (Tuple.make_materialized ~relation:"rational" ~attributes, generator)
| None -> Generator.Error "Cannot produce a randomly enumerated value for the rational domain."
in
let membership_criteria : Tuple.t -> bool = function
| Tuple.Materialized m -> (
match Tuple.AttributeMap.find_opt "denominator" m.attributes with
| Some attr -> (Obj.magic attr : int) <> 0
| None -> false )
| Tuple.NonMaterialized _ -> false
in
(* Note: Additionally to the membership criteria, we add the constraint here as this relation
can be further dismembered, rendering the denominator into a lone integer, that should
propagate the constraint `integer <> 0`. If we declared a domain `integer_without_zero`,
then this constraint would not be needed as the domain would not be dissociated from the value. *)
let constraints : Relation.RelationConstraint.t option =
let denominator_not_zero =
let open Constraint in
Not {body= Eq {left= Var "denominator"; right= Const (Obj.magic 0)}; universe= "integer"}
in
Some (("DENOMINATOR_DIFFERENT_FROM_ZERO", denominator_not_zero) |> List.singleton)
in
let schema =
Schema.empty |> Schema.add "numerator" "integer" |> Schema.add "denominator" "integer"
in
new Relation.domain
~name:"rational" ~generator ~membership_criteria ~schema ~constraints
~cardinality:Conventions.Cardinality.AlephZero ~lineage:None ~provenance:None
(** String domain: any string value. Not enumerable. Schema: [value: string]. *)
let string : Relation.domain =
let generator (_ : int option) : Generator.result =
Generator.Error "Strings are not enumerable."
in
let membership_criteria : Tuple.t -> bool = function
| Tuple.Materialized _ -> true
| Tuple.NonMaterialized _ -> false
in
let schema = Schema.empty |> Schema.add "value" "string" in
new Relation.domain
~name:"string" ~generator ~membership_criteria ~schema
~constraints:None
(* TODO: Implement some form of enumeration for strings with a bijection on naturals. *)
~cardinality:Conventions.Cardinality.Continuum ~lineage:None ~provenance:None
