datatype term = Constant of int
| Variable of ad_number
| Compound of int*(term list)
val make_p = 0
val make_q = 1
val make_x = (Base 2.0)
fun same (Variable variable1) (Variable variable2) =
variable1<=variable2 andalso variable2<=variable1
fun binds [] _ = false
| binds ((variable1, _)::substitution) variable2 =
if same variable1 variable2 then true else binds substitution variable2
fun lookup_value variable1 ((variable2, value)::substitution) =
if same variable1 variable2
then value
else lookup_value variable1 substitution
fun apply_substitution substitution (term as (Constant _)) = term
| apply_substitution substitution (term as (Variable _)) =
if binds substitution term then lookup_value term substitution else term
| apply_substitution substitution (Compound (function, terms)) =
Compound (function, (map (fn term => apply_substitution substitution term)
terms))
fun apply_substitution_to_clause substitution (p, term, terms) =
(p,
(apply_substitution substitution term),
(map (fn term => apply_substitution substitution term) terms))
fun member variable [] = false
| member variable1 (variable2::set) =
if same variable1 variable2 then true else member variable1 set
fun union ([], set) = set
| union ((element::set1), set2) =
if member element set2
then union (set1, set2)
else element::(union (set1, set2))
fun variables_in (Constant _) = []
| variables_in (term as (Variable _)) = [term]
| variables_in (Compound (function, terms)) =
foldl union [] (map variables_in terms)
fun variables_in_clause (p, term, terms) =
union ((variables_in term), (foldl union [] (map variables_in terms)))
fun occurs variable term = member variable (variables_in term)
fun unify (Constant constant1) (Constant constant2) =
if constant1=constant2 then SOME [] else NONE
| unify (term1 as (Constant _)) (term2 as (Variable _)) =
SOME [(term2, term1)]
| unify (Constant _) (Compound _) = NONE
| unify (term1 as (Variable _)) (term2 as (Constant _)) =
SOME [(term1, term2)]
| unify (term1 as (Variable _)) (term2 as (Variable _)) =
SOME [(term1, term2)]
| unify (term1 as (Variable _)) (term2 as (Compound _)) =
if occurs term1 term2 then NONE else SOME [(term1, term2)]
| unify (Compound _) (Constant _) = NONE
| unify (term1 as (Compound _)) (term2 as (Variable _)) =
if occurs term2 term1 then NONE else SOME [(term2, term1)]
| unify (Compound (function1, terms1)) (Compound (function2, terms2)) =
if function1=function2
then let fun loop [] [] substitution = SOME substitution
| loop (term1::terms1) (term2::terms2) substitution =
(case unify (apply_substitution substitution term1)
(apply_substitution substitution term2) of
NONE => NONE
| SOME substitution1 =>
loop terms1 terms2 (substitution1@substitution))
| loop _ _ _ = NONE
in loop terms1 terms2 [] end
else NONE
fun map_indexed f l =
let fun loop [] i = []
| loop (h::t) i = (f h i)::(loop t (i+(Base 1.0)))
in loop l (Base 0.0) end
fun alpha_rename clause offset =
apply_substitution_to_clause
(map_indexed
(fn variable => (fn i => (variable, (Variable (offset+i)))))
(variables_in_clause clause))
clause
fun proof_distribution term clauses =
let val offset = foldl (fn (x, y) => if x<=y then y else x)
(Base 0.0)
(map (fn (Variable variable) => variable)
(foldl union
[]
(map variables_in_clause clauses)))
in foldl op @
[]
(map (fn clause =>
let val (p, term1, terms) = alpha_rename clause offset
in let fun loop p substitution terms =
case substitution of
NONE => []
| SOME substitution =>
case terms of
[] => [(p, substitution)]
| term::terms =>
foldl op @
[]
(map (fn (p1, substitution1) =>
loop (p*p1)
(SOME (substitution@
substitution1))
terms)
(proof_distribution
(apply_substitution
substitution
term)
clauses))
in loop p (unify term term1) terms end end)
clauses) end
fun likelihood substitution_distribution =
foldl op + (Base 0.0) (map (fn (p, _) => p) substitution_distribution)
fun example _ =
(gradient_ascent_R
(fn [p0, p1] =>
let val clauses = [(p0, (Compound (make_p, [(Constant 0)])), []),
(((Base 1.0)-p0),
(Compound (make_p, [(Variable make_x)])),
[(Compound (make_q, [(Variable make_x)]))]),
(p1, (Compound (make_q, [(Constant 1)])), []),
(((Base 1.0)-p1),
(Compound (make_q, [(Constant 2)])),
[])]
in foldl op *
(Base 1.0)
(map (fn observation =>
likelihood
(proof_distribution
(Compound (make_p,
[(Constant observation)]))
clauses))
[0, 1, 2, 2]) end)
[(Base 0.5), (Base 0.5)]
(Base 1000.0)
(Base 0.1))
val _ =
let fun loop i result =
if i<=(Base 0.0) andalso (Base 0.0)<= i
then result
else let val ([p0, p1], _, _) = example ()
in loop (i-(Base 1.0)) [(write_real p0), (write_real p1)] end
in loop (Base 1000.0) [(Base 0.0), (Base 0.0)] end