Transforming a symbolic variable into a function

Question

I have a variable of the general form

variable = 
  Subscript[f, {a, b}] Subscript[f, {c, d}] + 
    Subscript[f, {a, c}] Subscript[f, {b, d}]

I would like to transform it something of the form:

f[x_, y_]:= x + y
variable[a_, b_, c_, d_] := f[a, b] f[c, d] + f[a, c] f[b, d]

After seeing some of the answers, perhaps I should pinpoint my problem, the variable I used here is just the general form of a long expression I got after some other calculations and eventualy a Series expansion, I tried to use replacement rules but got the next notice: "SetDelayed: Tag SeriesData in (a lot of symbols) is Protected"

Will appreciate any help!

Answer 1

First you must correct your definition of f to

f[x_, y_] := x + y

Then given

var =
  Subscript[f, {a, b}] Subscript[f, {c, d}] + 
    Subscript[f, {a, c}] Subscript[f, {b, d}];

you can write

 v[a_, b_, c_, d_] = var /. Subscript[f_, {u_, v_}] -> f[u, v];

Then

 v[a, b, c, d]

gives

(a + c) (b + d) + (a + b) (c + d)

If really want use the name var to refer to v, you can write

var = v

making var an alias for v, but I don't recommend that because you will lose the original definition of var

Answer 2

If I'm understanding your question, a replacement should do well.

variable = Subscript[f,{a,b}] Subscript[f,{c,d}] + Subscript[f,{a,c}] Subscript[f,{b,d}]
result = ReplaceAll[Subscript[f_,{x_,y_}] -> f[x,y]][variable]

For a more functional approach:

variable[a_,b_,c_,d_]:= Subscript[f,{a,b}] Subscript[f,{c,d}] + Subscript[f,{a,c}] Subscript[f,{b,d}]
g[a_,b_,c_,d_]:= variable[a,b,c,d]//ReplaceAll[Subscript[f_,{x_,y_}] -> f[x,y]]

Substitute any function of your choice for f using the following:

g[a,b,c,d] /. {f -> MyFunction}

Answer 3

Update: For more general inputs:

ClearAll[makeFunction]
makeFunction[v_] := Module[{vars = Sort@DeleteDuplicates@
       Flatten@Cases[v, Subscript[_, x_] :> x, {0, Infinity}]},
       v /. Subscript -> (# @@ #2 &) /. Thread[vars -> {##}]] &

Examples:

variable = Subscript[f, {a, b}] Subscript[f, {c, d}] + 
   Subscript[f, {a, c}] Subscript[f, {b, d}];
variable2 = Subscript[f1, {a, b, c}] + Subscript[f2, {c, d, e}] + 
   Subscript[f3, {e, f, g}] Subscript[f4, {g, h, i}];
variable3 = Subscript[f1, {a, b, c}] + Subscript[f2, {c, d, e}] + 
   Subscript[f3, {e, f, g}] Subscript[f4, {g, h, i}]^2;

Column[{variable, makeFunction[variable][a, b, c, d]}]

Column[{variable2, makeFunction[variable2] @@ (Symbol["x" <> ToString[#]] & /@ Range[9])}]

Column[{variable2, makeFunction[variable2] @@ Range[9]}]

Column[{variable3, makeFunction[variable3] @@ Range[9]}]

Original answer:

ClearAll[vF]
vF[w_, x_, y_, z_] := variable /. Subscript->(# @@ #2 &) /. 
      Thread[{a, b, c, d} -> {w, x, y, z}]

vF[a, b, c, d]

f[a, c] f[b, d] + f[a, b] f[c, d]

vF[u, v, w, x]

f[u, w] f[v, x] + f[u, v] f[w, x]

vF[1, 2, 3, 4]

f[1, 3] f[2, 4] + f[1, 2] f[3, 4]

If f is pre-defined:

f[x_, y_] := x + y;
vF[a, b, c, d]

(a + c) (b + d) + (a + b) (c + d)

vF[u, v, w, x]

(u + w) (v + x) + (u + v) (w + x)

vF[1, 2, 3, 4]

45