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Logic programming is an odd beast. As I lounged around over the holidays I tried to work out how to write a Secret Santa algorithm using core.logic. It seemed like the obvious choice but I quickly realised I’d bitten off more than I could chew.

Simple Secret Santa

Your common or garden Secret Santa consists of putting names in a hat and picking them out again, ensuring you don’t pick yourself. Suitable for offices and other groups of awkward strangers and acquaintances.

We need core.logic imported, so here’s the namespace declaration (note that both clojure.core and clojure.core.logic define == for totally different purposes. I have never used clojure.core/== so I’m a bit hazy as to what it’s for - we exclude it so they don’t clash):

(ns secret-santa.core
  (:refer-clojure :exclude [==])
  (:require [clojure.core.logic :refer :all]
            [clojure.core.logic.pldb :refer :all]))

We’ll start by defining a relation:

(db-rel santa p)

This is so that we can constrain values to only those names we’ve defined are santas (otherwise you get weird logic variables and nobody wants to deal with that noise).

Now for the Secret Santa function. It takes a list of friend names (not actually a requirement that they are friends, they may not be after exchanging presents) and returns a list of lists. Each inner list is a pair of giver and receiver of presents.

(defn simple-secret-santa
  (let [facts (coll->db (for [friend friends]
                           [santa friend]))
        num (count friends)
        givers (repeatedly num lvar)
        receivers (repeatedly num lvar)]
    (->> (with-db facts
           (run* [q]
             (everyg santa givers)
             (everyg santa receivers)
             (distincto givers)
             (distincto receivers)
             (pairupo givers receivers q)))
      (map sort)

If we call it thus:

(simple-secret-santa ["Tommen" "Gregor" "Daenerys" "Arya"])

Then we should get the following as a result:

(["Arya" "Gregor"]
 ["Daenerys" "Tommen"]
 ["Gregor" "Daenerys"]
 ["Tommen" "Arya"])

Please note my chronic lack of imagination evidenced by my lazily using characters from Game of Thrones who would likely not be involved in this kind of tomfoolery.

We need a helper function for pairing people up now:

(defne pairupo
  [givers receivers pairs]
  ([() () ()])
  ([[g . gs] [r . rs] [p . ps]]
   (!= g r)
   (== p [g r])
   (pairupo gs rs ps)))

The mind-bending nature of that function can be rather confusing. In core.logic (as with miniKanren and others), it’s common for an “output value” to be one of the parameters, in this case the 3rd parameter (pairs). This function defines a relationship between the parameters provided so you can call it with givers and pairs and it’ll fill in the blanks for the receivers value (effectively unzipping the pairs value), which is pretty neat.

Unify what?

It’s crucial to understand the clojure.core.logic/== unification function to understand how any of this works. It looks like an equality test from one of those other programming languages we shall not mention here, but it isn’t like that. Sometimes I thought of it as a kind of wishful-thinking-equality - a sort of “wouldn’t it be just lovely if these things were the same”.

Taking an example from the core.logic wiki:

(run* [q]
  (fresh [a]
    (membero a [1 2 3])
    (membero q [3 4 5])
    (== a q)))

In the above we are asking core.logic for the possible values of q (the output variable). We start using a new logic variable a which we unify with q using the == unification function. That is to say, all solutions must satisfy the equation a = q, they must have the same value. Core.logic tends to add ‘o’ to the end of common function names so (membero a [1 2 3]) isn’t a boolean predicate function for testing for list membership, it attempts to make that statement true. With simply that statement - a can be 1 or 2 or 3. The next line asserts that q is a member of the list [3 4 5] so its value can be 3 or 4 or 5. The last line unifies a and q so they must have the same value. We can tell that the only cross-over between the 2 lists already mentioned is the value 3, so q can only be 3. run* returns a list of all possible solutions so in fact we will get a single element list: (3) as the result from that code.

The input of our Secret Santa function will be a list of names (strings), so it’d be useful to turn that list of names into a database that we can constrain our logic functions on:

(defn coll->db
  (apply db coll))

clojure.core.logic.pldb/db takes a variable number of parameters and returns a database with those facts contained. We want to call it with a collection so apply comes in handy here. Effectively you can use apply to turn this: (apply somefn [arg1 arg2 arg3]) into (somefn arg1 arg2 arg3)

How does it work again?

Let’s concentrate on the core part of the function:

(with-db facts
  (run* [q]
    (everyg santa givers)
    (everyg santa receivers)
    (distincto givers)
    (distincto receivers)
    (pairupo givers receivers q)))

We are asking for all the solutions where every item in the givers collection is a santa and every item in the receivers collection is a santa (lines 3 and 4). This ensures that those collections contain only valid santa names and not logic variables or numbers or anything else. There’s something subtle about this - it’s not simply an assertion. core.logic will make those statements true in every way they can. One possible version of that (assuming there were 4 names input to the function) is that givers = ["Arya" "Arya" "Arya" "Arya"]. That’s nonsensical for our purposes (remember we mean to zip up these collections so the first giver is giving to the first receiver, the 2nd giver to the 2nd receiver and so on). So then we ensure that the collection of givers has distinct elements in it (line 5), saying the same about receivers on line 6. If you imagine core.logic has assembled all the combinations of values that are santas first - imagine now it throws away any where values repeat. One of the remaining values for givers will be [“Arya” “Gregor” “Tommen” “Daenerys”]. There will be more. How many?

Derangements, subfactorials, oh my!

The mathematical name for the number of unique permutations of elements in a set (that are different from their original arrangement) is the number of derangements or the subfactorial.

Roughly speaking, Secret Santa is a special case of finding a derangement of names. When you have 4 names, there are 9 derangements.

Here is some clojure that calculates that:

(defn derangements
  "Calculate the number of derangements of n elements."
  (condp = n
    0 1
    1 0
    (* (dec n)
       (+ (derangements (dec n))
          (derangements (- n 2))))))

This algorithm is primarily useful for checking your results in unit tests for example. Another way to use it is as the parameter to run - you can ask for all the possible derangements of friends. I didn’t have much luck with that because I misunderstood the values that my logic code was producing, also there is no need to know beforehand how many combinations are possible since you only want one.

Further improvements

One weakness of the solutions described above is that they model the relationship between the gift giver and receiver as a binary - allowed or not. It would be useful to be able to first attempt to solve the problem under ideal conditions before falling back to less and less ideal solutions. For example, it’s not ideal to have people paired up symmetrically - it’s just a bit boring that way. It’s more optimal to assign Santas in a circle, which makes it slightly more difficult to identify who’s assigned who.

Finding all possible solutions is extremely slow in core.logic (at least, the way I’ve written it) so this could do with a fair amount of optimisation.

I punted the problem of picking a solution out of possible solutions to the combination of (map sort), distinct and rand-nth. This isn’t really necessary, I could have told core.logic what constitutes a distinct solution (it doesn’t realise that order of pairings doesn’t matter) and then simply picked one. My brain hurt so much by that point that I decided to call it a day and move on to more interesting problems, like

Final thoughts

I had a great deal of difficulty writing the pairupo function, largely because all the various defne/defnu/defna confused me - I still couldn’t tell you what they do. This was partly due to me moving from a real problem (Secret Santa) to the implementation in core.logic on the basis of sketchy logic programming knowledge, so I missed a lot of subtlety related to conde which is crucial for understanding this stuff. The official documentaion for core.logic is extremely sparse also, you are very much on your own if the problem you want to solve isn’t Sudoku or the Typed Lambda Calculus (it boggles my mind that that is on the Examples wiki page, I’m not sure if the intention is to educate or obfuscate there).

I hope that a bit more blogging from mere mortals like myself might help others grok this mind-mending area of programming.