Using Clojure’s core.logic to Solve Simple Number Puzzles

I’ll start by saying I’m relatively new to the ideas of Logic Programming and Declarative Programming in general, having spent most of my developer life so far writing (imperative) PHP code.

The idea of both Logic and Declarative programming paradigms is simple: your job as a developer is to write what results you want your code to give you, rather than how it should go about calculating those results.

I’m going to use Clojure’s core.logic library to run through some very basic examples—I won’t go into how to run the Clojure code—finishing by solving some very basic number puzzles with our new Logic Programming skills.

With a fresh Leiningen project, let’s first get hold of core.logic by updating our project.clj :

:dependencies [[ org.clojure/clojure "1.5.1" ] [ org.clojure/core.logic "0.8.7" ]]

Our ns will need to look something like this to pull in core.logic:

( ns number-puzzles.core ( :refer-clojure :exclude [ == ]) ;; Prevent ns conflict ( :require [ clojure.core.logic :refer :all ]))

This is what core.logic’s syntax looks like (I recommend reading the primer):

( run* [ q ] ;; Logic expressions here )

A core.logic program is written inside a call to run* . We also need to give our main logic variable a name, q seems to be the convention. This logic variable (or lvar) is special: whatever value we give it later will be the output from the program.

Underneath this, our logic expressions will be any number of constraints used to determine what our results are.

We could, for example, write a constraint that says the value of q must always be 10. For this we’d use core.logic’s unify operator ==

( run* [ q ] ( == q 10 )) ;; -> (10)

When this program is run, core.logic will return all values that satisfy all of the given constraints. Here we’ll have the result (10) as 10 is the only possible value that can satisfy our constraint.

If we added a second constraint stating that q must equal 20…

( run* [ q ] ( == q 10 ) ( == q 20 )) ;; -> ()

…our result list would be empty—no value could be found that satisfies both of our constraints (because no value can be both 10 and 20).

In order to describe the logical constraints of our program, we often require more logic variables than just the q we start with. For this, we can use core.logic’s fresh to create new lvars:

( run* [ q ] ( fresh [ a ] ( == a 10 ) ( == q a ))) ;; -> (10)

Here we have two constraints: firstly that a must equal the value 10; secondly that the value of q must equal the value of a . The only value of q that can satisfy both these constraints is 10 , so again the output is a list with a single value (10) .

If we run the following, which unifies our main lvar q with a fresh lvar a , but makes no further constraints…

( run* [ q ] ( fresh [ a ] ( == q a ))) ;; -> (_0)

…we’re given the strange output _0 . This means that our lvar could take on any value, and nothing more specific can be said than that.

Solving a Newspaper Number Puzzle

Here’s a puzzle I found recently in the i newspaper, which we’re going to solve:

The rules: the empty boxes must be filled with the numbers 1–9 to satisfy the horizontal and vertical calculations; each number can only appear once; calculations should be performed left-to-right and top-to-bottom (no BODMAS).

The puzzle is simply a set of logical constraints, so rather than thinking about how we would imperatively write an algorithm to solve it, let’s see how we can use Logic Programming to sidestep this entirely.

Our first step will be to create all the lvars we’ll need, and set up the output format:

( run* [ q ] ( fresh [ a0 a1 a2 ;; Top row b0 b1 b2 ;; Middle row c0 c1 c2 ] ;; Bottom row ;; Unify q with our lvars in the output format we want ( == q [[ a0 a1 a2 ] ;; Top row [ b0 b1 b2 ] ;; Middle row [ c0 c1 c2 ]]))) ;; Bottom row

Here you can see we’re unifying q with some nested vectors containing our lvars, which is how we set up the format of the results. If we run this, we get the following output:

([[ _0 _1 _2 ] [ _3 _4 _5 ] [ _6 _7 _8 ]])

As we’ve not set up any constraints, we just get nine of the _0 , _1 , etc. symbols we saw earlier.

Let’s start adding the rules of the puzzle. Firstly, our values must be in the range 1-9. For this we can use core.logic’s finite domain (FD) tools, which can be found under the clojure.core.logic.fd namespace:

( ns logic.core ( :refer-clojure :exclude [ == ]) ( :require [ clojure.core.logic :refer :all ]) ( :require [ clojure.core.logic.fd :as fd ]))

Here’s our first constraint:

;; State that every one of our lvars should be in the range 1-9 ( fd/in a0 a1 a2 b0 b1 b2 c0 c1 c2 ( fd/interval 1 9 ))

Next, we need to ensure that each of our lvars contains a different number. core.logic’s distinct does the trick:

;; State that each of our lvars should be unique ( fd/distinct [ a0 a1 a2 b0 b1 b2 c0 c1 c2 ])

Now we can go ahead and add all of the mathematical constraints. core.logic has FD operators for this purpose, such as fd/+ , fd/- , fd/* , etc., or we can use fd/eq , which is a helpful macro that lets us write our FD constraints with normal Clojure operators ( + , - , * , etc.):

( fd/eq ;; Horizontal conditions for the puzzle ( = ( - ( * a0 a1 ) a2 ) 22 ) ( = ( - ( * b0 b1 ) b2 ) -1 ) ( = ( + ( * c0 c1 ) c2 ) 72 ) ;; Vertical conditions for the puzzle ( = ( * ( + a0 b0 ) c0 ) 25 ) ( = ( - ( - a1 b1 ) c1 ) -4 ) ( = ( + ( * a2 b2 ) c2 ) 25 ) ;; And finally, in the puzzle we are told that the top left ;; number (a0) is 4. ( = a0 4 ))

Putting it all together:

( ns logic.core ( :refer-clojure :exclude [ == ]) ( :require [ clojure.core.logic :refer :all ]) ( :require [ clojure.core.logic.fd :as fd ])) ;; Use run* to retrieve all possible solutions ( run* [ q ] ;; Create some new logic vars (lvars) for us to use in our rules ( fresh [ a0 a1 a2 ;; Top row b0 b1 b2 ;; Middle row c0 c1 c2 ] ;; Bottom row ;; Unify q with our lvars in the output format we want ( == q [[ a0 a1 a2 ] [ b0 b1 b2 ] [ c0 c1 c2 ]]) ;; State that every one of our lvars should be in the range 1-9 ( fd/in a0 a1 a2 b0 b1 b2 c0 c1 c2 ( fd/interval 1 9 )) ;; State that each of our lvars should be unique ( fd/distinct [ a0 a1 a2 b0 b1 b2 c0 c1 c2 ]) ;; fd/eq is just a helper to allow us to use standard Clojure ;; operators like + instead of fd/+ ( fd/eq ;; Horizontal conditions for the puzzle ( = ( - ( * a0 a1 ) a2 ) 22 ) ( = ( - ( * b0 b1 ) b2 ) -1 ) ( = ( + ( * c0 c1 ) c2 ) 72 ) ;; Vertical conditions for the puzzle ( = ( * ( + a0 b0 ) c0 ) 25 ) ( = ( - ( - a1 b1 ) c1 ) -4 ) ( = ( + ( * a2 b2 ) c2 ) 25 ) ;; And finally, in the puzzle we are told that the top left ;; number (a0) is 4. ( = a0 4 ))))

And here are our results. As you can see, all nine cells now have values, puzzle solved!

([[ 4 6 2 ] [ 1 7 8 ] [ 5 3 9 ]])

For me, the next direction to take this in will be to look further into use cases for Rich Hickey’s Datomic database. Datomic uses a Logic Programming language called Datalog to query its data, and queries are written as a set of logical constraints against lvars in the same way we’ve just discovered.

Being able to describe the results you want and the constraints of your problem in this manner, and not worrying about the underlying implementation, really is very exciting.