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Lambda Calculus - Boolean logic.

In this post I will introduce some of the basic concepts of the Lambda Calculus and use them to define basic terms and operators of the boolean logic.

Recently, I was challenged to write a Clojure’s macro called IF which behaves like the clojure.core/if but doesn’t use anything that expands to it. This means that you can exclude pretty much all the usual suspects in the core: cond, condp, and, or, case, cond->, etc.

After feeling a bit lost for about a minute or so, I understood that to solve this challenge I had to go back the primitive element of computation. For example on x86 the if it is implemented as a combination of a comparison operation cmp and a jump operation jz (jump if zero) 1.

My best guess to solve the challenge was to artificially jump to a location. However in Clojure there is no jump instruction so the only way to simulate something similar is to encode the jump location into a map. Therefore my solution was something like:

   (defmacro IF [test t f]
       `(({true  (fn [] ~t)
           false (fn [] ~f)} (boolean ~test))))

In other words:

  • evaluate the test expression and convert the result into a boolean value.

  • create a map with true and false as key and a thunk function as a value which wraps the truthy and falsey expressions.

  • use the map as function to lookup the result of the test.

  • evaluate the resulting thunk by wrapping the expression with an additional pair of brackets ().

You can use this macro pretty much like the clojure.core/if.

   (IF (= 0 1) "OK" "KO")
   ;;=> "KO"

   (IF true "OK" "KO")
   ;;=> "OK"

   (IF false "OK" "KO")
   ;;=> "KO"

Although this works, I wasn’t too happy with the solution. I thought there must be a more elegant solution. Since Clojure is a functional language, I searched inspiration on the foundation of functional programming languages and went back to Alonzo Church and the Lambda Calculus (2-3). The Lambda Calculus defines the concept of functions as computational boxes made only of very, very, very simple elements.

The Lambda Calculus defines the following elements:

  • the λ sign to denote a function.
  • followed by a parameter name
  • then a dot .
  • and followed by an expression which is the body of the lambda.
  • a set of parenthesis can wrap the expression to make it unambiguous.
  • lambdas can optionally be labelled, in which case the label when found in another expression it expands to the lambda which it labels.
      (λx. M)
   ;;   |  \-> body
   ;;   \----> parameter

For example a λ-abstraction (or λ-expression) which increments a number by one would be defined as:

      (λx. x + 1)
   ;;   |  ----
   ;;   |  \-> body
   ;;   \----> variable

Every time the λ-expression is applied to an argument the expression is expanded by replacing the term with its body, and replacing the variables with their values. For example:

   ((λx. x + 1) 3)
   (     3 + 1   )
   ;;=> 4

A λ-expression can also be labelled, once labelled the label can be used in place of the expression and, if applied, it is replaced with its definition.

   INC = (λx. x + 1)
   (INC 3)
   ((λx. x + 1) 3)
   (     3 + 1   )
   ;;=> 4

λ-expressions can also have multiple parameters.

   (λx. λy. x + y)

   ;; which can be simplified as:

   (λxy. x + y)

Boolean logic.

Let’s define TRUE as a λ-expression which takes two parameters and returns the first.

   TRUE = (λxy. x)

Similarly, FALSE takes two parameters, but returns the second one:

   FALSE = (λxy. y)

Once we defined the basic boolean values then we can define logical operators such as: AND, OR and NOT.

   AND = (λxy. (x (y TRUE FALSE) FALSE))
   OR  = (λxy. (x TRUE (y TRUE FALSE)))
   NOT = (λb.  (b FALSE TRUE)

Now let’s try to test this logical operators:

   (AND TRUE FALSE)

   ;; expanding AND
   ((λxy. (x    (y     TRUE FALSE) FALSE)) TRUE FALSE)

   ;; replacing values into expansion
   (       TRUE (FALSE TRUE FALSE) FALSE)

   ;; expanding TRUE
   ((λxy. x) (FALSE TRUE FALSE) FALSE)

   ;; replacing values into expansion
   (FALSE TRUE FALSE)

   ;; expanding FALSE
   ((λxy. y) TRUE FALSE)

   ;; replacing values into expansion
   ;; which is also the final result.
   FALSE

Once we have these three basic operators we can implement the entire boolean logic including something that behaves like IF.

   IF = (λbxy. ((b TRUE FALSE) x y)

Where b is the result of the boolean logic predicate expression, x is the value to return when the b is true, y otherwise.

   (IF TRUE  "OK" "FAIL")
   ;;=> "OK"

   (IF FALSE "OK" "FAIL")
   ;;=> "FAIL"

Clojure is unsurprisingly very similar to the Lambda Calculus definition

For example:

   ;; Lambda Calculus' λ-expression
   (λxy. x + y)

   ;; Clojure's λ-expression
   (fn [x y] (+ x y))

and with the label:

   ;; Lambda Calculus' λ-expression
   SUM = (λxy. x + y)

   ;; Clojure
   (def SUM (fn [x y] (+ x y)))

So if we redefine everything using Clojure’s lambdas we get something like:

   (def TRUE  (fn [x y] x))
   (def FALSE (fn [x y] y))

   (def NOT   (fn [b] (b FALSE TRUE)))

   (NOT TRUE) ;;=> FALSE

   (def AND (fn [x y]
              (x (y TRUE FALSE) FALSE)))

   (AND FALSE TRUE) ;;=> FALSE
   (AND TRUE TRUE)  ;;=> TRUE

   (def OR (fn [x y]
             (x TRUE (y TRUE FALSE))))

   (OR FALSE FALSE) ;;=> FALSE
   (OR FALSE TRUE)  ;;=> TRUE

   (def IF
     (fn [b x y]
       ((b TRUE FALSE) x y)))

   (IF (NOT FALSE)      "OK" "FAIL") ;;=> "OK"
   (IF (AND FALSE TRUE) "OK" "FAIL") ;;=> "FAIL"

At this point we pretty much have everything we need, with the only difference that the evaluation of the λ-expressions in Lambda Calculus is always delayed (lazy evaluation). In Clojure, function evaluation is done by first evaluating all the parameters, and then evaluating the function itself. However, Clojure macros (and special forms) take in input the forms (rather than the values) allowing more fine grained control.

For example, our new IF function will evaluate both branches of the expression before calling the function itself.

   ;; note: both branches are evaluated.
   (IF TRUE (println "OK") (println "FAIL"))
   OK
   FAIL
   ;;=> nil

To match the behaviour of clojure.core/if we need to evaluate only the branch which is returned. So like I did in my first IF implementation I have to turn it into a macro and wrap each branch into a thunk.

   (defmacro IF [b x y]
       `(((~b TRUE FALSE) (fn [] ~x) (fn [] ~y))))

   ;; note: only the correct branch is evaluated.
   (IF TRUE (println "OK") (println "FAIL"))
   OK
   ;;=> nil

Conclusions

It has been an interesting journey to the origins of computational theory and functional programming theory. It is fascinating to see that it is possible to build pretty much anything out of very very simple elements. Lambda Calculus has no concept of boolean logic or branching operations, and yet we managed to build all common boolean logic operators and the if special form.

The Lambda Calculus has much more to offer. Reduction logic and combinators deserve a post on their own.

As final note there is to say that clojure.core/case 4 doesn’t expand to if but, actually, it uses a technique which is similar to my first solution. A map is built for every case and a thunk is associated with every key. A the time I wrote my solution I was unaware of this.

Final code.

Here all the final code.

   (def TRUE  (fn [x y] x))
   (def FALSE (fn [x y] y))

   (def NOT   (fn [b] (b FALSE TRUE)))

   (NOT TRUE) ;;=> FALSE

   (def AND (fn [x y]
              (x (y TRUE FALSE) FALSE)))

   (AND FALSE TRUE) ;;=> FALSE
   (AND TRUE TRUE)  ;;=> TRUE

   (def OR (fn [x y]
             (x TRUE (y TRUE FALSE))))

   (OR FALSE FALSE) ;;=> FALSE
   (OR FALSE TRUE)  ;;=> TRUE

   (defmacro IF [b x y]
       `(((~b TRUE FALSE) (fn [] ~x) (fn [] ~y))))

   (IF TRUE (println "OK") (println "FAIL"))
   OK
   ;;=> nil

References

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