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Lambda Calculus and Lambda Calculators

 

Lambda-calculators


 

Non-Kleene predecessors: a representation workout

Lambda-calculus is banal in its operation -- and yet is an unending source of delightful puzzles. One of the first one was the predecessor: applied to the term representing a natural number n, it should reduce to the representation of n-1. If the number n is represented as the n-times repeated application, the predecessor amounts to an un-application -- which is not the operation lambda-calculus provides or supports. As Church was about to give up the hope that lambda-calculus could express arithmetic, his student Kleene was getting his wisdom teeth extracted, and under anesthetic (or so Barendregt says) saw the solution. The tooth-wrenching story and Kleene's predecessor have become the part of the Functional Canon, told and retold in tutorials and textbooks. The Kleene predecessor is so familiar that we hardly even think there may be others. There are: I found one back in 1992 after a day of thinking. (Incidentally, I also had a tooth extracted that year, at a different time though.) This article shows that by looking at the puzzle the right way we easily see more and more solutions -- some of which simpler (nearly half the size!) and more intuitive than the canonical Kleene predecessor. We can even represent the un-application.

Recall, the standard and the simplest representation of natural numbers is Church numerals, listed below. (As a common shorthand, we write cn for the term representing the number n.):

     c0: \f\x.x
     c1: \f\x.f x
     c2: \f\x.f (f x)
     ...
The successor -- the term whose application to cn reduces to c(n+1) -- has two equivalent forms:
     succ: \n\f\x. n f (f x)
     succ: \n\f\x. f (n f x)
The problem is to find the predecessor -- the term pred such that the application pred cn reduces to c(n-1). (What should be the result of pred c0 is an open choice; often it is c0.)

The fundamental tautology of Church numerals is easy to overlook:

     cn === cn succ c0
where the equality is understood as beta-convertibility. Its importance appears when viewed as the template to build other representations of natural numbers; let's call these terms pn:
     pn:  cn succp p0        (*)
where succp and p0 are two arbitrary terms. This is no longer a tautology: it is the closed-form expression for pn in lambda-calculus.

So far, the new numerals are the empty formality. They become useful once we relate them to Church numerals. For example, take p0 to be cm, for some m; take p1 to be c(m+1), etc. Clearly, succp is just the ordinary succ. Then pn has the meaning of c(n+m); definition (*) immediately gives the term for adding two numerals:

     add:  \n\m.n succ m

To obtain the Kleene predecessor, let's think of pn as a point between two consecutive numbers cn and c(n-1) on the number line. It can be represented as a pair (c(n-1),cn):

     p0: (cm1, c0)
     p1: (c0, c1)
     p2: (c1, c2) ...
using the standard encoding of pairs in lambda-calculus (the projections are fst and snd). Here, cm1 is the term that we want as the result of applying the predecessor to c0. For example, one may choose cm1 as c0. The successor on those `midpoint numbers' is easy to define:
     succp:  \p.(snd p, succ (snd p))
Definition (*) then gives the closed-form expression for pn, from which we can extract c(n-1) as the first component:
     pred: \n.fst (n succp p0)
or, in the desugared form
     \n. n (\p s. s (p (\x y. y)) (\f x. f (p (\x y. y) f x))) (\s. s (\s z. z) (\s z. z)) (\x y. x)

Albeit slick, the Kleene trick is a bit contrived. For example, succp does not even look at the first component of the pair. The `define new numeral' standpoint immediately brings to mind the simpler:

     p0: Nothing    or, desuraged: \g\y.y
     p1: Just c0    or, desuraged: \g\y.g (\f\x.x)
     p2: Just c1    or, desuraged: \g\y.g (\f\x.f x)
     ...
using the standard encoding of the Maybe type. The succp is then
     succp: \p.Just (p succ c0)
which gives the predecessor as
     pred: \n. (n succp p0) id c0
or, in the desugared form
     \n. n (\p g y. g (p (\n f x. f (n f x)) (\s z. z))) (\g y. y) (\x. x) (\s z. z)
which is a bit shorter. There is room for improvement. Let us factor out the \f\x. `prefix' of any Church numeral, giving us
     pred: \n\f\x. (n (\p.Just (p f x)) p0) id x
Where we had c0 before, we now have x, and succ is replaced with an application of f. The desugared form is:
     \n f x. n (\p g y. g (p f x)) (\g y. y) (\y. y) x
which is one of the shortest predecessors, nearly half the size of Kleene's.

There is more! What can be simpler than

     p0: cm1
     p1: c0
     p2: c1, etc.
We just have to choose cm1 in such a way so to distinguish it from all other Church numerals. That is easy: for any Church numeral cn, the application cn (\z.false) false reduces to false. Thus we may take as cm1:
     cm1: \f\x.true
that returns true no matter to what it is twice applied to. (The discrimination test takes constant time, in the normal evaluation order). With the p-successor defined as
     succp: \p.p (\z.false) false c0 (succ p)

the predecessor becomes

     pred: \n.n succp (\f\x.true)
Or, in the desugared form
     \n. n (\p. p (\z x y. y) (\x y. y) (\s z. z) (\f x. f (p f x))) (\f x x y. x)

Finally, consider a different choice of cm1, based on a different, albeit slower to use, property of Church numerals: for any cn, cn (\x.x) reduces to \x.x. In other words, the identity combinator is the fixpoint of Church numerals. As cm1, we chose something that, when applied to identity, is other than the identity; for example, \x.c0:

     cm1:   \f\x.c0
     succp: \p.p (\x.x) (succ p)
The predecessor then is
     pred: \n.n (\p.p (\x.x) (succ p)) (\f\x.c0)
Or, in the desugared form
     \n. n (\p. p (\x. x) (\f x. f (p f x))) (\f x s z. z)
This is nearly half the size of the Kleene predecessor. This was the predecessor that I found in 1992.

Even the initial puzzle of predecessor as an un-application is solvable. To be sure, lambda-calculus has no un-application primitive, letting us only apply terms but not examine them. However, lambda-calculus can represent all computations, including itself. The representations can be examined and deconstructed to our heart's content. For example, consider

     mynoapp: \x.   \g\y. y x
     myapp:   \f\x. \g\y. g f x

which is an encoding of the data type:

     type myterm = App myterm myterm | NoApp term
representing lambda-applications. If cn represents the natural number n by n repeated applications, let pn represent those repeated applications:
     pn: \f\x. cn (myapp f) (mynoapp x)
More insightfully, pn can be written as reify cn where
     reify: \n\f\x. n (myapp f) (mynoapp x)
gives us the representation of cn (goes up the level of representation). The inverse is reflect, which recovers the numeral from its representation:
     reflect: \p\f\x. fix (\s r. r (\f\x. f (s x)) (\y.y)) (p f x)
For example, if p3 is reify c3 then reflect p3 reduces back to c3. The advantage of representing applications is that we can examine and unapply them, as shown in OCaml notation:
     let unapply = fun t -> match t of | App (_,x) -> x | x -> x
or, in lambda-calculus
     unapp: \t.t (\d m. m) (\d.t)
We obtain the predecessor of cn by first reifying the numeral, removing one application and reflecting back.
     pred: \n. reflect (\f\x. (unapp ((reify n) f x)))

Thus, once again we have seen how important it is to contemplate representations and what they represent, and be able to move between the two. (Bonus to the reader who can find/make a Matrix quote.) Kleene numerals gave us an example that canonical, canonized approaches are not the only ones possible -- and sometimes are not the best. Hence, no matter how much something appears settled, it is worth probing. Finally, one cannot help but wonder at the delicate behavior that arises from trite rules.

Version
The current version is July 2016.
References
Predecessor and lists are not representable in simply typed lambda-calculus

Numerals and lists in lambda-calculus: P-numerals
A different representation of numerals, with the predecessor being just as simple as the successor.

The bluff combinator

 

Numerals and lists in lambda-calculus: P-numerals

This article derives a term in the pure untyped lambda-calculus that efficiently computes the symmetric difference of two Church numerals. We introduce other numerals, so called P-numerals, which are more convenient for arithmetic. We show the connection between the P-numerals and the list fold.

P-numerals are the functional equivalent of lists: increment, decrement, and zero testing operations on P-numerals all take constant time -- just as cons, car, and null? do on lists.

Peter G. Hancock has observed that P-numerals are closely related to Goedel's recursor R.

Paul Brauner has pointed out that P-numerals have been introduced by Michel Parigot back in 1988, with different motivations and in a typed setting. They are since called ``Parigot integers''. Paul Brauner has pointed out yet another way to come to P-numerals, from the definition of integers in deduction modulo.

Andrew Hodges, in `Alan Turing: The Enigma', wrote: ``But underneath there lay the same powerful idea that Goedel has used, that there was no essential distinction between numbers and operations on numbers.'' And so, lambda-calculus and Turing machine are duals: In a (universal) TM, an integer encodes an operation. In lambda-calculus, an operation encodes an integer.

Version
The current version is 1.5, November 2010.
References
LC_Pnum.lhs [13K]
The article itself. This is also a complete, literate Haskell code that defines the numerals and the symmetric difference, gives an example of their use, and compares Church and P-numerals.
The article was originally posted as ``Symmetric difference in LC, P-numerals, and fold'' on comp.lang.functional on Fri Aug 30 01:49:31 2002

Michel Parigot: Programming with proofs: A second order type theory.
LNCS 300, pp. 145-159, 1988.

Peter G. Hancock: The AMEN combinators and their reduction.
Combinatorially complete arithmetic operations, and a rewriting calculus for arithmetical expressions built entirely upon addition, multiplication, exponentiation, and naught.

 

for-any vs for-all

Contrasting Church-numerals with P-numerals gives an opportunity to show the difference between for-any and for-all. Lambda calculus lets us demonstrate the truth of a proposition -- such as the predecessor being the left inverse of the successor -- for any Church numeral, and yet unable to demonstrate it for all numerals. For P-numerals however, we can prove the proposition for all numerals, within the lambda calculus itself. In other words, in stark contrast to Church numerals, for P-numerals the composition pred . succ normalizes to the identity function.

In the notation of the Haskell lambda-calculator, Church-numerals, the successor and the predecessor are defined as follows:

     c1 = f ^ x ^ f # x                    -- Church numeral 1
     succ = n ^ f ^ x ^ f # (n # f # x)
     pred = n ^ f ^ x ^ n # (g ^ h ^ h # (g # f)) # (y ^ x) # (y ^ y)
Taking c1 as an example, we demonstrate taking the successor, then taking the predecessor, and checking the final result being beta-eta-equivalent to the original numeral c1:
     *LC_Pnum> eval (succ # c1)
     (\f. (\x. f (f x)))
     *LC_Pnum> eval (pred # (succ # c1))
     (\f. f)
     *LC_Pnum> eval (pred # (succ # c1)) == eval c1
     True

We can do these operations for any concrete numeral. However, when we try to prove that taking the successor followed by the predecessor being the identity for all numerals, we get stuck. It is easy to define the composition of the predecessor and the successor, and to normalize it:

     *LC_Pnum> let predsucc_c = n ^ pred # (succ # n)
     *LC_Pnum> eval predsucc_c
     (\n. (\f. (\x. n (\g. (\h. h (g f))) (\y. x) f)))
Alas, the normalization result does not look like the the identity function at all. We must use the external, to lambda calculus, induction principle. Here is the induction step:
     *LC_Pnum> eval (predsucc_c # (succ # n))
     (\f. (\x. f (n (\g. (\h. h (g f))) (\y. x) f)))
     *LC_Pnum> eval (succ # (predsucc_c # n))
     (\f. (\x. f (n (\g. (\h. h (g f))) (\y. x) f)))
     *LC_Pnum> eval (succ # (predsucc_c # n)) == eval (predsucc_c # (succ # n))
     True
One is reminded of Robinson Arithmetic, or Q, which is essentially Peano Arithmetic without the induction schema. One can prove in Q that the addition of any two concrete natural numbers is commutative, but one cannot prove the commutativity of addition for all natural numbers.

We now turn to P-numerals, with the following successor and predecessor:

     p0 = c0                         -- the same as Church 0
     succp = n ^ f ^ x ^ f # n # (n # f # x)
     predp = n ^ n # combK # combI
     p1 = succp # p0
(one can use anything in place of the I combinator, combI, in predp. More properly, one should use -1). As before, we check that the predecessor is the left inverse of the successor, for a concrete numeral, p1 in our example:
     *LC_Pnum> eval p1 -- The P-numeral 1
     (\f. f (\f. (\x. x)))
     *LC_Pnum> eval (succp # p1)
     (\f. (\x. f (\f. f (\f. (\x. x))) (f (\f. (\x. x)) x)))
     *LC_Pnum> eval (predp # (succp # p1))
     (\f. f (\f. (\x. x)))
     *LC_Pnum> eval (predp # (succp # p1)) == eval p1
     True
We can likewise show that for any numeral pn, (predp # (succp # p1)) is beta-eta-equivalent to pn. We now prove a stronger claim -- that the predecessor is the left inverse of the successor for all numerals -- within the lambda-calculus itself. We define the composition of predp and succp:
     predsucc = n ^ predp # (succp # n)
and normalize it
     *LC_Pnum> eval predsucc
     (\n. n)
obtaining the identity function, syntactically. Thus the predicate predsucc n == n holds uniformly, that is, for all n.

Although for-any vs for-all distinction is emblematic of intuitionism, Whitehead and Russell were already aware of it when writing Principia Mathematica. They insisted on separate notations for `any' vs. `all' (although admitting the equivalence of these notions in their theory) [Chap 1, p18, paragraph ``Ambiguous assertion and the real variable''].

The article was prompted by a question from Benedikt Ahrens.

Version
The current version is 1.5, November 2010.
References
Numerals and lists in lambda-calculus: P-numerals

 

The bluff combinator

A Bluff combinator problem in the pure untyped lambda-calculus was posed by Mayer Goldberg in the 1990s:

Let c_i denote a Church numeral for the number i. We should design two terms beq and bluff so that the following reductions hold:

     beq c_i c_j     ==> \x\y.x (that is, boolean 'true')
                         iff i == j
                     ==> \x\y.y (that is, boolean 'false') otherwise
     
     beq bluff c_i   ==> \x\y.x (that is, boolean 'true') for all i
     beq c_i bluff   ==> \x\y.x (that is, boolean 'true') for all i
     beq bluff bluff ==> \x\y.y (that is, boolean 'false')

In other words, beq should act as equality on Church numerals. The bluff combinator should be beq-equal to any numeral. However, the combinator should not be bequal to itself. The gist of the problem is discriminating any Church numeral from some other term.

Version
The current version is 1.2, Oct 14, 2002.
References
bluff.lhs [6K]
The complete code that shows one solution to the problem. Other solutions are described in an article below.

Mayer Goldberg and Mads Torgersen: Circumventing Church Numerals. Nordic Journal of Computing, v9, January 2002, pp. 1-12.

 

Alpha-conversion is inevitable

Of all the (three) operations of lambda-calculus, alpha-conversion feels bothersome -- whether we program reductions or do them by hand. We are forced to do it when an existing variable is shadowed -- locally rebound -- by a lambda-subexpression. For example, in ((\x\y.x y) y) the sub-expression \y.x y shadows the (free) variable y. Alpha-conversion renames the locally-bound variable and eliminates the shadowing: (\x\z.x z) y. One often hears that variable shadowing makes programs harder to understand and error-prone, and should be avoided whenever possible, by using distinct variable names. OCaml and GHC Haskell compilers, among others, take this advice to the heart: GHC has a special initial compiler phase that uniquely renames all variables within the program.

This prompts the question: Suppose we take trouble to consciously avoid variable shadowing when writing lambda-terms, choosing fresh (unique) names for bound identifiers. Do we still need alpha-conversion when evaluating such terms? In other words, can we apply alpha-conversion to a term at the very beginning, making all binders distinctly named -- and then forget about alpha-conversion for the rest of the evaluation? How necessary is the alpha-conversion really?

The unique naming of binders indeed makes alpha-conversion unnecessary -- in very many cases. Yet in general it is unavoidable. We show several subtle examples of terms with uniquely named binders that still demand alpha-conversion in the middle of evaluation. In fact, in some cases such `dynamic' alpha-conversion is needed indefinitely many times.

The answer satisfies the old curiosity: why all formal introductions to lambda-calculus start with the phrase like `Let V be an infinite countable set of variable names'. We shall see why lambda-calculus needs infinitely many names. Surprisingly, there are practical implications, for the implementation of staged languages. In fact, one of the examples below is patterned after a bug accidentally discovered in the first implementation of MetaScheme.

To start with, recall that variables in (closed) lambda-terms play the role of pointers, between the binder (lambda) and the places where a future application should insert its argument. Therefore, names per se do not matter, only their equality: \x.x and \y.y denote the same identity function. Formally, identity and other relations between lambda-terms are understood modulo alpha-conversion, the renaming of bound identifiers.

Alpha-conversion is not just an formalization artifact: it is something we may have to really do during the evaluation. Consider the following beta-reductions:

     (\x\y.x y) y  --> \y. y y  -- Wrong!
     (\x\z.x z) y  --> \z. y z
     (\x\u.x u) y  --> \u. y u
The terms to reduce are all equivalent, differing only in the name of one bound variable. The last two reduction results are also equivalent: the eta-expanded version of y. But the first reduction breaks the equivalence. It is therefore prohibited. The term (\x\y.x y) y has to be first alpha-converted, say, to (\x\z.x z) y, before the beta-reduction may be carried out. On the other hand, alpha-conversion feels rather similar to beta-reduction: in either case, we traverse the body of a lambda-abstraction, looking for a particular variable and replacing it. It seems bothersome and inelegant to having to precede a beta-reduction with alpha-conversion, the operation of roughly the same complexity.

In the above -- and many other cases -- alpha-conversion can be averted had we given bound variables unique names to start with. But not in all cases. No matter how variables are named at the beginning, there may come a time during the evaluation when shadowing appears. This happens when substituting a term into itself:

     (\y. y y) (\z.\x. z x)
     -->  (\z.\x. z x) (\z.\x. z x)
     -->  (\x. (\z.\x. z x) x)
To proceed further, alpha-conversion is unavoidable.

In fact, we may have to do alpha-conversion infinitely many times, e.g., when reducing the following term:

     (\y.y s y) (\t\z\x. z (t x) z)
     --> (\t\z\x. z (t x) z) s (\t\z\x. z (t x) z)
     --> (\z\x. z (s x) z) (\t\z\x. z (t x) z)
     --> \x. (\t\z\x. z (t x) z) (s x) (\t\z\x. z (t x) z)
     --> {alpha-conversion is necessary}
     --> \x. (\t\z\x1. z (t x1) z) (s x) (\t\z\x. z (t x) z)
     --> \x. (\z\x1. z (s x x1) z) (\t\z\x. z (t x) z)
     --> \x. \x1. (\t\z\x. z (t x) z) (s x x1) (\t\z\x. z (t x) z)
     --> {another alpha-conversion is necessary, renaming x to something that
          is different from x and x1}
     --> \x. \x1. (\t\z\x2. z (t x2) z) (s x x1) (\t\z\x. z (t x) z)
     --> ...
     

Our examples so far relied on self-applications and were written in the untyped lambda-calculus. One may wonder if the simply-typed lambda-calculus, where self-application is ruled out, might let us escape from alpha-conversion. It does not. The indefinite number of alpha-conversions may be needed even here, as the following example with the second Church numeral demonstrates:

     fun t -> (fun y -> y (y t)) (fun u z x -> u x z);;
     ->
     fun t -> (fun u z x -> u x z) (fun z x -> t x z)
     ->
     fun t -> (fun z x -> (fun z x -> t x z) x z)
The example is written in the OCaml notation, to double-check it is simply typed.

Thus typed or untyped, the alpha-renaming step may have to be done during the evaluation, arbitrarily many times. We may need an inexhaustible supply of fresh names. This is something to keep in mind when implementing staged languages, for example.

Version
The current version is July 2016.

 

Miscellaneous curious equalities

This section collects curious properties of particular lambda-terms. Some of the properties have been used in proving theorems or solving puzzles described on this site. Throughout this section, c_i means a Church numeral for integer i, true stands for the term \x\y.x and false stands for \x\y.y . The symbol ==> indicates reduction in one or several steps.
c_i (\x.false) false ==> false

This property of Church numerals helped to solve the bluff combinator problem.

c_i combI ==> combI

where combI is the identity combinator \x.x. Thus the identity combinator is a fixpoint of any Church numeral.

c_i c_j c_k ==> c_n where n=k^(j^i)

and the operator ^ means exponentiation. The property says that an application of a Church numeral to two other Church numerals is itself a Church numeral, which corresponds to the exponentiated integers.

c_1 === combI, true === K, c_0 === false

Trivial statements that c_1 is the identity combinator and true is the K (a.k.a. const) combinator. In Lambda calculus -- as in C, Perl and other programming languages -- numeral 0 and boolean false are the same.

c_i (\y.x) u ==> x, c_0 (\y.x) u ==> u, i>0

This property gives a simple test for discriminating c_0 from any other Church numeral.

cons a x (cons b y) ==> a b y x

where cons is \x\y\p.p x y.

Version
The current version is 1.2, Dec 24, 2002.

 

switch in lambda-calculus

We implement the switch combinator, a lambda-term with the following reduction rules:
     switch 0 -> id
     switch n test1 stmt1 test2 stmt2 ... test_n stmt_n default
       -> stmt_i    for the first i such that test_i -> true
       -> default   otherwise
Here n is a Church numeral, test_i are Church Booleans, stmt_i and default are arbitrary terms. We impose an additional constraint of avoiding general recursion.

For example, assuming c2 stands for Church numeral 2, etc., and true, false, and zerop are combinators with the obvious meaning,

     ((switch c2) true  x true y z)               ; reduces to: x
     ((switch c2) false x true y z)               ; reduces to: y
     ((switch c3) false x false y (zerop c1) z h) ; reduces to: h
Version
The current version is 1.1, Mar 25, 2003.
References
LC_switch.lhs [4K]
The complete code with the explanation and examples
The code was first posted in a message ``Re: Lambda Calculus Question'' on comp.lang.functional on Tue, 25 Mar 2003 14:13:34 -0800

 

Negative numbers, subtraction and division in lambda-calculus

This article will demonstrate basic arithmetic operations -- comparison, addition, subtraction, multiplication, and division -- on non-negative and negative integer numbers. Both the integers and the operations on them are represented as terms in the pure untyped lambda-calculus. The only building blocks are identifiers, abstractions and applications. No constants or delta-rules are used. Reductions follow only the familiar beta-substitution and eta-rules. We also show two approaches of defining a predecessor of a Church numeral. Curiously, none of the arithmetic operations involve the Y combinator; even the division gets by without Y.

Addition, multiplication, squaring, and exponentiation are easily expressed in lambda-calculus. We discuss two approaches to their ``opposites'' -- subtraction, division, finding of a square root and of a discrete logarithm. The opposite operations present quite a challenge. One approach, based on fixpoints, is more elegant but rather impractical. The dual approach, which relies on counting, leads to constructive and useful, albeit messy, terms.

Version
The current version is 1.2, Jul 14, 2002.
References
lambda-calc-div-neg.txt [15K]
The article itself. It was originally posted as ``Negativism and division in lambda-calculus'' on the newsgroup comp.lang.functional on Sun, 13 May 2001 23:12:25 GMT

lambda-arithm-neg.scm [14K]
Negative numbers and four arithmetic operations on integers
The source code that defines and tests all four arithmetic operations on positive and negative integers, in lambda-calculus. The code is meant to be evaluated by the practical lambda-calculator in Scheme.

LC_neg.lhs [17K]
Negative numbers and the four arithmetic operations on integers [Haskell version]

lambda-calc-opposites.txt [4K]
The article ``Subtraction, division, logarithms and other opposites in Lambda calculus'' posted on comp.lang.functional on Tue, 29 May 2001 20:58:02 -0700

lambda-arithm-div-term.scm [8K]
Lambda abstraction term for division An unadorned lambda term that implements division of two signed integers (also represented as lambda terms). The article explains in detail how this division term has been derived.

 

Basic Lambda Calculus terms

Definitions of basic lambda terms: Church numerals, combinators, Booleans, etc. These are basic lambda-calculus terms, needed to run more advanced examples. Two versions of the terms are given, in the notation of the Scheme and Haskell calculators.
Version
The current version is 1.3, Aug 18, 2002.
References
lambda-arithm-basic.scm [4K]
A version for the Scheme calculator.

LC_basic.lhs [4K]
A version for the Haskell calculator. This is a literate Haskell code that defines the terms and shows examples of their use.

 

Predecessor and lists are not representable in simply typed lambda-calculus

The predecessor of a Church-encoded numeral, or, generally, the encoding of a list with the car and cdr operations are both impossible in the simply typed lambda-calculus. Henk Barendregt's ``The impact of the lambda-calculus in logic and computer science'' (The Bulletin of Symbolic Logic, v3, N2, June 1997) has the following phrase, on p. 186:

Even for a function as simple as the predecessor lambda definability remained an open problem for a while. From our present knowledge it is tempting to explain this as follows. Although the lambda calculus was conceived as an untyped theory, typeable terms are more intuitive. Now the functions addition and multiplication are defineable by typeable terms, while [101] and [108] have characterized the lambda-defineable functions in the (simply) typed lambda calculus and the predecessor is not among them [the story of the removal of Kleene's four wisdom teeth is skipped...]
Ref 108 is R.Statman: The typed lambda calculus is not elementary recursive. Theoretical Comp. Sci., vol 9 (1979), pp. 73-81.

Since list is a generalization of numeral -- with cons being a successor, append being the addition, tail (aka cdr) being the predecessor -- it follows then the list cannot be encoded in the simply typed lambda-calculus.

To encode both operations, we need either inductive (generally, recursive) types, or System F with its polymorphism. The first approach is the most common. Indeed, the familiar definition of a list

     data List a = Nil | Cons a (List a)
gives an (iso-) recursive data type (in Haskell. In ML, it is an inductive data type).

Lists can also be represented in System F. As a matter of fact, we do not need the full System F (where the type reconstruction is not decidable). We merely need the extension of the Hindley-Milner system with higher-ranked types, which requires a modicum of type annotations and yet is able to infer the types of all other terms. This extension is supported in Haskell and OCaml. With such an extension, we can represent a list by its fold, as shown in the code below. It is less known that this representation is faithful: we can implement all list operations, including tail, drop, and even zip.

References
How to zip folds: A library of fold transformers (streams)

list-representations.txt [9K]
Detailed discussion of various representations of lists in Haskell, and their typing issues
The message was originally posted as ``Re: Lists vs. Monads'' on the Haskell-Cafe mailing list on Thu, 21 Jul 2005 19:30:38 -0700 (PDT)

 

A practical Lambda-calculator in Haskell

The present calculator implements what seems to be an efficient and elegant algorithm of normal order reductions. It is almost identical to that employed by yacc sans one critical difference. The calculator also takes a more functional approach to the hygiene of beta-substitutions, which is achieved by coloring of identifiers where absolutely necessary. This approach is ``more functional'' because it avoids a global counter or the threading of the paint bucket through the whole the process. The integration of the calculator with Haskell lets us store terms in variables and easily and intuitively combine them.

The traditional recipe for normal-order reductions includes an unpleasant phrase ``cook until done''. The phrase makes it necessary to keep track of reduction attempts, and implies an ugly iterative algorithm. We're proposing what seems to be an efficient and elegant technique that can be implemented through intuitive re-writing rules. Our calculator, like yacc, possesses a stack and works by doing a sequence of shift and reduce steps. The only significant difference from yacc is that the lambda-calculator ``reparses'' the result after the successful reduce step. The source and the target languages of our ``parser'' (lambda-calculator) are the same; therefore, the parser can indeed apply itself.

The parsing stack can be made implicit. In that case, the algorithm can be used for normalization of typed lambda-terms in Twelf.

The following examples show that lambda-calculus becomes a domain-specific language embedded into Haskell:

     > c0 = f ^ x ^ x                -- Church numeral 0
     > succ = c ^ f ^ x ^ f # (c # f # x)            -- Successor
     
     > c1 = eval $ succ # c0   -- pre-evaluate other numerals
     > c2 = eval $ succ # c1
     > c3 = eval $ succ # c2
     > c4 = eval $ succ # c3

It is indeed convenient to store terms in Haskell variables and pre-evaluate (i.e., normalize) them. They are indeed terms. We can always ask the interpreter to show the term. For example, show c4 yields (\f. (\x. f (f (f (f x))))).

     let mul = a ^ b ^ f ^ a # (b # f)                 -- multiplication
     eval $ mul # c1 ---> (\b. b), the identity function
     eval $ mul # c0 ---> (\b. (\f. (\x. x))), which is "const 0"

These are algebraic results: multiplying any number by zero always gives zero. We can see now how lambda-calculus can be useful for theorem proving, even over universally-quantified formulas.

The calculator implements Dr. Fairbairn's suggestion to limit the depth of printed terms. This makes it possible to evaluate and print some divergent terms (so-called tail-divergent terms):

     Lambda_calc> let y_comb = f^((p^p#p) # (c ^ f#(c#c))) in eval $ y_comb#c
     c (c (c (c (c (c (c (c (c (c (...))))))))))
It is amazing how well lambda-calculus and Haskell play together.

A monadic version of the calculator gives us the trace of the reduction steps.

Version
The current version is 2.3, May 5, 2005.
References
Lambda_calc.lhs [23K]
The complete literate Haskell code for the calculator.
The code also includes convenience functions to list free variables of a term and to compare two terms modulo alpha-renaming. The latter function is extensively used in built-in regression tests. To run the tests, evaluate all_tests.
The code was first posted in a message ``ANN: Normal-order evaluation as bottom-up parsing'' on Sun, 28 Apr 2002 14:11:59 -0700 on the Haskell mailing list.

Basic Lambda Calculus terms

Reynald Affeldt. Abstract machines for functional languages. A Survey.
< http://web.yl.is.s.u-tokyo.ac.jp/~affeldt/study/abstract_machines.ps >
The survey describes SECD, Krivine, CAM and ZAM machines and several of their variations.
Incidentally, Lambda_calc.lhs appears to have little in common with Krivine, SECD or other machines. The Lambda_calc.lhs machine has stack, but it has no concept of environment. This is indicative of a larger difference: in Lambda_calc.lhs variables have 'no meaning'. To this machine, normalization of lambda terms is a pure syntactic transformation, akin to a transformation of (((a+b))) into a+b by a regular parser.

 

A practical Lambda-calculator in Scheme

The code below implements a normal-order interpreter for the untyped lambda-calculus. The interpret permits ``shortcuts'' of terms. The shortcuts are not first class and do not alter the semantics of the lambda-calculus. Yet they make complex terms easier to define and apply.

The code also includes a few convenience tools: tracing of all reduction, comparing two terms modulo alpha-renaming, etc.

Version
The current version is 1.1, Mar 30, 2001.
References
A practical lambda-calculator in Scheme

Basic Lambda Calculus terms

 

Lambda-calculator as a CPS R5RS macro

Another normal-order interpreter for untyped lambda-calculus, implemented as a syntax-rule macro. The source and the object languages of the calculator are regular Scheme terms. Lambda-abstractions for the calculator are ordinary lambda-forms of Scheme:
     (?lc-calc
      ((lambda (x) (x x)) (lambda (y) (y z))) ; term to reduce
      (??!lambda (result) (display '(??! result)))
     )
expands to (display '(z z)). The funny ??!lambda is a macro-lambda -- the abstraction of the (meta-)calculator.

The calculator normalizes regular Scheme terms. The result can be compiled and evaluated in a call-by-value lambda-calculus with constants (i.e., Scheme). The lambda-calculator acts therefore as a partial evaluator.

Version
The current version is Dec 16, 2001.
References
Lambda-calculator as a R5RS macro

 

Lambda-calculator as a short, direct-style R5RS macro

Yet another normal-order interpreter for untyped lambda-calculus, implemented as a direct-style syntax-rule macro. This implementation is the shortest and the fastest one. It does no alpha-renaming directly and no substitutions directly. Everything is integrated with normalization, and everything is delayed until the latest possible moment. The source and the object languages of the calculator are regular Scheme terms. Lambda-abstractions for the calculator are ordinary lambda-forms of Scheme:
     (((NORM ((lambda (c)     ; succ
                 (lambda (f) (lambda (x) (f ((c f) x)))))
              (lambda (f) f)  ; c1
             ))
       (lambda (u) (+ 1 u))) 0)
normalizes the application of the succ lambda-term to c1, Church numeral 1. The result, which is a Scheme procedure, is then applied to the Scheme values (lambda (u) (+ 1 u)) and 0 to yield 2 -- which shows that the result of the normalization was Church numeral 2.

The calculator normalizes regular Scheme terms. The result can be compiled and evaluated in a call-by-value lambda-calculus with constants (i.e., Scheme). The lambda-calculator acts therefore as a partial evaluator.

Version
The current version is December 2004.
References
Lambda-calculator as a R5RS macro

 

Type-level call-by-value lambda-calculator in three lines

We present a type-level interpreter for the call-by-value lambda-calculus with booleans, natural numbers, and case discrimination. Its terms are Haskell types. Using functional dependencies for type-level reductions is well-known. Missing before was the encoding abstractions with named arguments and closures.

The core of the interpreter indeed takes only three lines:

     instance                                   E (F x)           (F x)
     instance (E y y', A (F x) y' r)         => E ((F x) :< y)    r
     instance (E (x :< y) r', E (r' :< z) r) => E ((x :< y) :< z) r
The first line says that abstractions evaluate to themselves, the second line executes the redex, and the third recurs to find it. The representation of abstractions is apparent from the expression for the S combinator, which again takes three lines:
     instance A (F CombS)     f (F (CombS,f))
     instance A (F (CombS,f)) g (F (CombS,f,g))
     instance E (f :< x :< (g :< x)) r => A (F (CombS,f,g)) x r
The instances of the type class A f x r define the result r of applying f to x. The last line shows the familiar lambda-expression for S, written with the familiar variable names f, g, and x. The other two lines build the closure `record'. The closure-conversion is indeed the trick. The second trick is taking advantage of the instance selection mechanism. When the type checker selects a type-class instance, the type checker instantiates it, substituting for the type variables in the instance head. The type checker thus does the fundamental operation of substitution, which underlies beta-reduction. No wonder why our lambda-calculator is so simple.

The encoding of the applicative fixpoint combinator is even simpler:

     data Rec l
     instance E (l :< (F (Rec l)) :< x) r => A (F (Rec l)) x r
expressing the reduction of the application (fix l) x to l (fix l) x. The article shows several examples of using the fixpoint combinator, e.g., to compute Fibonacci numbers. Incidentally, the present code constitutes what seems to be the shortest proof that the type system of Haskell with the undecidable instances extension is indeed Turing-complete.

The type-level lambda-calculator albeit trivial has the expected properties, e.g., handling of closures and higher-order functions. It uses named function arguments, rather than De Bruijn indices and is based on explicit substitutions. We again rely on the Haskell type checker for appropriate alpha-conversions.

Version
The current version is 1.1, Sep 14, 2006.
References
TypeLC.lhs [9K]
The complete literate Haskell code with explanations and tests, including Fibonacci, S and K combinators and the combinatorial arithmetic
The code was originally posted in a message ``On computable types. I. Typed lambda and type closures'' on Thu, 14 Sep 2006 19:07:29 -0700 (PDT) on the Haskell mailing list.

Applications of computable types



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