Created by Chad Kainz of the Multimedia and Visualization Center at the University of Chicago.
The pictorial logo for the Chicago Journal of Theoretical Computer Science represents the combinatory expression SII(SII) (left foreground) reducing to I(SII)(I(SII)) (right background). In two more steps, the expression reduces further to SII(I(SII)) and SII(SII), which is the same as the starting expression, showing that reduction will go on infinitely. In the pictorial version, red squares represent the binary operation of application, blue squares represent the symbol S, and green squares represent the symbol I. Arrows link application squares to a function (left arrow) and its argument (right arrow). The repeated subexpression SII is shared, rather than represented twice.
Theoretical computer science, which studies the abstract mathematical nature of computation, has provided several key insights into the structural properties that must hold in all computing systems, no matter how they are designed and implemented. Several of these insights reveal inherent and unavoidable limitations on computing systems. Desirable properties of a computing system may ineluctably entail other, undesirable properties. For example, in order to be completely general-purpose, that is, to have the desirable capability of computing all of the functions that are in principle computable, a computing system must also provide programs that compute forever uselessly, producing no output. In fact, all programming languages in widespread use today have chosen generality, at the cost of making each programmer face the risk of accidentally producing an infinite computation.
Theoretical computer science has also clarified the programming resources that are required in order to provide the power to compute all computable functions. They are amazingly simple. One of the simplest systems that is powerful enough for general-purpose computing is the Combinator Calculus, which is carried out in a language involving only one binary operator and two constants.
The binary operator of the Combinator Calculus is called "application," intended to capture the intuitive idea of applying a function to an argument, or a program to an input. An expression of the form MN represents the application of the subexpression M (function or program) to the subexpression N (argument or input). Every expression in the language of combinators is allowed to serve as a function/program and also as an argument/input (this is sensible since programs are written out as special sorts of texts, and those texts are easily read as inputs by programs). The application operation itself does not require a symbol: it is represented by the juxtaposition of its operands, just as multiplication is represented by juxtaposition in conventional numerical mathematics. Unlike multiplication, application is not associative. That is, (MN)P is not always equal to M(NP). To avoid proliferation of parentheses, we agree to associate applications to the left whenever possible. So, MNPQ is read as ((MN)P)Q. The alternative (MN)(PQ) can be written MN(PQ), but the rightmost parentheses cannot be omitted.
The two constants required by the Combinator Calculus are written S and K. The language of the Combinator Calculus contains precisely those expressions that can be written by combining Ss and Ks, using the binary application operation. For example, K(SK) and S(KS)(SK) are expressions in the language of the Combinator Calculus. There are no variables or other sorts of symbols or operations besides the constants S and K and the implicit application operation.
Finally, the Combinator Calculus has two rules of calculation, one for each of its constants:
Perhaps S and K appear rather weak at first, but together they are capable of computing every function that is in principle computable. For a very simple example, notice that the identity function may be defined as SKK. That is, for every expression M, SKKM reduces to KM(KM), which reduces to M. The identity function is so important that it is often given its own symbol, I, and the rule that IM reduces to M. In the infinite loop in the logo, I have used the third constant I for brevity, but it may be replaced everywhere by SKK and the same computation works, with a few more steps.
Whether we regard I as a third constant, or merely as an abbreviation for SKK, the expression SII(SII) is important, because it yields the simplest example of an infinite computation in the Combinator Calculus. The mere presence of infinite computation does not prove that the Combinator Calculus can define all of the computable functions, but it is one crucial requirement for that generality. Notice that SII(SII) reduces to I(SII)(I(SII)) (by the S rule with M=I, N=I, and P=SII), then to SII(I(SII)), and then to SII(SII), and so on ad infinitum. The first step in that infinite reduction is shown by the logo. SII may be understood as an operator that takes one argument and applies that argument to itself. So, SII(SII) is self-application self-applied.
SII(SII) is called the "paradoxical combinator," because it demonstrates the same self-applicative schema that drives Russell's "paradox" (more properly, it is an antinomy). Bertrand Russell defined the set of all sets that do not contain themselves and asked whether that set contains itself. Asking whether a set contains itself has the same formal structure as applying a function to itself. Haskell B. Curry took advantage of the particular simplicity of SII(SII) to construct another antinomy (also popularly called "Curry's paradox"), involving only implication and self-reference, without sets or negation. Curry defined the moon-cheese proposition that says, "if this proposition is true, then the moon is made of cheese."
A lot more work is required to demonstrate that the Combinator Calculus can define all of the computable functions. One key element is the encoding of the integers 0, 1, 2, 3, etc. by the expressions KI, S(S(KS)K)(KI), S(S(KS)K)(S(S(KS)K)(KI)), etc. The expression KI represents 0, and S(S(KS)K) represents the incrementation function that adds 1 to its argument. The expression representing the integer i may also be understood as a function which, taking another function f as its argument, composes f with itself i times.
Another key element in general-purpose computing is a mechanism for performing recursive computation. Given an expression M to be applied recursively, construct S(KM)(SII)(S(KM)(SII)). This expression reduces in the sequence
The reduction sequence above gives the essential ideas in a proof of the famous and crucial recursion theorem. A pictorial version of that proof could be very elegant, but a bit too big for a logo. In principle, a whole operating system could be written as a combinatory expression. Pictorial versions of the workings of that operating system could provide murals for all the buildings in the world.
Michael J. O'Donnell <email@example.com>
Last modified: Fri Dec 6 1996