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2013-01-17

Vale of tears.


Consider a system describing possible computations (e.g., a programming language or a state machine formalism) including interactions with the external world, that is, input and output facilities.

A pair of dual sub-classes of computational elements (values, objects, functions, &c.) can be defined:
  • finitary elements that are known to not depend on input and
  • ideal elements that output is known to not depend on.
The rest of elements may depend on input and may affect output. Let's call such elements laic ("temporal" might be a better word).

The class of finitary elements is well-known: because they can be computed without input, they can be computed before the program starts, i.e., they correspond to various constants, including static entities like types (in statically typed languages), classes, function bodies and so on. Some languages have powerful finitary computations, for example, C++ template specialisation is Turing complete.

Laic elements are the most usual things like variables and objects.

Ideal elements are less known. They have a long history of use in the area of formal program verification where they are called ghost or auxiliary variables. 

There is an obvious restriction of data and control flow between various types of elements:

  • finitary element may depend only on finitary elements;
  • laic element may depend on laic and finitary elements (e.g., normal function can take a constant as a parameter, but constant cannot be initialised with the value of a variable or function call);
  • ideal element may depend on any element (e.g., ideal variable can be assigned the value of a laic variable, but not other way around).

The most important property of ideal elements is that, because they don't affect observable program behaviour, there is no need to actually compute them! Yet, they are useful exactly because of this property: ideal elements are not computed and, hence, are not constrained by the limitations of actual computational environments. For example, an ideal variable can represent an infinite (even uncountable) collection or a real number (real real number, not  approximation); an ideal function can be defined by the transfinite induction or by a formula involving quantifiers.

To use ideal elements, one assumes that they follow normal rules of the language (for example, axiomatic or denotational semantics). This assumption doesn't burden the implementors of the language precisely because the ideal elements are not computed. Under that assumption, one can reason about properties of ideal elements.

As a simplest example, an ideal variable can be used to record the sequence of calls to a certain function:

ideal f_seq = {};
function f(arg) {
        f_seq := f_seq ++ arg;
        ...
};

and then reason about f_seq using whatever method is used to reason about laic elements (e.g., weakest preconditions, Hoare triples or usual hand-waving), for example, to prove that messages delivered to a receiver were sent by the sender (that is, deliver_seq is a sub-sequence of send_seq).

It is interesting that both finitary elements (specifically, static types) and ideal elements help to reason about the behaviour of the laic world sandwiched between them.

Nothing in this short article is new, except for the (obvious) duality between ideal and finitary elements.

Exercise 0: implement linear types by casting laic elements to ideal.
Exercise 1: implement garbage collection similarly.

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