Iota and Jot


In formal language theory and computer science, [|Iota] and [|Jot] are languages, extremely minimalist formal systems, designed to be even simpler than other more popular alternatives, such as the lambda calculus and SKI combinator calculus. Thus, they can also be considered minimalist computer programming languages, or Turing tarpits, esoteric programming languages designed to be as small as possible but still Turing-complete. Both systems use only two symbols and involve only two operations. Both were created by professor of linguistics Chris Barker in 2001. Zot is a successor to Iota that supports input and output.

[|Universal iota]

Chris Barker's universal iota combinator has the very simple λf.fSK structure defined here, using denotational semantics in terms of the lambda calculus,
From this, one can recover the usual SKI expressions, thus:
Because of its minimalism, it has influenced research concerning Chaitin's constant.

Iota

Iota is the LL language that prefix orders trees of the aforementioned Universal iota combinator leafs, consed by function application,
iota = "1" | "0" iota iota
so that for example denotes, whereas denotes.

Jot

Jot is the regular language consisting of all sequences of 0 and 1,
jot = "" | jot "0" | jot "1"
The semantics is given by translation to SKI expressions.
The empty string denotes,
denotes,
where is the translation of,
and denotes.
The point of the case is that the translation satisfies for arbitrary SKI terms and.
For example,
holds for arbitrary strings.
Similarly,
holds as well.
These two examples are the base cases of the translation of arbitrary SKI terms to Jot given by Barker,
making Jot a natural Gödel numbering of all algorithms.
Jot is connected to Iota by the fact that and by using the same identities on SKI terms for obtaining the basic combinators and.

Zot

The Zot and Positive Zot languages command Iota computations, from inputs to outputs by continuation-passing style, in syntax resembling Jot,

zot = pot | ""
pot = iot | pot iot
iot = "0" | "1"
where produces the continuation,
and produces the continuation,
and consumes the final input digit by continuing through the continuation.