I have been musing a bit about combinatory and probabilistic
logic, and what they have to teach us philosophically. It’s perhaps an obvious sort of point, but
to me it’s interesting to reflect on these basic foundations and try to
understand what are the core, minimum requirements for being able to perform
computations and think about the world.

The conclusion of my musing here will be list of
commonsensical semantic primitives, so that, roughly speaking: Any mind that
has mastered these, should be able to do universal computing and universal
probabilistic logic … limited only by its available resources.

**The Crux of Combinatory Logic**

Let us start with computation; and what is in a sense the
simplest formalism for universal computation, combinatory logic.

One thing combinatory logic tells us is that to do any kind
of computation, all we need is to be able to do the following things:

- ·
**Sequential ordering**: arranging two or more things in a list with a specific order - ·
**Grouping**: drawing a boundary around a subsequence of two or more things, to distinguish the subsequence from other things before or after it - ·
**Insertion**: draw an arrow connecting one entity or sequence, with a position in another sequence - ·
**Removal**: note that one entity or sequence, consists of a certain other sequence with its {beginning, middle or end} removed. We could decompose this into the primitive concepts of beginning middle and end; and the primitive concept of removal. - ·
**Action:**Actually carry out an insertion (placing one entity in a certain position in another sequence) or removal (create a copy of a sequence with its beginning, middle or end removed)

An elegant way to represent insertion and removal is to
think about there as being four sorts of arrows: simple arrows,
remove-beginning arrows, remove-middle arrows and remove-end arrows. Enacting a remove-end arrow pointing from a
sequence to a position in another sequence, results in said position receiving
a version of the first sequence with its end removed.

This set of basic operations is enough to do general-purpose
computation. To see, this, note that
the S and K combinators, defined as

**K x y = x**

**S x y z = (x y) (x z)**

are enough to do general-purpose computation. For,

**· K is just a remove-end arrow**

**· S is a sequencing of a grouping of a remove-end arrow, and a grouping of a remove-middle arr**ow

(Note that, according to the sufficiency of S and K,
remove-beginning is not strictly necessary; we can make do with just
remove-middle and remove-end, and derive remove-beginning therefrom.)

We can depict this graphically by adopting some simple
conventions:

- · Entities written on a horizontal row, connected by thick arrows pointing from left to right, are considered as a sequence to be read from left to right
- · Arrows denoting insertion point down
- · Removals are represented as arrows labeled by –e (remove end), -m (remove middle) or –b (remove beginning)
- · A star indicates the result of doing an action (where an action is an arrow)
- · Grouping of a subsequence is indicated by drawing a box around the subsequence
- · An arrow pointing into a box, means that upon enaction, the result of the action denoted by the arrow is placed into the box
- · An arrow pointing from a bracket around a subsequence, takes the bracketed subsequence as input when enacted
- · An arrow pointing from a grouping (box) around a subsequence, takes the box and its contents as input when enacted

The following diagrams illustrate these conventions:

In this formalism, an arbitrary computation can be
represented as a series of rows, where each row consists of a sequence of boxes
(some containing other sequences of boxes) or spaces; with each lowest-level
box and each space containing the pointy end of an arrow, whose root is some
box or bracketed subsequence on the row above.

We can support concurrency in this formalism by interpreting
two sequences A and B so that neither A nor B is below each other, as
concurrently enactable. We say that X is
immediately below Y if there is an arrow pointing from somewhere in X to
somewhere in Y. We say that X is below Y
if you can reach Y from X by following a series of immediately-below
relationships.

Conceptually, and to be a bit repetitive, what this means
is: to do ANY computation whatsoever, all you need to be able to do is…

- · Arrange things in a linear sequence
- · Put a boundary around a subsequence, to mark it off from the rest of the sequence
- · Indicate to yourself that one sequence (with a boundary or not) should be inserted in a certain place in another sequence
- · Indicate to yourself that one sequence is obtained by removing the beginning, middle or end of some sequence
- · Enact some insertion or removal that you have conceptualized

That’s it. In a very basic sense, that’s what computing
is.

**Boolean and Probabilistic Logic**

Now, none of the above tells us what things should be
arranged in sequence, or grouped in boxes, in the first place. Combinatory
logic as formulated above can only do purely mathematical computing. To do computing relative to some body or
other practical system, we need to add at very least one more primitive:

- · Enact some primitive action, based on some sequence of (primitive or composite) perceptions

To handle perceptions and primitive actions

*elegantly*, however, it seems this is not enough. It seems very useful to add a few additional components to our basic model of computation, to wit:- · Un-ordered grouping (sets)
- · Join (on sets)
- · Direct product (on sets)
- · Set difference
- · Comparison (a partial order operation)
- · Valuation: mapping sets of primitive perceptions into numbers

As Knuth and Skilling have shown if we add these operations,
then we get not only Boolean logic, but also, fairly directly, probability
theory. If we let the valuation map into
pairs of real numbers instead of single real numbers, then we get quantum
probability theory.

On the other hand, what Schonfinkel showed when originally
formulating combinatory logic, was that he could emulate quantifier logic using
combinatory logic simply by adding one more combinator, the U combinator. What the U combinator does is says “UAB means
not both A and B.” But this comes
immediately as a side-effect of the set operations introduced just above. So if we put the basic combinator operations
together with the basic lattice operations, we get quantifier logic.

So we then get general-purpose probabilistic-logical
computation.

Graphically, to represent these additional algebraic
operators, we would add the potential for unordered grouping. For instance, in this diagram

the outputs of the first two arrows can be in either order
inside their grouping; and this grouping can be in either order as compared to
the third arrow. The left-to-right
ordering on the page has no semantics here.

One would then add join (+) and direct product (x) operators
between unordered groups. Unordered
groups can contain sequences;a nd sequences can contain unordered groups; or
operator expressions built up from unordered groups using + or x. A box around an unordered group may be used
to allow + or x to act on that group as a whole.

(To emphasize, these additions are not strictly needed,
because pure combinatory logic can be used to emulate these lattice
operations. I just think the latter is
counterintuitive and messy, and it’s better to add the lattice operations as
primitives. This is certainly
debatable.)

So, to be a bit repetitive again -- philosophically, what
this suggests is that to do computing nicely about observations in a world, you
also want to have a few other processes at your disposal besides those strictly
required for computation:

- · The ability to group things into unordered sets
- · The ability to join two sets together
- · The ability to combine two sets (by combining each element of one, with each element of the other)
- · The ability to compare two sets, to see if one is part of the other
- · The ability to measure how big a set is – i.e. to map sets to numbers representing “size”, in a way that works nicely with the above set operations
- · The ability to compare two size-measurements, so as to say if one set is bigger than another

These basic operations let you do probabilistic logic; and
put together with combinators, they let you do quantifier logic … and this
logic can then interoperate complexly with computation.

**Musing a Bit**

One nice thing is that all these operations are pretty
simple and natural conceptually. They are not weird, obscure mathematical
operations. They are all quite basic,
intuitive operations.

Looking at these basic operations, it’s not hard to see how
the ability for general-purpose computation and logical reasoning would emerge
from simpler organisms lacking this
ability. Each of the basic operations
needed to support computation or reasoning seems likely to be useful in
isolation, in various situations. The
ability to identify middles and ends, or the ability to insert sequences into
other sequences, or the ability to mentally join together two sets of
observations, etc. – each of these things is going to be useful in practice for
many organisms that cannot do general logic or computation. But then once these basic capabilities have
evolved, due to their partly-separate, partly synergetic-in-various-groupings,
utility … then they can come together to do general-purpose computation and
logic … and then the utility of this general-purpose logic will reinforce all
these capabilities and cause them to pass on through the generations.

Thinking about general intelligence, it seems that if we
wanted to take a reasonably good stab at making an unbiased sort of general
intelligence, we would make a system that tried to learn concise programs
computing patterns in its observations, using a programming language comprising
the simple operations enumerated above.
Such an AGI would aim to learn patterns in the world that were as simple
as possible, when expressed in terms: of basic lattice operations,
corresponding numerical size comparisons; and basic sequence groupings,
insertions and beginning/middle/end removals.

In terms of teaching baby AGIs, it would perhaps be wise to
include in the curriculum lessons focused on each of the elementary operations
mentioned above. Via combining together
practical facility with these elementary operations, quite general practical
facility becomes possible.

**Semantic Primitives**

Finally, it is a little bit interesting to take these basic
mathematical operations and reframe them less formally as semantic primitives.

The verbal primitives we arrive at, from the above math
operations, look like:

- · Before
- · After
- · Together with
- · Put into
- · Take out of
- · Beginning
- · Middle
- · End
- · Do
- · Or
- · Combine with
- · More than
- · Graded indication of how much more than

Every human language, so far as I know, contains good methods
of referring to all of the above except for perhaps the latter. The latter I suspect is handled gesturally
and/or in terms of phonological variation, in languages which have limited
verbal mechanisms for expressing magnitudes.

At a basic level, it is mastery of these semantic primitives
that enables universal computation and universal uncertain reasoning.

**An Alternative Language for Automated Programming?**

These considerations also suggest a concise elementary
programming language, a bit different from existing ones. What we would have are:

- · Basic types for: primitive action, primitive perception, list, set, group
- · The notion of an arrow from a set, group, list or sub-list to a specific location in another list
- · The notion of removing the beginning, middle or end of a list (or, for good measure, perhaps removing any specified sub-list between positions m and n in a list).
- · The notion of enacting a set or list (thus indirectly enacting some primitive actions, after following some arrows)
- · Join and direct-product as set operations
- " "Greater than" and "less than", for sets based on inclusion, and for numbers produced as probabilities
- Addition and multiplication of probabilities; and evaluation of probabilities via normalizing set sizes

Such a language would not be very suitable for human
programming, but could be an interesting alternative for automated programming
such as evolutionary learning or probabilistic program inference. It would seem amenable to a highly efficient
implementation, leveraging nice data structures for holding and manipulating sets
and lists.

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