Preamble over, let's jump in.

-- the ket-length function:

ket-length |abcde> == |number: len(abcde)>

-- the apply-value function:

apply-value |a: b: n> == n |a: b: n> (if n is a float)

apply-value |a: b: n> == |a: b: n> (otherwise)

-- the extract category/data-type function:

extract-category |a> == |>

extract-category |a: b> == |a>

extract-category |a: b: c> == |a: b>

-- the extract value function (the opposite of extract-category):

extract-value |a> == |a>

extract-value |a: b> == |b>

extract-value |a: b: c> == |c>

-- the category depth function:

cat-depth |> == |number: 0>

cat-depth |a> == |number: 1>

cat-depth |a: b> == |number: 2>

cat-depth |a: b: c> == |number: 3>

cat-depth |a: b: c: d: e: f: g> == |number: 7>

-- the expand-hierarchy function:

sa: expand-hierarchy |a: b: c: d: e>

|a> + |a: b> + |a: b: c> + |a: b: c: d> + |a: b: c: d: e>

-- pop-float and push-float

-- Here are some examples:

-- NB: this is not |>, there is a space in there, an important distinction!

pop-float |3.2> == 3.2| >

pop-float 5|7> == 35| > -- NB: the multiplication of 5 and 7

pop-float |x: 2> == 2|x>

pop-float 5.1|x: y: 2> == 10.2|x: y> -- NB: the multiplication of 5.1 and 2

pop-float |x: y> == |x: y>

push-float n|> == |> for all n

push-float 3| > == |3> (NB: the space in there, | > not |>)

push-float |x> == |x: 1>

push-float 3|x> == |x: 3>

push-float 3.2|x: y > == |x: y: 3.2>

-- a couple of example usages:

-- action man reached a height 4 times that of everest

-- first, learn height of everest:

height |everest> => |km: 8>

-- learn height of "action man", noting that the units of height for everest are irrelevant.

height |action man> => push-float 4 pop-float height |everest>

-- "some mountain" is 1/3 the height of everest

height |some mountain> => push-float 0.3333 pop-float height |everest>

-- the to-coeff function

-- kind of a dual to the clean sigmoid

-- clean sets all coeffs to 1

-- to-coeff sets all labels to | >

-- (excluding the identity operator, which we leave intact)

to-coeff n|> == |> for all n

to-coeff n|a> == n| > for all a

-- the to-number function

-- eg, as used in the algebra() code

-- idea, is to map all types of kets to the form "n | >", where n is a float

to-number |7.2> == 7.200| >

to-number 3|9> == 27| >

to-number |number: 3.1415> == 3.142| >

to-number 8|number: 3> == 24.000| >

-- NB: this code treats the "number" data-type differently than other types:

to-number |number: not-a-float> == 0| >

-- when you use a data-type other than "number" we just return the input ket:

to-number |a: b> == |a: b>

to-number 27|a: b: c: d: e> == 27.000|a: b: c: d: e>

-- the round[t] function

-- rounds floats to t decimal places

-- round[t] |a: b: n> == |a: b: round(n,t)> if n is a float, else |a: b: n>

-- eg:

round[2] |pi: 3.14159265> == |pi: 3.14>

round[7] |a: b: c> == |a: b: c>

-- the range function (this one is very useful in defining lists to work on):

-- categories/data-types must be equal:

-- in this case "a" != "b"

sa: range(|a: 2>,|b: 5>)

|>

-- default is step of size 1

sa: range(|5>,|11>)

|5> + |6> + |7> + |8> + |9> + |10> + |11>

-- specify a data-type (here "x"):

sa: range(|x: 1>,|x: 6>)

|x: 1> + |x: 2> + |x: 3> + |x: 4> + |x: 5> + |x: 6>

-- step size of 2

sa: range(|5>,|11>,|2>)

|5> + |7> + |9> + |11>

-- float step size of 0.25

sa: range(|5>,|7>,|0.25>)

|5.00> + |5.25> + |5.50> + |5.75> + |6.00> + |6.25> + |6.50> + |6.75> + |7.00>

-- negative step sizes is currently broken!

range(|5>,|8>,|-1>) == |>

range(|8>,|5>,|-1>) == |8> + |7>

-- the arithmetic function:

-- categories/data-types must be equal (to prevent mix type errors):

-- in this case "a" != "b"

arithmetic(|a: 5>,|+>,|b: 3>) == |>

-- this is one way to ensure data-types are equal:

-- NB: the to-km operator applied to the ket using miles.

arithmetic(to-km |miles: 5>,|+>,|km: 3>) == |km: 11.047>

-- more generally (assuming "a" and "b" have to-X defined correctly):

arithmetic(to-X |a>,|op>,to-X |b>)

Final note, arithmetic supports these operators: +, -, *, /, %, ^

(addition, subtraction, multiplication, division, modulus, exponentiation)

-- the algebra function:

-- (13x + 17)*(19y + 2z + 5)

sa: algebra(13|x> + |17>,|*>,19|y> + 2|z> + |5>)

247.000|x*y> + 26.000|x*z> + 65.000|x> + 323.000|y> + 34.000|z> + 85.000| >

-- (a + b)^6

sa: algebra(|a> + |b>,|^>,|6>)

|a*a*a*a*a*a> + 6.000|a*a*a*a*a*b> + 15.000|a*a*a*a*b*b> + 20.000|a*a*a*b*b*b> + 15.000|a*a*b*b*b*b> + 6.000|a*b*b*b*b*b> + |b*b*b*b*b*b>

And note that algebra currently supports these operators: +, -, *, ^

(addition, subtraction, multiplication, exponentiation)

Also note that currently algebra is Abelian,

ie, labels commute: |x*y> == |y*x>

-- set union and intersection:

-- if coeffs are in {0,1} it works like standard union and intersection:

sa: union(|a> + |c> + |d>,|a> + |b> + |c> + |d> + |e>)

|a> + |c> + |d> + |b> + |e>

sa: intersection(|a> + |c> + |d>,|a> + |b> + |c> + |d> + |e>)

|a> + |c> + |d>

-- if coeffs are not strictly {0,1} then union is max(a,b) and intersection is min(a,b)

-- eg, the simplest possible example:

sa: union(3|a>,7|a>)

7.000|a>

sa: intersection(3|a>,7|a>)

3.000|a>

-- extends in the obvious way for more interesting superpositions:

sa: union(2|a> + 0.3|b> + 0|c> + 13|d> + 0.9|e>,|a> + 11|b> + 23|c> + 0.5|d> + 7|e>)

2.000|a> + 11.000|b> + 23.000|c> + 13.000|d> + 7.000|e>

sa: intersection(2|a> + 0.3|b> + 0|c> + 13|d> + 0.9|e>,|a> + 11|b> + 23|c> + 0.5|d> + 7|e>)

|a> + 0.300|b> + 0.500|d> + 0.900|e>

-- using the same back-end code, we can implement other examples of foo(a,b).

-- eg, multiplication and addition, and so on.

sa: multiply(2|a> + 3|b> + 5|c>,7|a> + 5|b> + 0|c> + 9|d>)

14.000|a> + 15.000|b> + 0.000|c> + 0.000|d>

sa: addition(2|a> + 3|b> + 5|c>,7|a> + 5|b> + 0|c> + 9|d>)

9.000|a> + 8.000|b> + 5.000|c> + 9.000|d>

-- now a couple of really simple ones:

-- spell and read:

sa: spell |word: frog>

|letter: f> + |letter: r> + |letter: o> + |letter: g>

-- NB: since it is a superposition, the duplicate letters get added together.

-- plan is to eventually have a sequence type, where this doesn't happen

-- in that case we would instead have:

-- |letter: l> . |letter: e> . |letter: t> . |letter: t> . |letter: e> . |letter: r>

sa: spell |word: letter>

|letter: l> + 2.000|letter: e> + 2.000|letter: t> + |letter: r>

-- NB: read ignores case and punctuation, as we can see:

sa: read |text: I don't know about that!>

|word: i> + |word: don't> + |word: know> + |word: about> + |word: that>

-- now, spell assumes the "word" data-type, and read assumes the "text" data-type

-- and returns |> if they are not, but if it turns out this isn't useful (I think it will be),

-- it is trivial to change.

-- now, their inverse, which I had totally forgotten about (heh, that's how useful they are :).

sa: read-letters spell |word: letter>

|word: letter>

sa: read-words read |text: I don't know about that!>

|text: i don't know about that>

-- again, they would work better using sequences, not superpositions.

-- now code wise simple, but useful:

-- merge-labels()

sa: merge-labels(|a> + |b> + |c> + |d> + |e>)

|abcde>

-- now a couple of simple number related functions:

is-prime |number: n> == |yes> (if n is prime)

is-prime |number: n> == |no> (if n is not prime)

is-prime |blah> == |> (since we require the "number" data-type)

is-prime |blah: n> == |>

factor |number: n> returns list of prime factors, and again requires the "number" data-type.

sa: is-prime |number: 21>

|no>

-- as far as I know the python is using arbitrary precision integers:

sa: is-prime |number: 90214539181246357>

|yes>

sa: factor |number: 210>

|number: 2> + |number: 3> + |number: 5> + |number: 7>

sa: factor |number: 398714527>

|number: 521> + |number: 765287>

sa: factor |number: 987298762329>

4.000|number: 3> + |number: 11> + |number: 1108079419>

-- convert numbers into the word equivalent

-- (and eventually we would want the inverse too)

-- currently unimplemented!

-- though it would look something like this:

number-to-words |number: 7> => |text: seven>

number-to-words |number: 35> => |text: thirty five>

number-to-words |number: 137> => |text: one hundred and thirty seven>

number-to-words |number: 8,921> => |text: eight thousand, nine hundred and twenty one>

number-to-words |number: 54,329> => |text: fifty four thousand, three hundred and twenty nine>

number-to-words |number: 673,421> => |text: six hundred and seventy three thousand, four hundred and twenty one>

number-to-words |number: 3,896,520> => |text: three million, eight hundred and ninety six thousand, five hundred and twenty>

-- convert decimal number to another base:

sa: to-base(|350024>,|2>)

0.000|1> + 0.000|2> + 0.000|4> + |8> + 0.000|16> + 0.000|32> + |64> + 0.000|128> + |256> + |512> + |1024> + 0.000|2048> + |4096> + 0.000|8192> + |16384> + 0.000|32768> + |65536> + 0.000|131072> + |262144>

sa: to-base(|350024>,|8>)

0.000|1> + |8> + 5.000|64> + 3.000|512> + 5.000|4096> + 2.000|32768> + |262144>

sa: to-base(|350024>,|10>)

4.000|1> + 2.000|10> + 0.000|100> + 0.000|1000> + 5.000|10000> + 3.000|100000>

-- now a couple of functions to swap between temperature and distance units

-- proof of concept really, in practice we would want more (for other unit types),

-- and a cleaner way to implement them

-- F operator maps Celcius and Kelvin to Fahrenheit:

sa: F |C: 0>

|F: 32.00>

sa: F |C: 100>

|F: 212.00>

sa: F |K: 0>

|F: -459.67>

-- C maps Fahrenheit and Kelvin to Celcius:

sa: C |K: 0>

|C: -273.15>

sa: C |F: 0>

|C: -17.78>

sa: C |F: 100>

|C: 37.78>

-- K maps Fahrenheit and Celcius to Kelvin:

sa: K |C: 18>

|K: 291.15>

sa: K |C: 0>

|K: 273.15>

sa: K |F: 100>

|K: 310.93>

-- now similar, but for distances:

-- to-km maps meters or miles to km:

sa: to-km |miles: 1>

|km: 1.609>

-- to-meter maps km or miles to meters

sa: to-meter |miles: 7>

|m: 11265.408>

sa: to-meter |km: 5.213>

|m: 5213.000>

-- to-mile(s) maps km or m to miles

sa: to-miles |km: 42>

|miles: 26.098>

sa: to-miles |m: 800>

|miles: 0.497>

-- now a fun one! This should be useful in a bunch of places:

-- the list-to-words function:

list-to-words |x> == |x>

list-to-words (|x> + |y>) == |x and y>

list-to-words (|x> + |y> + |z>) == |x, y and z>

list-to-words (|x> + |y> + |z> + |u> + |v>) == |x, y, z, u and v>

and so on.

-- a practical example:

-- learn Eric's list of friends:

sa: friends |person: Eric> => |person: Fred> + |person: Sam> + |person: Harry> + |person: Mary> + |person: liz>

-- output Eric's list of friends:

sa: list-to-words extract-value friends |person: Eric>

|Fred, Sam, Harry, Mary and liz>

-- the "common" function (a type of intersection)

-- (though intersection is currently limited to 2 or 3 parameters, common can handle any number)

common[op] (|x> + |y> + |z>)

-- expands to:

intersection(op|x>, op|y>, op|z>)

-- some common usages:

common[friends] (|Fred> + |Sam>)

common[actors] (|movie-1> + |movie-2>)

-- or indirectly:

|list> => |Fred> + |Sam> + |Charles> + |Liz>

common[friends] "" |list>

-- next, we have an if statement in BKO.

-- really does require its' own post, to explain best how to use it. Perhaps later.

-- raw details are just:

if(|x>,|a>,|b>) returns |a> if |x> == |True>, |b> otherwise

-- and its more useful brother (since we try to avoid just living in the {0,1} world):

-- the weighted-if function:

wif(|x>,|a>,|b>)

eg:

wif(0.7|x>,|a>,|b>)

if |x> == |True>, returns 0.7|a> + 0.3|b>

if |x> != |True>, returns 0.3|a> + 0.7|b>

-- next, the map function, again this one is very useful!

-- we need this since we don't have multi-line for loops, so we use this to map operators to a list of kets.

map[op] (|x> + |y> + |z>)

runs:

op |x> => op |_self>

op |y> => op |_self>

op |z> => op |_self>

map[fn,result] (|a> + |b> + |c> + |d>)

runs:

result |a> => fn |_self>

result |b> => fn |_self>

result |c> => fn |_self>

result |d> => fn |_self>

-- most common usage is:

fn |*> #=> ... some details here

map[fn,result] "" |some list>

-- the exp function:

exp[op,n] |x>

maps to:

(1 + op + op^2 + ... + op^n) |x>

-- the exp-max function:

exp-max[op] |x>

maps to:

(1 + op + op^2 + ... + op^n) |x>

for an n such that exp[op,n] |x> == exp[op,n+1] |x>

-- ie, we have found every "child node" of |x>

-- with a warning that we have no idea how big the result is going to be, or how many steps deep.

-- a common usage is to find 6 degrees of separation:

exp[friends,6] |Fred>

exp-max[friends] |Fred>

-- the apply() function:

-- again, this is one of those very useful ones!

eg: apply(|op: age> + |op: friends> + |op: father>,|Fred>)

maps to:

age |Fred> + friends |Fred> + father |Fred>

-- a common usage is to define a list of operators separately:

|op list> => |op: mother> + |op: father> + |op: dob> + |op: age>

-- then apply them:

apply("" |op list>,|Fred>)

-- eg, maybe use like this:

|basic info op list> => |op: mother> + |op: father> + |op: height> + |op: age> + |op: eye-colour>

basic-info |*> #=> apply("" |basic info op list>,|_self>)

basic-info |Fred>

basic-info |Sam>

-- here is a toy function, maps dates to day of the week:

sa: day-of-the-week |date: 2015/01/24>

|day: Saturday>

-- here is one that saves typing, the split operator:

sa: split |a b c d e>

|a> + |b> + |c> + |d> + |e>

sa: split |word1 word2 word3 word4>

|word1> + |word2> + |word3> + |word4>

-- currently only splits on space chars, but maybe useful to specify the split char(s).

-- the clone ket function (not yet sure of a use case):

-- clone(|x>,|y>) copies rules from |x> and applies them to |y>

-- hence the name, clone().

-- say we have:

-- age |x> => |27>

-- mother |x> => |Jane>

-- after clone(|x>,|y>) we have:

-- age |y> == |27>

-- mother |y> == |Jane>

-- eg, if |x> and |y> are twin sisters.

--

-- thought of a use case:

-- say we have just learnt "elm" is a type of tree.

-- well, load that up with some default values we know about all tree's:

-- (cf. inheriting from a parent class in OO programming)

-- clone(|plant: tree>,|plant: tree: elm>)

-- then fill in more specific data as we learn more.

-- the relevant-kets[op] function

-- returns a list of all the kets in the current context that have "op" defined.

-- relevant-kets[op] is frequently useful for generating lists we can apply the map function to.

-- eg, learn some data:

sa: friends |Fred> => |Sam> + |Liz>

sa: friends |Rob> => |Jack> + |Tom>

sa: age |Fred> => |22>

-- now find who knows what operator types:

sa: relevant-kets[friends] |>

|Fred> + |Rob>

sa: relevant-kets[age] |>

|Fred>

-- there is a variant on this.

-- returns: intersection(relevant-kets[op],SP)

intn-relevant-kets[op] SP

-- eg, we can chain them and find all kets that support both friends, and age:

-- (NB: one has "intn" prefix, and one doesn't!)

intn-relevant-kets[age] relevant-kets[friends] |>

-- the pretty print rules as a matrix function.

-- first define some rules:

sa: op |a> => |a> + 2.000|b> + 3.000|c>

sa: op |b> => 0.500|b> + 9.000|c> + 5.000|e>

sa: op |c> => 7.000|e> + 2.000|b>

-- now take a look:

sa: matrix[op] [ a ] = [ 1.00 0 0 ] [ a ] [ b ] [ 2.00 0.50 2.00 ] [ b ] [ c ] [ 3.00 9.00 0 ] [ c ] [ e ] [ 0 5.00 7.00 ] |matrix>

-- and we finish with a slightly more interesting function, the train-of-thought function.

-- this code makes heavy use of supported-ops, pick-elt and apply().

-- and will work much better with a big knowledge base, but even a small one gives hints of what a large example will be like.

sa: load early-us-presidents.sw -- load up some knowledge

sa: create inverse -- needed, else we run into dead ends.

sa: train-of-thought[13] |Madison> -- take 13 steps, starting with |Madison>

context: sw console

one: |Madison>

n: 13

|X>: |Madison>

|early US Presidents: _list>

|Adams>

|year: 1797>

|Washington>

|early US Presidents: _list>

|Adams>

|number: 2>

|Adams>

|year: 1801>

|Jefferson>

|early US Presidents: _list>

|Adams>

|year: 1799>

Anyway, I guess the summary of this post is that we have some proof of concept functions trying to map our BKO scheme towards a more general purpose knowledge engine. Don't take the above functions as finished, take them as hints on where we could take this project.

Home

previous: introducing sigmoids

next: foaf vs sw

updated: 19/12/2016

by Garry Morrison

email: garry -at- semantic-db.org