a big collection of function operators
I have been putting this off, as it will be quite a bit of work. But now is the time to try and describe some of the more interesting function operators. There is a whole collection of them (a quick grep says about 150 of them) in this file. And unlike the functions built into ket/sp classes, the plan is to add as many of these as we want or need. Indeed, get other people to write them too (if I can get others interested in this project). Note that behind the scenes, once you have a new function you need to "wire it in" to the processor. This currently means adding an entry into the appropriate hash-table (a black art that, on first try, I frequently get wrong! Though the console debugging info on similar functions is often helpful.)
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