Indexing
Indexing maps between data structures. It is deceptively powerful.
Indices of a list
The indices of a list are originzero ordinals.
q)`cow`duck`python`snail [3 1]
`snail`duck
Above, the index expression is the vector 3 1
.
Indices of a dictionary
The indices of a dictionary are its keys.
q)show d:`ATW`KEI`SJT!("Whitney";"Iverson";"Taylor")
ATW "Whitney"
KEI "Iverson"
SJT "Taylor"
q)d[2 3#`SJT`ATW]
"Taylor" "Whitney" "Taylor"
"Whitney" "Taylor" "Whitney"
Index notation
Select items from a list or dictionary by suffixing it with zero or more index expressions, separated by semicolons, and embraced in square brackets.
 Index expression

An atom index or a list or dictionary in which the atoms are indices.
Indexing is rightatomic
The result of indexing a list mirrors the index expression.
q)1 type\" #"[1] / atom index
"#"
10h
q)1 type\" #"[1 0 1] / vector index
"# #"
10h
q)" #"[5 3 0 4 2#'1] / nestedlist index
"#####"
"###"
""
"####"
"##"
A matrix is a list of samelength lists of uniform type.
q)show A:(11111b;10001b)[0 1 0 1 1]
11111b
10001b
11111b
10001b
10001b
q)" #"[A] / matrix index
"#####"
"# #"
"#####"
"# #"
"# #"
Indexing in depth
A matrix is a list of samelength lists of uniform type.
q)show m:3 5#15?100
12 10 1 90 73
90 43 90 84 63
93 54 38 97 88
Its items are its rows.
q)m[2 1]
93 54 38 97 88
90 43 90 84 63
In a matrix the columns are its second axis or dimension. A second index expression indexes the columns.
q)m[2 1;2]
38 90
Omitted index expressions
Omitting an index expression selects all indices in the corresponding dimension.
q)m[2;]
93 54 38 97 88
q)m[;]
12 10 1 90 73
90 43 90 84 63
93 54 38 97 88
A general null is the same as an omitted index.
q)m[;2]
1 90 38
q)m[::;2]
1 90 38
Indices of a table
A table is a list of like dictionaries. Its first axis is indexed by nonnegative integers. Its second axis is indexed by the column names as symbols.
q)a
species genus feet

cow mammal 4
duck bird 2
python reptile 0
snail mollusc 1
q)a[3 2] / rows
species genus feet

snail mollusc 1
python reptile 0
q)a[1;`genus`species] / rows and columns
`bird`duck
q)a[;`genus`species] / columns
mammal cow
bird duck
reptile python
mollusc snail
Syntactic sugar allows you to index a table with just column names. It returns the columns as lists. (Table columns are most often, but not always, vectors.)
q)a[`genus`species]
mammal bird reptile mollusc
cow duck python snail
Invalid indices
An invalid (out of range) index to a vector returns a null of the same type as the vector.
q)(til 4)[4] / long null
0N
q)(0.5*til 4)[4] / float null
0n
q)now:.z.d+.z.t / current timestamp
q)first 0#now
0Np
q)`a`b`c`d[4] / symbol null
`
There is no boolean null. Instead you get 0b
.
q)1001b[4] / false
0b
For a list of strings you get an empty string.
q)("quick";"brown";"fox")[4]
""
For an empty general list you get – an empty general list.
q)() ~ ()[4]
1b
For any other general list you get … ah, that’s tricky. We’ll come back to that in the exercises.
q)(42;3.14159;"string";`symbol)[4]
0N
q)(42 43;3.14159;"string";`symbol)[4]
`long$()
q)(`symbol;42;3.14159;"string")[4]
`
A list of functions is not a vector, but a general list.
An invalid index returns the Identity operator ::
.
q)type (+;;*;%)
0h
q)(::) ~ (+;;*;%)[4]
1b
For a dictionary, the above rules are applied to the value list.
q)d
ATW "Whitney"
KEI "Iverson"
SJT "Taylor"
q)d[`JHS] / empty string
""
q)show e:`ATW`KEI`SJT!(`Arthur`Whitney;`Ken`Iverson;`Stephen`Taylor)
ATW Arthur Whitney
KEI Ken Iverson
SJT Stephen Taylor
q)e[`JHS] / empty symbol vector
`symbol$()
For a table, the above rules are applied to each of the column lists.
q)a[4 2]
species genus feet

python reptile 0
q)a[4]
species `
genus  `
feet  0N
Invalid indices to general lists
When handing long vectors, mapping invalid indices to nulls is immensely valuable. Similarly for dictionaries and tables.
Q has corresponding rules for general lists, but the rules are by no means intuitive. The exercises below are not intended to help you to master and employ these rules for general lists, but to alert you to the difficulties.
Let your code use only valid indices to general lists.
Multiple dimensions
 Rectangular list

A list in which the items are either all atoms or all rectangular lists of the same length.
Vectors and matrices are rectangular.
 Rank

The rank of a list is the number of index expressions with which you can index it.
A vector can be indexed by only one expression; a matrix by two. Their respective ranks are 1 and 2.
In other contexts the rank of a rectangular list is known as the number of its axes or dimensions.
Do not confuse rank with the
rank
keyword, which concerns sorting.  Shape

The shape of a rectangular list is a vector of its counts on each axis.
The shape of a vector is a 1item vector; the shape of a matrix is a 2item vector.
You can form a rectangular list of shape
shp
withshp#
.
Functional indexing
The Index and Index At operators make indexing a function.
. / Index
@ / Index At
Index
You can select any item in a rank\(N\) list with the Index operator and a length\(N\) vector.
q)show m:3 4#til 12
0 1 2 3
4 5 6 7
8 9 10 11
q).[m;1 2]
6
q)m . 1 2
6
You can make a rectangular selection of a rank\(N\) list with a length\(N\) list.
q)m . (2 0;1)
9 1
Index At
The items of the right argument of Index are index expressions in successive dimensions of a list.
The items of the right argument of Index At @
are index expressions in its first dimension.
That is, Index At indexes only items of the first axis; it does not ‘dive deep’.
q)m
0 1 2 3
4 5 6 7
8 9 10 11
q)m . 2 1
9
q)m @ 2 1
8 9 10 11
4 5 6 7
Exercises
No peeking.
Attempt each exercise before looking at its answer.
The exercises clarify your thinking. Reading answers does not.

All data structures can be indexed. But what data structures cannot be used as index expressions? For example, can a dictionary be used as an index expression? A list of dictionaries? A table? In other words, in
x@y
what cany
not be?Answer
q)" #" ( 1 0 1; 1 1 0 1 1) / nested integers "# #" "## ##" q)" #" ( 1 0 1; 111011b) / indices of mixed type "# #" "### ##" q)" #" d:`tom`dick!01b / dictionary with index values tom  dick # q)" #" ( 1 0 1; 111011b; d) / list of mixed types including dict "# #" "### ##" `tom`dick!" #" q)" #" (d;d;d) / list of like dictionaries (table) 'type [0] " #" (d;d;d) ^ q)e:`tom`dick!(101b;11011b) / list of unlike dictionaries q)" #" (d;e) 'type [0] " #" (d;e) ^ q)" #" ( 1 0 1; 111011b; e;d) "# #" "### ##" `tom`dick!("# #";"## ##") `tom`dick!" #" q)show f:`one`two`three!(11100b;d;e) one  11100b two  `tom`dick!01b three `tom`dick!(101b;11011b) q)" #" f one  "### " two  `tom`dick!" #" three `tom`dick!("# #";"## ##") q)" #" f`two`three 'type [0] " #" f`two`three ^
A list of dictionaries – like or unlike – cannot be an index expression.

In your q session:
q)"abcdef" 1 0 3 6 1 4 4 5 "bad beef"
Explain the space in the result.
Answer
The space corresponds to index 6, which is invalid – out of range. An invalid index returns a null. The char null is a space.

In your q session:
Explain what you see.b:6 cut 011100100010111100111110100000000000b / boolean matrix a:" a"!(5 5 5 5 5;0 1 3 1 1) / alphabet dictionary b a "a" " #" b a "a"
Answer
q)b a "a" 011100b 100010b 111110b 100010b 100010b q)" #" b a "a" " ### " "# # " "##### " "# # " "# # "
Above:
b
is a matrix of rows in a 6column digital display.a
is a dictionary mapping each letter to a list of rows fromb
.

In dictionary
a
make entries for keys"b"
,"c"
,"d"
,"e"
, and"f"
.Evaluate
" #" b a "c"
and" #" b a "e"
.Answer
q)a["bcdef"]:(3 1 2 1 3;0 1 4 1 0;3 1 1 1 3;3 4 2 4 3;3 4 2 4 4) q)a  5 5 5 5 5 a 0 1 3 1 1 b 3 1 2 1 3 c 0 1 4 1 0 d 3 1 1 1 3 e 3 4 2 4 3 f 3 4 2 4 4 q)" #" b a "c" " ### " "# # " "# " "# # " " ### " q)" #" b a "e" "##### " "# " "#### " "# " "##### "

Evaluate:
(,'/) " #" b a "bad beef"
If you see a length error signalled, check the previous exercise.
Describe in English what the above expression does. How would you say
,'/
?Answer
q)(,'/)" #" b a "bad beef" "##### ### ##### ##### ##### ##### ##### " "# # # # # # # # # # # " "#### ##### # # #### #### #### #### " "# # # # # # # # # # # " "##### # # ##### ##### ##### ##### # "
Expression
" #" b a "bad beef"
maps each letter to its list of rows
 each list of rows to a bitmap
 each bitmap to a char matrix of spaces and hashes
Finally,
,'/
(Join Each Over) joins corresponding rows from each char matrix into the result char matrix. 
Show different ways you can set the type of an empty list.
Answer
"b"$() / Cast 0#0b / zero Take

Make an array (rectangular list) of rank 3 by using its shape as the left argument of Take.
Answer
q)show a3:2 3 4#.Q.a "abcd" "efgh" "ijkl" "mnop" "qrst" "uvwx"

Write a function
shp
that returns the shape of a rectangular list. Write a lambda or compositionrnk
that returns its rank. What is the shape and rank of an atom? Why is that?Answer
q)shp:{1_ count each first scan x} q)shp a3 2 3 4 q)shp "string" ,6 q)rnk:count shp :: q)rnk a3 3 q)rnk 3 0 q)shp 3 `long$() q)shp a3 / 3 axes 2 3 4 q)shp (101b;010b) / 2 axes (matrix) 2 3 q)shp 42 43 / 1 axes (vector) ,2 q)shp 42 / no axes (atom) `long$() q)rnk 42 0
With each step above, we remove an indexable dimension (axis) and the shape shortens. The atom
42
has no indexable dimensions; its shape is thus an empty vector and its rank is zero. 
How could you determine the rank of a nonrectangular list?
Answer
The concepts of rank and shape are inherited from languages in which arrays are only rectangular. A q list is an array only while it happens to be rectangular. It’s not obvious that the concepts of shape and rank are helpful with a nested list.
However, you might usefully determine whether a list is an array:
And haveq)same:{all x=first x} q)rect:{$[0>type x;0b;same count each x]} q)array:{$[rect x;all rect each x;0b]} q)array a3 1b q)show n3:(("abc";"def");("ghi";"jklm")) "abc" "def" "ghi" "jklm" q)array n3 0b
shp
signal a type error if its argument is not an array.q)shp:{$[array x;1_ count each first scan x;'`type]} q)shp a3 2 3 4 q)count shp a3 3 q)count shp n3 'type [0] rnk n3 ^
 Define a 4×5 matrix
m
and a variableind
so thatm . ind
returns the third column ofm
.
Answer
The right argument of Apply is a list of argument values; the right argument of Index is a list of index expressions. In either case, the general nullq)show m:4 5#.Q.a "abcde" "fghij" "klmno" "pqrst" q)m[;2] "chmr" q)ind:(::;2) q)m . ind "chmr"
::
corresponds to an omitted item.  Define a 4×5 matrix

The
@
key on your keyboard has stopped working. Write a lambdaat
to use instead. (Assume the indices are all valid.)Answer
The right argument of Index
.
is a list of expressions indexing the left argument on successive axes.The right argument of Index At
@
is a list of expressions indexing the left argument on its first axis. That is to say, it selects only the items of the left argument and makes no deeper selections.Its right argument is the first and only right argument of Index:
x @ y <==> x . enlist y at:{x . enlist y}