J Labs

A Taste of J (2)

(1 of 18) First Steps

Look at the following J sentences; can you figure them out? The sentences are shown indented 3 spaces; the result, if any, is shown aligned to the left margin.

These simple sentences illustrate some of the core facilities of J, that will be described in this lab.

(2 of 18) First Steps (ctd)

To start, note that "verbs" (i.e. functions) in J can apply to several data items at once. Indeed, the basic datatype is an array, and J verbs are designed to apply to arrays. For example, the following adds two lists of numbers:

(3 of 18) Adverbs

The "adverb" in J takes a verb argument and returns another verb, typically related. The behaviour is much like in English: "run quickly", is the verb "run", modified by the adverb "quickly".

The J adverb / results in a verb that applies to each pair of elements, compare with above:

(4 of 18) Adverbs (ctd)

Thus, +/ forms an addition table, and similarly, */ forms a multiplication table:

(5 of 18) Adverbs (ctd)

Try entering:

```2 3 5 %/ 10 20 30```

(6 of 18) Adverbs (ctd)

The adverb ~ applies a verb with the same argument on the left as on the right, thus the following two expressions are equivalent:

(7 of 18) Adverbs (ctd)

We can combine two adverbs. The following applies the +/ addition table with the same argument on left and right:

(8 of 18) Adverbs (ctd)

Since /~ is to be used again, we give it a name "tab". This is not required, but is helpful in reading later expressions.

(9 of 18) Adverbs (ctd)

Try entering:

```^ tab 0 1 2 3```

(10 of 18) Integer

The verb i. (integer) generates the first n numbers.

Try entering:

```i. 3 4```

```i. 3 4 5```

(11 of 18) Negative Numbers

Negative numbers are shown with a leading underscore character, which is not the same as "-", the minus verb.

(12 of 18) Combining Verbs

Now lets look at the expression in parentheses:

```];*.tab;!tab```

Ignoring the semicolons for the moment, the expression contains 3 verbs:

] is the identity verb:

(13 of 18) Combining Verbs (ctd)

*. is the least common multiple verb, so *.tab is the corresponding table. For example, the LCM of 3 and 4 is 12:

(14 of 18) Combining Verbs (ctd)

! is the combinations verb. m!n is the number of ways of taking m combinations of n objects; so !tab is the corresponding table. For example, 2!4 is 6.

Note that if you look at the lower right part of the table, you can see a copy of the triangle of Pascal.

(15 of 18) Combining Verbs (ctd)

The expression in parentheses:

```];*.tab;!tab```

is therefore seen to be of the form:

```f;g;h```

for verbs f g and h. It happens that ; is also a verb, so this expression is a list of 5 verbs!

To understand this, consider a simpler list of 3 verbs commonly seen in mathematics. Suppose f and g are functions, then:

```(f + g) x```

is typically defined to be

```f(x) + g(x)```

In J, this concept is extended to any type of function. Given verbs f g and h, then:

```(f g h) x```

is defined as

```(f x) g (h x)```

Example:

(16 of 18) Combining Verbs (ctd)

A list of 3 verbs together is called a fork, and defines a new verb.

Here is another example. The verb ; links its arguments together, putting each in a box. The following is the fork: identity linked with reciprocal.

(17 of 18) Combining Verbs (ctd)

J interprets a list of 5 verbs by creating a fork from the rightmost 3 verbs, then another fork from the new verb and the two remaining verbs.

It may be seen that:

```(];*.tab;!tab)```

is a verb that returns the identity; the LCM table; and the combinations table; all linked together.

Lets try it with a different argument:

(18 of 18) Table Utility

Since verb tables are so useful for exploration, J has a standard utility adverb called "table" that creates a table bordered by its arguments.

For example:

End of Lab