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tags: cs1101s
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# CS1101S Studio S11 (T05J)

## Streams II

### Question 1
Describe the streams A and B produced by the following definitions. Assume that `integers` is the stream of positive integers (starting from 1):

function scale_stream(c, stream) {
    return stream_map(x => c * x, stream);
}

const A = pair(1, () => scale_stream(2, A));

function mul_streams(a,b) {
    return pair(head(a) * head(b),
                () => mul_streams(stream_tail(a), stream_tail(b)));
}

const B = pair(1, () => mul_streams(B, integers));

### Question 2
To get some initial practice with streams, write definitions for each of the following:

* `alt_ones`: the stream 1, -1, 1, -1, ... in as many ways as you can think of.
* `zeroes`: the infinite stream of 0's. Do this using `alt_ones` in as many ways as you can think of.

Now, show how to define the series:

$$
\begin{align}
S_1 &= 1 + x + x^2 + x^3 + \cdots \\
S_2 &= 1 + 2x + 3x^2 + 4x^3 + \cdots
\end{align}
$$

//seeu sim
const ones = pair(1, ()=> ones);
const alt_ones = pair(ones, () => pair(negate_series(ones), () => alt_ones));

const zeroes = add_series(ones, negate_series(ones));
const s1 = fun_to_series( x => 1);
const s2 = fun_to_series( x => x + 1);



// ai lin
const alt_ones = pair(1 , () => stream_map(x => -x, alt_ones));
const alt_ones = pair(1, () => pair(-1, () => alt_ones));
const zeros = stream_map(x => 0, alt_ones);
const S1 = pair(1, () => S1);
const S2 = pair(1, () => stream_map(x => x + 1, S2));

## Bonus Round

### Question 1
Given a stream `s` the following function returns a stream of pairs of elements from `s`:

function stream_pairs(s) {
    return is_null(s)
        ? null
        : stream_append(
            stream_map(
                sn => pair(head(s), sn),
                stream_tail(s)),
            stream_pairs(stream_tail(s)));
}

Suppose that `ints` is the (finite) stream 1, 2, 3, 4, 5. What is `stream_pairs(ints)`

Give the clearest explanation you can of how `stream_pairs` works.

Suppose that `integers` is the infinite stream of positive integers. What is the result of evaluating

const s2 = stream_pairs(integers);

(1,2), (1,3),.... , (1, n) as n -> inf
Hence recursion error (max call stack exceeded)

Hint: Note the function `stream_append` is defined Source S3 as follows:

function stream_append(xs, ys) {
    return is_null(xs)
        ? ys
        : pair(head(xs),
               () => stream_append(stream_tail(xs),
                                   ys))
}

Consider the following variant of `stream_append`, called `stream_append_pickle` and the function `stream_pairs2` which makes use of it.

function stream_append_pickle(xs, ys) {
    return is_null(xs)
        ? ys()
        : pair(head(xs),
               () => stream_append_pickle(stream_tail(xs),
                                          ys));
}

function stream_pairs2(s) {
    return is_null(s)
        ? null
        : stream_append_pickle(
            stream_map(
                sn => pair(head(s), sn),
                stream_tail(s)),
            () => stream_pairs2(stream_tail(s)));
}

const s2 = stream_pairs2(integers);

Why does the function `stream_pair2` solve the problem that arose in the previous question?

What are the first few elements of `stream_pairs2(integers)`? Can you suggest a modification of `stream_pairs2` that would be more appropriate in dealing with infinite streams?

### Question 2