February 11, 2016

Pivot table is a way to get summarized data in a spreadsheet or other data processing programs. Here’s a short introduction of pivot table on Wikipedia:

For typical data entry and storage, data usually appear in flat tables, meaning that they consist of only columns and rows, as in the following example showing data on shirt types:

While tables such as these can contain many data items, it can be difficult to get summarized information from them. A pivot table can help quickly summarize the data and highlight the desired information. The usage of a pivot table is extremely broad and depends on the situation. The first question to ask is, “What am I seeking?” In the example here, let us ask, “How many Units did we sell in each Region for every Ship Date?”

A pivot table usually consists of row, column and data (or fact) fields. In this case, the column is Ship Date, the row is Region and the datum we would like to see is (sum of) Units. These fields allow several kinds of aggregations, including: sum, average, standard deviation, count, etc. In this case, the total number of units shipped is displayed here using a sum aggregation.

Recently, I tried to build a substitute version of `pivot` in a logic programming language called miniKanren. The resulting program is capable of running backwards – given a summarized data, the function could try to guess the original raw data.

## Functional Logic Programming

miniKanren, in its pure form, only provides a few functions (read about it in a previous Thursdays post). It tries to be mini, while preserving all important features of a logic programming language. This is great, but with one particular drawback, that in miniKanren, predicates, the things that form all logical expressions, are not first-class citizens. I wrote a Thursdays post about solving the problem directly by implement a microKanren programming language inside a miniKanren programming language. That solution works beautifully, however, with some overhead. Because the programming language itself is now capable of running backwards, sometimes when a predicate is complicated, the solver would take a long time to find any solution.

There’s an easier solution. miniKanren’s predicates are implemented as functions on Lisp. Functions are first-class citizens in Lisp, so we can just add functions back to miniKanren.

A common usage of being first-class is to map over a list of things. For example, with the following function, I can apply a predicate over a list by adding functions back to miniKanren predicates:

``````(define (mapo* p in out)
(conde
((== in '())
(== out '()))
((fresh (inf inr outf outr)
(== `(,inf . ,inr) in)
(== `(,outf . ,outr) out)
(p inf outf)
(mapo* p inr outr)))))
``````

Notice that `p` in the above predicate is another predicate. If `in` is not empty, the above predicate would apply `p` to the first element of `in`, and then recursively call `mapo*`.

## Data Structures

The predicate, let’s call it `pivoto`, applies to a list of associative list of key value pairs with key `from`, `to`, and `val`, and returns another list of associative list of key value pairs with key `from`, and all `to` values in the input as keys.

Here’s an example input:

``````'(((from . a) (to . b) (val . 1))
((from . a) (to . c) (val . 2))
((from . b) (to . c) (val . 3)))
``````

And it’s output:

``````'(((from . b) (c . 3))
((from . a) (b . 1) (c . 3)))
``````

## Implementation

The `pivoto` predicate first finds all the distinct values in the `from` column, and then apply on those distint values:

``````(define (pivoto in out)
(fresh (distincts)
(distinct-valo 'from in distincts)
(pivot-distincto distincts in out)))
``````

For those distinct values, we first select it from the original table, and then function on all the `to` values.

``````(define (pivot-distincto distincts in out)
(conde
((== distincts '())
(== out '()))
((fresh (df dr df-selected toof outf outr)
(== `(,df . ,dr) distincts)
(== `(,outf . ,outr) out)
(selecto 'from df in df-selected)
(pivot-too df-selected toof)
(== `((from . ,df) . ,toof) outf)
(pivot-distincto dr in outr)))))
``````

In that `pivot-too`, we figure out each `to`, and `val`, and then adds then back to the result table.

``````(define (pivot-too in out)
(conde
((== in '())
(== out '()))
((fresh (inf inr outf outr to val)
(== `(,inf . ,inr) in)
(== `(,outf . ,outr) out)
(item-pairo 'to to inf)
(item-pairo 'val val inf)
(== `(,to . ,val) outf)
(pivot-too inr outr)))))
``````