17 August 2013

This is the first of five posts [1, 2, 3, 4, 5] on this subject.

PostgreSQL implementation of SQL is Turing Complete, as show by Andrew Gierth on PostgreSQL wiki, where a Cyclic Tag System (CTS), which is proven Turing-complete, is implemented using WITH RECURSIVE, a WINDOW function, CASE conditional expressions and an OUTER JOIN. Although the proof is sound, there is a long path from a CTS to an actual Turing Machine (TM).

The first question I want to answer is: how to build a Turing Machine with SQL only?

The second question I would like to investigate is: what SQL features are actually required to build a relational Turing Machine. For instance, can it be done without WITH RECURSIVE, WINDOW functions and OUTER JOIN? (Teaser for later posts: yes!)

## Turing Machine with Arrays

In this post I will show how to build a TM in SQL with the following features: WITH RECURSIVE to iterate till the machine stops, INNER JOIN to get transition and state informations, ARRAY operations to build and store the tape, one COALESCE function to deal with the end of the tape, and a sub-SELECT with an ORDER clause to initiate the tape.

Obviously the use of ARRAYs make this solution not really relational, so this is cheating, but it is in SQL, it works, and allows to introduce the general setup that I will use for other more relational solutions.

### Turing Machine description

First, the TM will be reprensented by a set of relations for the states, symbols, transitions, and the initial tape contents.

The TM run will be recorded in the following table:

### Turing Machine execution

Let us now record a run with a recursive query:

Note that on each iteration, the next iterations of all rows seems to be recomputed over and over. This is not really the case because of the peculiar behavior of WITH RECURSIVE discussed in the next post. Basically, only just-generated tuples are used to compute the next iteration.

The COALESCE call creates implicit blanks at the end of the tape.

You can try this self-contained SQL script which implements a Turing Machine for accepting the AnBnCn language using the above method.

In the next post, I show how to put the tape in a separate TABLE so that the TM is really relational, although it is at the price of using SQL functions.