Minds and Computers : An Introduction to the Philosophy of Artificial Intelligence

Figure 7.1
Two states.

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purely mechanically, without any appreciation of its significance. We
will call such a task an e
ffective procedure.
The application of [2] to state A to generate state B relies, for its
e
ffectivity, on the fact that states of a chess game are effectively dis-
tinguishable. The concept of e
ffectivity will be doing some work in the
next few chapters, so let’s discuss those obvious features of chess I
mentioned, by virtue of which states of chess are e
ffectively distin-
guishable.
Firstly, there is no ambiguity about where pieces begin or end, nor
about where squares begin or end. Pieces are discretely bounded
spatial objects and squares have clearly delineated borders. Secondly,
there is never any ambiguity about whether or not a piece is ‘in’ a
square. These conditions are necessary in order that there can be
formal rules which govern the legitimate ways in which pieces can
move – a characteristic feature of chess and all such games.
To say that states of chess are e
ffectively distinguishable is to say
that there is an e
ffective procedure which will decide the matter. You
should be getting some sense by now of what an e
ffective procedure
might be – [2] is an example of one. Let’s try to nail down a working
definition. Let us call a procedure e
ffective iff it can be achieved
by merely following a specified set of steps – a list of instructions, a
recipe – without any understanding of the significance or meaning of
the task. This list of instructions we can call an algorithm. Another
way of referring to the e
ffectivity of a procedure is to say that it
is algorithmic – that there is an algorithm for implementing the
procedure.
The term ‘algorithmic’ has common-parlance usage – you may
have heard someone call a task or chore purely algorithmic to
describe that it is simply a matter of going through the motions,
taking the prescribed steps – a mundane task. In other words, an
e
ffective procedure.
It is fairly clear that distinguishing states of chess is algorithmic.
Let’s specify the algorithm:
1. Begin with square a1.
2. Repeat step 3 sixty-four times unless instructed to halt. When
instructed to move on to the next square, move to the adjacent
square on the right-hand side if there is one, otherwise move to the
leftmost square of the row immediately above.
3. If the square in state A is empty then:
– if the corresponding square in state B is empty, move on to the
next square, otherwise halt and utter ‘the two states are di
fferent’.
 
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If the square in state A is occupied then:
– if the corresponding square in state B is occupied by a piece of
the same form, move on to the next square, otherwise halt and
utter ‘the two states are di
fferent’.
4. Utter ‘the two states are formally equivalent’.
Although specifying this algorithm amounts to a rather tedious
spelling-out of a task which is, for us, both simple and obvious to
execute, it does drive home the point that I need bring no under-
standing to bear in order to achieve the task. I do not need to know
that I am comparing states of chess. I do not need to know what chess
is. I do not need even need to know what a game is. I need only follow
the formally specified instructions.
You might think that this algorithm is not a good representation
of how you conceive of yourself going about the task. When pre-
sented with state A and state B and asked to compare them, it is
immediately obvious to us how they di
ffer (without beginning com-
parison at square a1, moving on to a2, etc.), even though they di
ffer
only in respect of the positioning of one piece. This is partially
because state A is a regular pattern on a su
fficiently small scale. If
the pieces were distributed more randomly on the board, no doubt
we would find ourselves more closely following the algorithm above.
If the board were ten times longer along each side – 6,400 squares
in area – and there were ten times as many pieces, our only hope for
a correct comparison of two states would be to follow the above
algorithm.
In any case, the point here is merely that the task can be achieved
by following an algorithm and, hence, is e
ffective.
The final aspect of e
ffectivity we need to appreciate is the finitude