18 KLEENE
the event does not occur. The possible inputs on k neurons
are described by k x p tables of 0's and 1 * s with columns for
> - - - > Jlyr and rows for t = p, . . ., 1 . As p varies over all positive
integers, these are all tables of 0's and 1*s with k columns and any
finite number of rows.
In Section 5 we used k x £ tables to describe inputs over the
last £ moments ending with the present.
Now that our time has an initial moment 1 , we must be careful
whenever we give a table with k colunns and a finite number, say £, of
rows to describe an input, to make it clear whether we intend it to describe
the input over the complete past (so p = £ ) or only over the last £ mo
ments of the past (so p > £ ). In the one case we call the table initial,
in the other non-initial. The table may be thought of as carrying a tag
saying, respectively, p = £ or p > £ . There was no necessity for this in
Sections b and 5, as there tables referring to the complete past were infinite.
A definite event of length £ in Section 5 was one in which the
partition of the inputs over the complete past is such that any two inputs
which agree in the upper £ rows of their tables always fall into the same
one of the subclasses. But now when p < £ there wonft be £ rows in the
table describing the input. For such a p, can the event occur? The con
vention we adopt is that the event shall not occur in this case. Thus the
inputs of the first subclass for a definite event of length
£
are those
described by a set of non-initial k x £ tables. If E 1 is the logical
formula we used in Section 5 to describe a definite event, the event is now
described by E 1 & p >
£
. The negation of this is E 1 V p < £, while the
formula for the complementary definite event of length £ is E 1 & p > £,
which is not equivalent, except for £ = 1 when the "& p > £ " and
" v p < £ " are superfluous.
The identical event (written 1^ or briefly I) which occurs no
matter what the input (the second subclass of the partition being empty) is
a definite event of length 1 ; the improper event (written 1^. or I)
which never occurs (the first subclass being empty) can be considered as a
definite event of length £ for every £ .
REPRESENTATION OP EVENTS IN NERVE NETS
19
With the sole exception of the improper event, a given event can
be definite of length i for only one i , and the set of the k x I tables
(all of them non-initial) which describes it is unique. This was not the
case in Section 5, as there a definite event of length i was also definite
of length m for each m > but now this would be absurd (except for the
improper event) as the extra specification that p > m would contradict
that the event can occur for p=i, ...,m-l. vp
them for a given k and & ) are those which arise most naturally from those
considered in Section 5 by taking into account that now the past may not
include i moments.
event on k neurons of length &, by changing the specification for all the
tables that p > l to p = l ; we do not include the improper event among
these. These definite events we call initial. For a given k and z , there
2k^
are 2 - 1 of them. An event can be an initial definite event for only one
i, and the set of the k x I tables (all of them initial) which describes
it is unique. If E 1 & p > l is a given non-initial definite event not im
proper, E 1 & p = l is the corresponding initial definite event.
but now the events can refer to the value of p. This may seem somewhat
unnatural; but, reversing the standpoint from which we were led to this In
6.1 and 6.2, if we are to analyze nerve nets in general, starting from arbi
trary initial states, we are forced to give p an absolute status. This is
illustrated in Figures 16-21, where the formula gives for each net the
"solution" for L 1, i.e., the condition for its firing. The "+" indicates
initial firing of the indicated neuron; inner neurons not bearing a " + " are
initially quiet.
' V w i l U U C U .1 w O LJJL -L ^ JL Jk-J — } * • • J i l l “ I •
The definite events we have just finished describing (22 of
We now find it advantageous to introduce also a new kind of definite
In Section 5 p entered the formulas for events only relatively
L 1(P) - p > 1. L,(p) = p > 3 .
L,(P) a P < 3 .
FIGURE 16
FIGURE IT
FIGURE 18
L ^ P ) = P = 1 • L 1 (p) s p = 3.
FIGURE 19 FIGURE 20
L,(P) =[P = 1 (mod 3 )]•
FIGURE 21
20
KLEENE
Our theory can now include the case k = 0. (In Sections b and 5
we were assuming k > 1 , which of course Is required for nets without
circles; Lemma 1 and Theorem 1 Corollary 3 hold for k = 0.) For k = 0
there are exactly three definite events of a given length Z , namely p > Z ,
p = z and p /= p; only p /= p Is positive. The nets of Figures 16 — 21
can be considered as representing events for k = 0.
6 A KEPRESENTABILITY OF DEFINITE EVENTS. In Section 5 we showed how to
construct nets which represent definite events on k > 0 input neurons of
length Z under the assumption of an infinite past. The proof there that
the nets represent the events is valid now for non-initial (initial) definite
events for values of p > Z (p = Z ) , when the nets are started with all
inner neurons quiet at time t = 1. To make the following discussion general,
we can take the representation to be by a property of the state of the net
at t = p (cf. Lemma 1).
Using these nets now can sometimes give rise to a "hallucination"
in the sense that the state of the net at t = p has the property without
the event having occurred. By reasoning similar to that in 6.2 in connection
with Figures 13 and 'k, this will happen for suitable inputs, when p > Z
in the case of an initial definite event, and when p < Z (so Z > 1 ) in
the case of a definite event which can occur without the firing of an Input
neuron at its first moment t = p - z + 1.
Conversely these are the only cases in which it will happen. For
consider any nerve net a property of which at t = p represents a non-initial
(initial) definite event correctly for p > Z (p = Z ) , when the net is
started at t = 1 with all inner neurons quiet. For there to be a hallucina
tion when p = m < z (so Z > 1 ) means that for some input jc^ r . . c over
t = 1, ..., m the net has at t = m a state having the property which goes
with occurrence of the event. Now let the input ... jc be assigned
instead to t = i - m + 1 , . . . , Z and an input consisting of only non
firings jCj ... to t =
1
, ...,
Z -
m. With the input £j ...
the state of the inner neurons at t = Z - m + 1 must consist of all non
firings, as it did before at t = 1 . So with the input jCJ * z~m ^ 1 * * * ^m
the net will have at t = z the state it had before at t = m, which shows
that C . . . C l C. . . . p constitutes an occurrence of the event,
i & —m I m
Call a definite event of length Z prepositive, if the event is
not initial, and either Z = 1 or the event only occurs when some input
neuron fires at t = p - Z + 1 . (For k - 0, then only p > 1 and p /= p
are prepositive.) Prepositiveness is a necessary and sufficient condition
for representability in a nerve net with all the inner neurons quiet initially.
This result suggests our first method for constructing nets to
represent non-prepositive definite events. Say first the event is non
initial and z > 1. We supply the St o f Figure 16, and treat it as though
REPRESENTATION OP EVENTS IN NERVE NETS
21
it were a k + 1 - st input neuron, required to fire at t = p - & + 1
(but otherwise not taken into account) in reconsidering the event as on the
k + 1 neurons jl^> . . ., . This event on k + 1 neurons is preposi
tive, so our former methods of net construction (Section 5) apply.
A second method is to use the net of Figure 17 or 18; e.g., if the
representation is by firing an inner neuron (quiet at t = 1 ) at
t = p + s, the inhibitory endbulb of in Figure 18 shall impinge upon
*P, and the figure is for I + s = 5 . If the representation is by a property
of the state at t = p, that property shall include that of Figure 17
fire (for i = 3) or that of Figure 18 not fire (for & = *0 .
For initial definite events, the respective methods apply using
Figures 19 and 20 instead of Figures 16 and 17, respectively.
The upshot is that only by reference to artifically produced firing
of inner neurons at t = 1 could an organism recognize complete absence of
stimulation of a given duration, not preceded by stimulation; otherwise it
would not know whether the stimulation had been absent, or whether it had
itself meanwhile come into existence.
As already'remarked in 6.2, instead of an initially fired inner
neuron as In Figures 16-20, we could use an additional Input neuron K sub
ject to continual environmental stimulation.
A hallucination of the sort considered would be unlikely to have a
serious long-term or delayed effect on behavior; but when definite events are
used in building indefinite ones, this cannot be ruled out without entering
into the further problem of how the representation of events is translated
into overt responses.
For organisms, the picture of the nervous system as coming into
total activity at a fixed moment t = 1 is implausible in any case. But
t
his only means that organisms (at least those which survive) do solve their
problems for their processes of coming into activity. For artificial auto
mata or machines generally it is familiar that starting phenomena must be
taken into account.
Of course our analysis need not apply to the whole experience and
the entire nerve net of an organism, but t = 1 can be the first moment of
a limited part of its experience, and the nerve net considered a sub-net of
its whole nerve net.
7 . Regular Events
7.1 REGULAR SETS OF TABLES AND REGULAR EVENTS. In this section as in 6.3
we shall use k x 1 tables (for fixed k and various I , with each table
tagged as either non-initial or initial) to describe inputs on k neurons
, . . ., over the time t = p - i + 1 , ...,p for which an event shall
occur. But we shall not confine our attention to the case of 6.3 that the
22
KLEENE
set of tables describing when the event occurs are all of them k x £ tables
for the same £ and either all non-initial or all initial.
First we define three operations on sets of tables. If E and
F are sets of tables, E V F (their sum or disjunction) shall be the set
of tables to which a table belongs exactly if it belongs to E or belongs
to F.
If E and F are sets of tables, EF (their product) shall be
the set of tables to which a table belongs exactly if it is the result of
writing any table of F next below any non-initial table of E; if the
table of E has £^ rows and that of F has £ rows, the resulting table
has £^ + £^ rows, is initial exactly if the table of F is initial, and
describes an occurrence of an event consisting in the event described by F
having occurred ending with t = p - £1, as evidenced by the input over
t = p - £j - £p + 1, ..., p - £^, followed by the event E having occurred
ending with t = p, as evidenced by the input over t = p - ^ +1, ..., p.
The notation EF is written so that we proceed backward into the past in
reading from left to right.
Obviously E V F and EF are associative operations. We may write
E°F for F, E 1 for E, E 2 for EE, Y ? for EEE, etc.
If E and F are sets of tables, E*F (the iterate of E on F,
or briefly E iterate F) shall be the infinite sum of the sets F, EF,
EEF, . . ., or in self-explanatory symbolism F V EF V EEF V . . . or
I
n=0
The regular sets (of tables) shall be the least class of sets of
tables which includes the unit sets (i.e., the sets containing one table each)
and the empty set and which is closed under the operations of passing from
E and F to E V F, to EF and to E*F.
An event shall be regular, if there is a regular set of tables which
describes it in the sense that the event occurs or not according as the input
is described by one of the tables of the set or by none of them.
To include the case k = 0 under these definitions, we shall under
stand that for k = 0 and each £ > 1 there are two k x £ tables, one non
initial and one initial.^
Any finite set of tables is obviously regular, in particular the
empty set, and the sets of k x £ tables all with a given £ and either
all non-initial or all initial; so every definite event is regular.
In writing expressions for regular sets or the events they describe
we may omit parentheses under three associative laws ((3) — (5) in 7.2),
^ McCulloch and Pitts [19*0 ] use a term "prehensible", introduced quite
differently, but in what seems to be a related role. Since we do not under
stand their definition, we use another term.
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