LOS LÍMITES AL DERECHO FUNDAMENTAL DE LIBERTAD

Even when a search is conducted in the space of only those concepts describable by the current generalisation language, the search space is large. If a is a vector of dimensionality p, where p is the number of attributes each object has, and each element, ay, of a contains the number of permissible attribute values (not including the wild card) for that attribute, then equation [2.1] shows the number, n, of possible different general object description vectors there are for the generalisation languages used here.

n=n(a;+1) [211

?=i.p

For example, in poker, there are two attributes: suit and card number. There are four suits, and thirteen cards in each suit. The vector a is [4 13]. The number of possible general object description vectors includes the possibility for a wild card being placed in either attribute. This is equivalent to saying that there is an extra suit, and an extra card number. Hence the number of possible general object description vectors, n is 70. Equation [2.2] shows the size of concept space for the generalisation languages used here, W, where there are q objects in an instance.

Mitchell's Symbolic Technique Searching For No Alternative

[2.2] where nCr is the notation for the number of combinations of n things taken r at a time, without repetitions.12 Since a given general object description vector may be repeated in a single concept, repetitions are allowed. The number of combinations of n things taken r at a time with repetitions is the same as the number of combinations of n + r-1 things taken r at a time, without repetitions.13 Hence (W + r - i)Cr is the number of combinations of n things taken r at a time, with repetitions.

In the poker concept space, there are five objects in an instance, since there are five cards in a hand of poker. The number of possible concepts is therefore 70+5-1C5, or 16 108 764. Note that this allows repetitions of cards, and hence this is not the number of possible poker hands from a single deck of cards. This figure is 52C5, or 2 598 960, for comparison.

Bidirectional search, or search that proceeds from two initial points in the search space, is part of the key to enabling the recognition of the no­ alternative situation. Bidirectional search alone cannot achieve this. To have that knowledge requires more than just searching from two initial states. It is also necessary to have only a single point where the two directions of the bidirectional search meet. Bidirectional search per se allows for the possibility of more than one meeting point of the two directions. In conjunction with search heuristics, however, to be discussed in the next section, and a partial ordering of the search space, there is the possibility for a single meeting point of the bidirectional search, and hence the assurance of the no-alternative situation.

Mitchell partially orders concept space using the more-specific-than relation. A concept, ci is more-specific-than another concept, C2 if and only if all the instances matched by ci form a proper subset of all the instances matched by C2-

For example,

12James & James, 1992, p. 66 13James & James, 1992, p. 67

Mitchell's Symbolic Technique Searching For No Alternative

ci = {[Viola odorata, *, New Forest]}

is more-specific-than

02 = {[*, *, New Forest]}

but not more-specific-than

q = {[*, April, *]}.

This is because the set of instances matched by ci is not a proper subset of the instances matched by C3. For example, the instance:

{[Viola odorata, March, New Forest] }+

is matched by ci (and ci}, but not by C3.

The partial ordering is used to guide the bidirectional search. Two sets of concept descriptions are used, S and G, the members of which form the maximal and minimal elements of all finite chains in the partial ordering. S is the set of the most specific concept descriptions that are consistent with all of the instances shown at any stage during training. That is to say, there is no concept description more specific than any member of S that is consistent with all the instances shown so far.

G is the set of the most general concept descriptions that are consistent with all of the instances shown at any stage during training. So all concept descriptions that are consistent with the instances and are not members of G are more specific than members of G, and no member of G is more specific than any other member of G.

The most specific concept descriptions are those which are only as general as is necessary such that they match all the positive instances. Conversely, the most general concept descriptions are those which are only as specific as is necessary such that they do not match all the negative instances. S and G, therefore mark the most extreme points (in terms of the partial ordering), or sets of points, that are consistent with the instances shown at a given time during training. The set of all concepts that lies between and including S and G in the partial ordering is called version space. Since

Mitchell's Symbolic Technique Searching For No Alternative

version space contains all concepts that are consistent with the instances shown so far, the search can be restricted to version space. Mitchell summarises the above as follows:

The advantage of the version space strategy lies in the fact that the set G summarises the information implicit in the negative instances that bounds the acceptable level of generality of hypotheses, while the set S summarises the information from the positive instances that limits the acceptable level of specialisation of hypothesis. Therefore, testing whether a given generalisation is consistent with all the observed instances is logically equivalent to testing whether it lies between the sets S and G in the partial ordering of generalisations.^

Version space is the essence of Mitchell's technique. At any time, during training, version space contains only those concepts that are consistent with the instances shown. This means that all the positive instances shown are instances of all concepts in version space, and all the negative instances shown are not instances of all concepts in version space. Therefore, all the possibilities for the final no-alternative concept are contained in version space at any given time, since all the possibilities for the final concept must match all the positive instances so far, and not match any of the negative instances so far.

To see how version space, V, cuts down the search for the instance and generalisation languages used here, it is worth showing the size of version space at the initial stage, after the presentation of the first positive instance. (Version space is undefined before this.) This is given in equation [2.3], where there are p elements in the general object description vector, and q general object description vectors in a concept:

#V0=2M [2.3]

Intuitively, this can be understood by realising that all the possible concepts in version space have either a * or not a * in each individual element of the set of general object description vectors. This is because, after the presentation of the first positive instance, the possible attribute

Mitchell's Symbolic Technique Searching For No Alternative

values for each attribute are known. In order to match the positive instance, every concept in version space must have either a wild card, or the same attribute value as in the instance, in each element of each general object description vector.

In the poker world, where the number of possible concepts is roughly sixteen million, the size of version space after the first positive instance is presented is 210, or 1 024, which represents a scale down factor of just under sixteen thousand. This is certainly a remarkably large reduction in the volume of search space. In the proverbial hunt for needles, it is roughly equivalent to cutting down the search from a haystack to a mere armful of hay.

S and G, and the partial ordering are the means by which version space is represented in Mitchell's technique:

In general, the number of plausible versions can be very large (possibly infinite) when the language of patterns for rules [concepts] is complex. The key to an efficient representation of version spaces lies in observing that a general-to-specific ordering is defined on the rule pattern space by the pattern matching procedure used for applying rules. The version space may be represented in terms of its maximal and minimal elements according to this ordering.15