## Combinatorics on Words. Progress and Perspectives

Recall from above that we defined the partition set of an inventory as the set of paradigms that can be derived from it. For the three minimal complete inventories of the two cell case, the partition sets can be read off the partitions column of Table 2.

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Typological evidence ultimately can inform us which partitions are attested. However, the typological evidence cannot alone decide between different inventories that both predict the same possible partitions like inventories 1 and 2 above. Despite the relatively trivial nature of the exercise with the two-cell paradigm space, the preceding discussion demonstrates that assumptions have consequences, and the the assumption that UG inventories be both minimal and valid has reduced the space of possible inventories from 7 or 8 with the empty set to 3.

We have shown how typological evidence can be brought to bear on the choice. Finally, we note that the two minimal valid inventories that are capable of generating both AA and AB patterns are in fact related to one another by a permutation of the cells. Since we have taken the order of cells in a list to be arbitrary, there is no way on our assumptions to distinguish among inventories that are permutations of one another in this way.

Minimality prefers one of the first three inventories; if there is syncretism in some paradigms, then one of the first two is to be preferred. But considerations of Minimality and Validity alone do not resolve the venerable debate about which value of number is marked Sauerland et al.

Our first result is that in a paradigm space that constitutes only a binary opposition, the only minimal valid analyses that also permit syncretism are the ones that takes UG to have a single feature that names one member of the opposition, and which is contrasted with a default feature, compatible with both members. In this way, there would be an empirically-grounded argument to be made that if Minimality is assumed, then Binarity should be rejected as a general condition on feature inventories. In the manner just noted, the two assumptions make contrasting predictions about the state of the world.

Turning to the three-cell paradigm space, we begin to see the growth in the space of analytical possibilities, and we also see how various assumptions such as Minimality and intrinsic, i. If features could be freely chosen to form inventories, then distinct feature inventories could in principle be constructed from these features including the empty set , i. Each inventory is in turn relatable to n! We now consider the degree to which the assumptions mentioned above restrict the space of possible grammars analyses. The first restriction we impose is Validity, as in For example, the inventory f , f , f is valid, in that it describes a three-way contrast, while the inventory f , f , f is invalid — it is not complete, as it provides no means to describe the third cell.

It turns out that 96 of the possible inventories of active features are valid in this sense in the three cell case see Table 3 below. With four cells, the ratio is 31, out of 32, see Table A2 below. Validity thus restricts the number of feature sets, but the restriction is not particularly strong.

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Table showing which 12 of the 16 logically possible three cell partition sets can be generated by feature inventories applying intersective closure and by how many inventories. Partition sets related by cell permutation are grouped together. The more interesting and less obviously empirically motivated requirement is Minimality. As defined above, a minimal valid feature inventory is an inventory that contains the minimal number of features needed to describe the maximally differentiated partition. For the two-cell space, the minimality requirement does not restrict the possibilities in any interesting way it excludes only one inventory out of the 4 valid ones , but for the three-cell space, the minimal number of features that is needed to describe the maximally differentiated partition is two, as we show presently.

Validity plus Minimality together thus restrict the choice from among different logically possible feature inventories to the following three:. To see that 28 are the minimal valid inventories, consider first that they are indeed each valid: 29 gives the rules of exponence that generate the maximally differentiated partition from the inventory in 28c. For 28a and 28b analogous sequences can be specified.

Now consider minimality: Obviously, no inventory with fewer than two features can be valid, hence we only need to show that the inventories in 28 are the only valid two cell inventories. Assume that there was another valid inventory I with only two features. Because 28 lists all combinations of the features f , f , and f , I would need to contain one of f , f and f , or f But it is easy to see that for any these features, it is impossible to satisfy validity by only adding one further feature to I.

Hence, 28 are the three minimal inventories for three cells. Note that in order to describe the ABC pattern, the rules of exponence must be partially ordered, such that the exponent of the conjoined features takes preference over the rules in 29b — 29c this holds for any of the three inventories in Because each of the inventories in 28 has two basic features that may be conjoined to define a third feature, the number of possible sequences for each inventory is 16, although many of these sequences will be redundant or incomplete.

To see this consider first that if the order of 29b and 29c is changed as in 30 , the resulting sequence still derives the complete partition ABC.

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Other orders of the rules in 29 render feature f redundant. This corresponds to an intrinsic order: if the conjoined rule is active, it must be ordered before its individual conjuncts. If the conjoined rule is omitted or not ordered first, it can be omitted and only the order between the two rules referring to the basic features matters. The following table shows, for one inventory, the six possible sequences six distinct orders of three rules and the three corresponding partitions that are derived.

As before, redundant elements in the sequences are in parentheses.

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The analogous table for the other two choices can be readily constructed. As an expository device, we use green text to indicate a feature that is derived as the intersection of the two basic features. As explained in section 2. What 34 shows is the following: There are only three minimal valid feature inventories that can generate a maximally differentiated three-celled paradigm space. The first two lines in 34 derive the same surface patterns partitions , since the ordering of the last two rules is irrelevant.

As the reader may verify, the other two minimal valid inventories in 28 have the same properties as The three inventories amount to permutations in the order of the cells, but are otherwise identical in their formal properties. Each inventory generates a partition set that contains only two of the three logically possible bifurcations of the paradigm.

Since we have not stipulated a meaningful order of the paradigm cells, the three are equivalent, up to linear order. At this point, we note two properties we believe to be of theoretical interest. However, imposing the conditions of Validity and Minimality on the UG feature inventories restricts the expressive power of the system, such that each inventory generates only 3 of the 5 possible partitions. The three inventories that are permitted are moreover linear permutations of one another.

We believe this is of interest since it appears to be true at least in some domains that the number of attested partitions is a small subset of the logically possible ones. Being able to predict restrictions on the space of possibilities is thus of potential theoretical interest, if the restrictions indeed line up with the data.

In the case at hand, the following restrictions obtain:. Of the five possible partitions of a three-cell space, four show some differentiation among the cells. However, each of the inventories in 28 generates only three of those partitions. As in the case of inventory 3 in the two-celled paradigms, we are now able to connect our formal results to potential empirical evidence.

If there is, as we have hypothesized, a fact of the matter for some domain, such that UG contains only one of the inventories in 28 , then this should show up as the following empirical generalization: across the relevant domain, only three of the four possible patterns of differentiated partition should be attested. No sequence from that inventory will generate a pattern in which the first and last cell share an exponent, to the exclusion of the middle cell.

The same holds for the other two inventories, up to the linear order of the cells: each inventory will fail to generate exactly one of the possible partly syncretic partitions. As discussed above, since the linear order of the cells is arbitrary, these inventories and partition sets are permutations of one another, and thus each can be reduced to the first via permutation of the cells. In this way, there is, in what we develop here, a formal equivalence among partition sets that differ only as a function of linear permutations of the cells.

In other words, what we have just shown has two parts. We have just done so. Before that discussion, we note one further point about these inventories. No valid, minimal inventory for a 3-cell paradigm space generates the maximally undifferentiated partition AAA.

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Curiously, it is not a general property of our assumptions that such undifferentiated partitions are universally excluded in the minimally valid inventories, and we show below that it does not hold for four cells. We note this, but leave it as an unexplored aspect of the system.

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Total syncretism appears to exist, of course, and we do not exclude it across the board. We return to this issue again in section 5. The upshot of that section will be that the equivalent of the 3 inventories considered to be minimal valid inventories with intersection become three among a larger class of minimally valid inventories including the containment patterns. Some inventories from among the larger class permit AAA, but the general result holds: no member of that larger class admits all three bifurcations of the paradigm space: any minimal valid inventory whose partition set contains ABB and AAB will necessarily exclude ABA.

Thus far, we have examined only the three minimal valid inventories that generate a three-cell paradigm. To evaluate the effect of minimality, we now look also at non-minimal inventories. In the two-cell case, we were able to present a complete discussion of all the possible inventories and of the partition sets described by each inventory. There were only 8 possible inventories for the features definable over a two-cell paradigm, and 4 inventories were invalid.

But for a three cell space, there are inventories, and numerous sequences to consider. Table 3 provides a summary of important aspects of the grammar of three-celled paradigms and the models that generate them. In the next paragraphs, we walk through this table in some detail, identifying various properties that are of potential interest.

Possibly of more interest, we note that there are some partition sets that do not arise under any constellation of the assumptions considered here.

follow url Even without Minimality, for example, feature inventories turn out to be somewhat restrictive. Table 3 is divided horizontally into two halves. We discuss the differences below. The columns in Table 3 represent possible partition sets of a three-cell paradigm space, using colour instead of letters, as in 35 above: the same colour in two cells indicates the same exponent syncretism.

The header of each column represents a distinct partition set, and the number in a given column represents the number of formally distinct valid inventories that can in principle generate that set. The three minimal valid inventories that we have discussed above are in the top row of the top half of the table columns 6—8. These are the only three valid, two-feature inventories.

[Mini-course]: Combinatorics on words. Lecture 2 - part 2

But the table provides a range of information about what happens if we do not include the minimality requirement. Assuming extrinsic rule ordering is allowed, there is exactly one choice of an inventory with three features, from among the 7 possible features, which yields only an ABC partition. We have seen that already; it was the inventory in 5. If that inventory is chosen, from among the possible inventories, then the only partition that can be generated is ABC.

On the same line, the number in the rightmost column is 3. There are exactly three distinct choices of feature inventories from each of which all five logically possible inventories can be derived. One such inventory is f , f , f , i.