Reducing Lattices to Languages

The main problem is that we cannot treat concepts like letters in a language. We will now show some preliminary results on how we might nonetheless reduce them in some way to simple languages. The technical side will be quite self-explaining, but one has to be careful what these results mean. They donotmean that results from string-based pre-theories can be carried over to concepts. They mean that the results can be transferred if we are happy with thinking of our concepts as the basic letters of a new language, that is, the basic language-theoretic objects, which only in the very final spell-out are substituted by strings. This might be quite desirable: we can think of our concepts as syntactic categories (or rather: pre-categories), and project not the string language, but the language of syntactic categories. I guess most linguists would be fine with this procedure, and many would prefer it over a procedure which simply works with strings. The main reason is: presumably, it makes language less messy than it is on the level of visible strings. And from a language-theoretic point of view, our procedure of pre-categorization is very well-defined, so on grounds of formality there is nothing to object. Still we have to be aware that we lose the very immediate touch with the “visible language” we had before, regardless of whether we consider this an advantage or rather a disadvantage.

Given a languageL⊆Σ^∗, we denote byS(L) the partially ordered monoid ([Σ^∗]_∼L,≤_L,•), where≤_L is the linguistic order, which is extended from strings, where it forms a pre-order, to equivalence classes of the form [w]~ _L:={~v:~v∼_Lw},~ where it forms a partial order; and• is defined by [w]~ L•[~v]L = [w~~v]L. Note that the monoid operation • is also not free, and in general, we might have [w]~ L[~v]L 6= [~w~v]L. (see [9] on this problem).

We now present the extensionality lemma, which correlates the extension of concepts (set-theoretically) with their combinatorial properties in the lattice.

This result does not hold in all residuated lattices, but only for syntactic concept lattices! The reason that this can be despite the completeness theorem in [72] is that it is not an inequation of the lattice itself, but a statement in our meta-language.

Lemma 102 (Extensionality Lemma)

1. ForX, Y ∈SCL(L), if X 6=Y, then there are X1, X2 ∈ SCL(L), such that X1◦X◦X2≤ C(L) andX1◦Y ◦X26≤ C(L)or vice versa.

2. If for allX1, X2 ∈SCL(L), we have X1◦X ◦X2 ≤X1◦Y ◦X2, then X ≤Y.

Proof. 1. Assume we haveX 6=Y. Then we also haveX^.6=Y^.. Assume wlog that there is (~x, ~y)∈X^.,(~x, ~y)∈/ Y^.. Then it follows thatC(~x)◦X◦ C(~y)≤ C(L), because~xX~y⊆L, and for all~x⁰∈ C(x),~x⁰ ≤_Lx, for all~y⁰∈ C(~y),~y⁰≤_L~y.

Conversely, as (~x, ~y)∈/ Y^., we have~xY ~y6⊆L, and soC(~x)◦X◦ C(~y)6≤ C(L).

2. Contraposition: assume we have X6≤Y. Then by the Galois-connection, we have Y^. 6⊆X^., and so there exists (~x, ~y) ∈Y^.,(~x, ~y)∈/ X^.. As in 1., we know that C(~x)◦Y ◦ C(~y) ≤ C(L), butC(~x)◦X◦ C(~y)6≤ C(L). From this it follows that C(~x)◦X◦ C(~y)6≤ C(~x)◦Y ◦ C(~y).

In the sequel, we will embed (monoid reducts of) concept lattices in languages.

In all of what is to follow, we are interested in (combinations of) concepts being

4.9. ANALOGIES AND INFERENCES WITH POWERSETS 117 smaller than C(L), the concept of the strings of the language with respect to which we form our concepts. There is one problem about this approach: let L ⊆ Σ^∗. There is always a largest conceptC(Σ^∗); and it can easily happen (though not necessarily) that we have (Σ^∗)^. = ∅. In that case, there are no conceptsX, Y such thatX◦ C(Σ^∗)◦Y ≤ C(L), and consequently, the techniques presented below do not work for this concept. In particular, there is no general way to translate this concept into an equivalence class, because there might be a language, where each string in Σ^∗ figures as some substring, yet the language does not equal Σ^∗ – just think of the palindrome or copy languages! So for all that is to follow, we exclude this concept from consideration, which does no harm as it is trivial and syntactically uninformative anyway.

Definition 103 We say a SCL over a language L⁰ is embedded in a language L⊆Σ^∗, if there is a injective mapi:SCL(L⁰)→[Σ]_∼L, such that i(C₁◦ C₂) = [i(C1)i(C2)]L, and C1≤ C2 ⇔i(C1)≤L i(C2). In other words, we require there to be an embedding of the reduct (BL⁰,◦,C()) in S(L). If in addition, i is surjective, we writeS(L)∼=SCL(L⁰).

A first reduction result goes as follows:

Theorem 104 For each finite language I, concept latticeSCL(I), there is a finite languageJ over a finite alphabet such that S(J)∼=SCL(I).

Proof. Let I be a finite language and SCL(I) its concept lattice. We construct Σ as follows: for everyX ∈SCL(I), we have a letterx∈Σ, such that there is a bijectioni:SCL(I)→Σ. We now defineI_C ⊆Σ^∗ byx₁x₂...x_i∈I_C, if and only if we havei⁻¹(x₁)◦i⁻¹(x₂)◦...◦i⁻¹(x_i)≤ C(I).

We now prove thatS(IC)∼=SCL(I). First, we extendi⁻¹ to strings in the usual fashion: i⁻¹(a ~w) =i⁻¹(a)◦i⁻¹(w); we thus have a homomorphism~ i⁻¹ from Σ^∗ onto the syntactic concepts.

1. There is a bijection from SCL(I) to [Σ^∗]∼IC.

Let Y ∈[Σ^∗]∼_IC be an equivalence class overIC, and letY^. be the set of its contexts in IC. Then by definition, for (x1...xi, xj...xk)∈Y^., xl...xn ∈Y, we havex1..xixl...xnxj...xk ∈IC, and therebyi⁻¹(x1..xixl...xnxj...xk)≤ C(I). In turn, by the extensionality lemma, this means that for everyY ∈[Σ^∗]_∼I

C, we have a separate concept. Now assume we have two conceptsX, X⁰. Then by the extensionality lemma, we have a distinguishing context inI_C. This shows that there is a bijection between concepts ofI and equivalence classes ofI_C.

2. ◦

We just have to show that ifX◦Y =Z, then we havei(X)i(Y)∼IC i(Z). This is straightforward, because we havei⁻¹(xy)≤ C(I) if and only ifi⁻¹(z)≤ C(I).

3. ≤

We show thatX ≤Y if and only ifi(X)≤ICi(Y).

Only if: assume X ≤Y. Then it follows that if haveX ◦Y ◦X⁰ ≤ C(I), thenX◦X◦X⁰≤ C(I). So ifi(X)i(Y)i(X⁰)∈I_C, then i(X)i(X)i(X⁰)∈I_C.

If: Assume we have ~z ≤IC ~u. Then we have ~x~u~x⁰ ∈ I_C ⇒ ~x~z~x⁰ ∈ I_C; consequently,i⁻¹(~x)◦i⁻¹(~u)◦i⁻¹~x⁰≤ C(I)⇒i⁻¹(~x)◦i⁻¹(~z)◦i⁻¹~x⁰≤ C(I).

By the extensionality lemma part 2, it follows thati⁻¹(~z)≤i⁻¹(~u).

Note that the proof works equally well with an infinite languageL; but if the language is not regular, we will have an infinite alphabet forLC. This is a very

118 CHAPTER 4. THE CLASSICAL METATHEORY OF LANGUAGE good result, but note that the usage of equivalence classes puts some difficulties to us:

Lemma 105 Let I⊆Σ^∗ be a finite language, and assume that we have~x∼_I

~

y, ~x1∼I~y1, ~x2∼I ~y2, where~x6=~yand~xi6=~yifori∈ {1,2}. Then~x6≈^{P r}_I ~x1~x~x2, and~y6≈^{P r}_I ~y1~y~y2.

Proof. Assume wlog that~x₁6=~y₁. First of all, we have to make a premiss explicit, which is implicit in the assumption thatI is finite, namely that~x6v~y, and~x₁6v~y₁, and vice versa. That follows from a basic lemma on finite languages.

By assumption, that ~x≈^{P r}_I ~x₁~x~x₂, it follows that we have some w~~x~v ∈I, where (~w, ~v) is not recursive for (~x, ~x₁~x~x₂). Then we have w~~x₁~x~x₂~v ∈I. By

~

x∼I ~y we can infer w~~x1~y~x2~v ∈I. Now, (~w~x1, ~x2~v) is not a recursive context for (~y, ~y1~y~y2) (otherwise~x1 would be a substring of~y1 or vice versa); therefore we havew~~x1~y1~y~y2~x2~v ∈I. Therefore, we have w~~x1~y1~x~y2~x2~v ∈I, and for the same reason as above, (w~~x1~y1, ~y2~x2~v) is not recursive for (~x, ~x1~x~x2), therefore we have...and so on, soI must be infinite, contradiction.

So this shows us: we cannot just talk about equivalence classes as we can talk about strings; the notion of pseudo-recursion is fundamentally at odds with the notion of equivalence classes. This result also puts us in guard: in interpreting concepts as equivalence classes we have to be very careful, and the first reduction theorem is not as useful as it seems.

Now the question is: assuming we can perform a reduction back to strings without any substantial loss, we are (almost) back where we have been before;

maybe we have not lost anything substantial with respect to the simple approach – but what have we gained? The answer to this question lies in the properties of our concept language, which has particular properties corresponding to the concept lattice. For example, we recognized the problem that with string-based pre-theories we cannot distinguish between different distributions of strings, for example: ~xin the position where both~x, ~y occur. Concepts obviously solve this problem. How is the solution preserved in the language reduction? The key is a property of the reduction language.

Definition 106 We say a languageL ⊆Σ^∗ is distributional, if every X ∈ SCL(L), there is aw~ ∈Σ^∗ such that X =↓Lw~ :={~v:~v≤Lw}.~

This means: for every distribution (closed set of contexts), which is charac-terized by the occurrence of some set of strings, we find a single word which characterizes it; more explicitly: for everyS ⊆Σ^∗, there is a wordw~ ∈Σ^∗ such that~x ~w~y∈Liff~xS~y⊆L(excluded the special case where there is no context forS). For distributional languages, we find the following nice property.

Given a setS⊆Σ^∗, we haveS^u:={~t: for all~s∈S,~s≤_L~t); and similarly, S^l := {~t : for all ~s ∈ S, ~t ≤L ~s). In the context of the order induced by a language, the setS^u denotes the set of all strings~t, such that for any~x, ~y∈Σ^∗, if~x~t~y∈L, then~x~s~y ∈Lfor all~s∈S. So it is the set of substrings, for which we can substitute all~s∈S. S^l conversely is the set of strings, which we can take as substitute for all~s∈S: ifw~ ∈S^l,~s∈S, then from~x~s~y∈Lit follows that~x ~w~y∈L.

The DM-completion of a partially ordered set (P,≤) is the lattice ({A⊆ P : A^ul = A},⊆). For a partially ordered monoid, it respects the monoid operation, and creates unique preserves meets and joins, where we define for

4.10. CONTEXT-FREENESS AND BEYOND:SCLN 119