Koskenniemi (1983a, 1983b, 1984) proposed a rule-system for describing morphological regularities in a language, depending centrally on the idea of matching two sequences of symbols—a lexical string (made up of the lexical forms of morphemes) and a surface string (the sequence of characters in the normal, inflected, form of the word). (In general, symbols could be orthographic or phonological; here we shall abstract from this linguistic issue, and merely consider strings of atomic symbols, which could be phonemes, typewritten characters, or any other separate entities).

Koskenniemi (1983a) originally described the rules in two alternative forms— high-level rules and finite-state transducers. The high-level rules were given only an informal interpretation, and were used as an expository device to state the linguistic regularities more perspicuously. The formalism that was actually used to write and im­plement two-level morphological grammars was parallel combinations of finite-state transducers. Koskenniemi's own implementation was an interpreter for parallel trans­ducers, which were directly written by the linguist as rules in their own right. Also, various linguistic analyses presented in Dalrymple et al. (1983) follow this approach, expressing rules as transition tables for transducers, and Antworth (1990) describes a recent implementation based wholly on transducers being written directly. Never­theless, Koskenniemi conjectured that an automatic compilation procedure could be

* An earlier version of part of this work was presented at the Fourth Conference of the European Chapter of the Association for Computational Linguistics, Manchester, April 1989.

f Department of Artificial Intelligence, 80 South Bridge, Edinburgh EH1 IHN, Scotland, g.d.ritchie@ed.ac.uk

devised to transform the more readable high-level form into the more directly im-plementable parallel transducer form. Koskenniemi (1985) refined the notation and sketched a compilation method, although he still did not provide a detailed declar­ative definition of the meaning of the high-level rules. Various implementations that compile variants of the notation into arrangements of transducers have been devel­oped (e.g. Karttunen et al. 1987; Ritchie et al. 1991).

The general view of this earlier work was that the rule notation was a mere "syntactic sugar" for parallel combinations of arbitrary transducers, and that there was no difference in meaning or power between the two formalisms.

This paper establishes the following:

(i) an alternative (declarative) statement of the meaning of the high-level rule notation is possible, without recourse to compilation into finite-state transducers;

(ii) the usual two-level morphological mechanism is more limited than arbitrary transducers in its ability to define relationships between strings;

(iii) the use of a special boundary symbol slightly increases the generative power of the model;

(iv) for any two-level morphological grammar, there is an equivalent one in a simpler normal form;

(v) the family of languages generated by two-level rules is not closed under union or complementation, but is closed under intersection.

The original notation proposed in Koskenniemi (1983a) included some rather complex notational conventions that have not survived into later versions. The formalization given here will deal only with the core ideas, as embodied in Koskenniemi (1985) (and other implementations such as Karttunen et al. 1987; Ritchie et al. 1991).

To understand how two-level rules operate, it is useful to consider four separate levels, as depicted in Figure 1. The topmost level contains lexical forms, which are strings of symbols in some suitable lexical alphabet. Each string can be thought of as the abstract representation of the phonology or orthography of an individual mor­pheme or lexeme. The second level consists of a notional automaton "tape" made up of any concatenation of strings from the first level (the lexicon), with special null symbols inserted anywhere between the symbols of the lexical alphabet (optionally). Hence a finite set of lexical strings characterizes an infinite set of possible "lexical tapes," produced by concatenation and insertion of nulls. The third level is a similar "surface tape" consisting of symbols in a surface alphabet, with the null symbol also appearing at arbitrary positions. The fourth level, at the bottom in Figure 1, represents the word as spelt (or phonetically represented) in the surface alphabet; the relation­ship between the surface tape and the surface form is that the latter can be produced from the former by removing all occurrences of the special null symbol. The ordinary

Graeme Ritchie Languages Generated by Two-Level Morphological Rules

Lexical Forms: move +ed Surface Form: moved

Figure 1

Lexical and surface tapes and forms.

surface form (such as moved) is related to a sequence of lexical forms (such as move and +ed) if a surface tape and a lexical tape exist such that:

1. the lexical tape is the concatenation of the lexical forms in question, possibly with the addition of nulls;

2. the surface tape contains the symbols of the surface form, possibly with the addition of nulls; 3. the two tapes have the same number of symbols, counting nulls; 4. the contents of the tapes are related by the two-level rules.

That is, the two-level rules define possible pairs of equal-length sequences (the two "tapes"), and the actual surface form and lexical forms are then related by general conventions acting on these sequences. The formalism as a whole can be used to describe phenomena involving a surface string of a different length from the lexical forms, via the conventions regarding null symbols, but the rules themselves are written as statements about symbol-strings of equal length. Hence, the rules can be viewed as statements about sequences of symbol-pairs, where each symbol-pair consists of one symbol from the lexical tape and its counterpart from the surface tape. Conventionally, symbol-pairs are written with an infix colon, rather than as a pair within parentheses. The rules specify which sequences of symbol-pairs are or are not valid, by supplying contexts (sequences of symbol-pairs) within which a particular symbol-pair may or must occur. For example, the rather trivial rule would specify that the pair e:0 could occur only if it were surrounded by v:v on the left and +:0 on the right, which is a rough approximation to the phenomenon illustrated by Figure 1.

There are three types of rule, but they all have the same overall form—a single symbol-pair whose behaviour is being specified, followed by an operator (=>, <= or <=>)/ a left context, a punctuation ("___" in this paper) to indicate where the symbol pair appears relative to the context, and a right context.

The two-level rule compiler of Karttunen et al. (1987) was based on sophisticated manipulation of regular expressions in order to define very precisely and rigorously the set of transducers produced by the compiler. Kaplan (1988) generalized some of this work further, by formalizing the algebraic manipulations of regular sets of sequences of pairs of symbols, which he called regular relations. These relations bear the same relationship to finite-state transducers that regular languages bear to finite-state machines, and his formal definition is exactly analogous to that of a regular language:

the empty set is a regular relation; the set consisting of the empty string is a regular relation; the set consisting of a single ordered pair of symbols, either of which may be the empty string, is a regular relation; if Ri and R2 are regular relations, so are U Ri, R\Rz, and R\ (i.e. the union of the two sets, the set consisting of concatenations of elements from the two

1 The material in this section is based solely on Kaplan (1988). For the results reported, there are no formally published details that I am aware of.

Having established the basis for discussion, we can now formulate a precise set-theoretic definition of two-level rules, so as to go on to investigate their formal prop­erties.

Given any two finite symbolic alphabets, A and A', a symbol-pair from A and A' is a pair (a, a') where a € A and a' e A'. Such symbol-pairs will normally be written as "a:a"'. A symbol-pair sequence from A and A' is simply a sequence (possibly empty) of symbol-pairs from A and A', and a symbol-pair language over A and A' is a set of symbol-pair sequences (i.e. a subset of (A x A')*).

Given two alphabets A and A', and a symbol-pair sequence E from A and A', a sequence (Pi,..P„) of symbol-pair sequences from A and A' is said to be a partition of E iff (i.e. E is made up of the concatenation of the P,).

Given two alphabets A and A', a two-level morphological rule over A and A' consists of a pair (P, C) where P is a symbol-pair from A and A', and C is a nonempty set of pairs (LC,RC) where LC and RC are sets of symbol-pair sequences from A and A'; each such set of symbol-pair sequences is called a context set. The reason for having C be a set of pairs of context sets, is that we must cater, in the general case, for there being a disjunction of pairs of context sets, as in the illustrative rule given in Section 2 earlier. In the case where the set is a singleton, this reduces to the simple (nondisjunctive) case. Only one of the formal proofs that follow depends upon C being a finite set, but if infinite sets of context-pairs were required, some suitable notation would have to be devised for expressing such infinite sets. There seems to be no linguistic motivation to go beyond finite sets of pairs of context sets. Each individual context set (i.e. LC or RC in the above notation) can be an infinite set, and often is.

The above definition is the generalization beyond regular context expressions men­tioned in the previous section. To specialize it to the traditional two-level case, we need a further definition:

A two-level morphological rule is said to be regular if all the context sets in it are regular sets. (Notice that even here we have abstracted from the actual notation used to represent regular sets).

A set 6 is said to match at the right-end a symbol-pair sequence E iff there is a partition (Pi, P2) of E such that P2 € 9.

A set 6 is said to match at the left-end a symbol-pair sequence E iff there is a partition (Pa, P2) of E such that Pi e 6.

A set R of two-level morphological rules contextually allows a symbol-pair se­quence E iff, for every partition (Pi, a:a', P2) of E, either there is no rule of the form (a:a', C) G R, or there is at least one rule (a:a', C) e R such that C contains a pair (LC, RC) such that LC matches Pi at the right end and RC matches P2 at the left end.

A two-level morphological rule ((a,a'), C) coercively allows a symbol-pair se­quence E iff for every possible partition (Pi, b:b', P2) of E and every element (LC, RC) € C such that LC matches Pi at the right end, and RC matches P2 at the left end, if b = a, then b' = a'.

An alternative but equally useful variation on the last definition would be that a two-level morphological rule ((a,a')z C) coercively disallows a symbol-pair sequence E iff there is a possible partition (Pi, b:b', P2) of E and an element (LC, RC) e C such that LC matches Pi at the right end, RC matches P2 at the left end, b = a and b' ^ a'.

Real implementations of two-level morphology have to consider the issue of what counts as a feasible pair within a two-level grammar. Roughly speaking, the set of feasible pairs is the set of symbol-pairs that have to be considered as potential elements of symbol-pair sequences. The normal approach is not to allow the whole cross-product A x A' as possible symbol pairs, but to define a subset of this as being the effective alphabet under consideration. This is normally done in three ways:

• Any symbol a that appears in both the lexical alphabet and the surface alphabet gives rise to a feasible pair a : a.

• Any symbol-pair that is explicitly mentioned in a context-expression anywhere in a rule is feasible.

• Any symbol-pair that is explicitly declared to be feasible (in a list supplied along with the rules) is feasible.

The set of feasible pairs is then used in two ways—any variables or sets occurring in the statement of rules are deemed to range only over feasible pairs, not over arbitrary symbol-pairs; also, any feasible pair may occur freely in a symbol-pair sequence if not otherwise constrained by the rules of the grammar.

It can be seen that these definitions and conventions are to some extent bound up with the concrete textual representation of rules, and the way of stating symbolically the contents of the grammar. At the set-theoretic level of abstraction here, a slightly different (but compatible) set of definitions is necessary. We are assuming that no variables or symbol-set-mnemonics appear in our rules, so the question of using the feasible pair set to expand or give meaning to such abbreviations is irrelevant, but there is still the question of freedom of occurrence.

The notion of implicit definition of feasibility can be altered so that instead of referring to occurrence within a context-expression, it refers to occurrence with a set (for regular sets specified using disjunction, concatenation and indefinite iteration, the two are equivalent). Explicit listing of additional feasible pairs can be represented by including context restriction rules of the form where e denotes the empty sequence. A trivial rule like this allows the symbol-pair to occur freely, since any adjacent material will match the empty string. (The exact definition of matching is important here—matching the empty string or sequence means that there is some partition of the surrounding string in which the portion of the partition next to the symbol-pair of interest is the empty sequence; it does not mean that there are no adjacent symbol-pairs).

That is, no extra mechanism is needed for adding feasible pairs—what would ap­pear in a practical implementation as a declaration of an enumerated list can be viewed as a convention for defining rather degenerate context restriction rules. The point is that the original textually-based definitions of feasibility are merely a notational con­venience for conveying (to the human reader or a software interpreter/compiler) a set of symbol-pairs that includes at least the pairs from the contexts (and the identity pairs), and we shall abstractly regard that set as being part of the definition of the grammar, regardless of the notation used to make it manifest.

Given a two-level morphological grammar R = (CR, SC), a symbol-pair sequence £ is generated by R iff all the following hold:

3. the set CR of rules contextually allows E.

As mentioned earlier, the two classes of rules are treated slightly differently— surface coercion rules are conjoined, forming a set of constraints all of which must be met, and the context restriction rules for a given symbol pair are disjoined, giving a set of possible licensing contexts. If no rules apply to a particular symbol-pair, it is acceptable if and only if it is feasible.

With the above definitions, it is now possible to ask what sorts of symbol-pair languages can be characterized using a two-level morphological grammar.

Let CR be a set of two-level morphological rules. Let E\ and E2 be symbol-pair se­quences such that CR contextually allows Ei, and CR contextually allows E2. Then CR contextually allows the concatenation EiE2.

Let a:a' be a symbol-pair occurring in EiE2, such that (Pi, a:a',P2) is a partition of EiE2. Assume, without loss of generality, that a:a' occurs in Ei. That is, Pi is a proper initial subsequence of Ei and P2 = S2E2 for some sequence S2. Since CR contextually allows Ei, for the partition (Pi, a:a', S2) of Ei there is at least one rule C in CR that contains at least one context-pair (LC, RC) such that LC matches Pi at the right end and RC matches S2 at the left end. If RC matches S2 at the left end, then RC will also match S2E2 = P2 at the left end. Hence, for the partition (Pi, a:a', P2) of EiE2 there is at least one rule C in CR that contains at least one context-pair (LC, RC) such that LC matches Pi at the right end and RC matches P2 at the left end. A similar argument can be given for the occurrence of a:a' being in E2. Since this will be true for any such a:a' in EiE2, CR contextually allows EiE2. ■

Graeme Ritchie

Corollary

If a two-level grammar is of the form (CR, 0) (i.e. it contains no surface coercion rules), then the concatenation of any two strings in its language is also in its language.

Let R = (a:a', C) be a two-level morphological rule. Let E\, E2l e3 be symbol-pair sequences such that E1E2E3 is coercively allowed by R. Then E2 is coercively allowed by R.

If E2 were not coercively allowed by R, it would mean that there is a partition (Si, a:b, S2) of E2 such that for some (LC, RC) in C, LC matches Si at the right end, RC matches S2 at the left end, and b^ a'. If this were the case, there would be a corresponding partition (E1S1, a:b, S2E3) of E1E2E3, with LC matching E1S1 at the right end, and RC matching S2E3 at the left end. This would (by definition) mean that R does not coercively allow EiE2E3, which is not the case by hypothesis. ■ Corollary (a)

Let C be a set of two-level morphological rules, all of which coercively allow a symbol-pair sequence E. Then all of the rules in C coercively allow any subsequence of E.

Corollary (b)

If all the context restriction rules in a two-level morphological grammar are trivial, in the sense of having vacuous contexts (see Section 5 above), then any substring of an element of its language is also in the language.

Lemma 3 (The Concatenation Property)

Let G be a two-level morphological grammar (CR, SC), and let L(G) be the set of symbol-pair sequences generated by G. Suppose that there are sequences Ei, E2, E3, E4 such that E2 G L(G), E3 G L(G), and E1E2E3Ei G L(G). Then E2E3 G L(G).

(i) Since E1E2E3E4 G L(G), all the symbol-pairs in it are feasible with respect to G, hence all the symbol-pairs in E2E3 are feasible.

(ii) Since E2 and E3 G L(G), it follows that CR contextually allows E2 and E3 (by definition). By Lemma 1 above, this means that CR contextually allows E2E3.

(iii) Since E1E2E3E4 G L(G), it follows (by definition) that all of the rules in SC coercively allow EiE2E3E4. Hence, by Corollary (a) to Lemma 2 above, all of the rules in SC coercively allow E2E3.

This establishes the three defining conditions for E2E3 G L(G). ■ 7. Comparison with Transducers

As mentioned in the introduction, two-level grammars have historically been written in two different ways—as rules as defined here, and as sets of finite-state transduc­ers. In the latter case, each transducer deals with some linguistic phenomenon, and a sequence of symbol-pairs is generated by the grammar if every transducer in the gram­mar accepts it. That is, the symbol-pair sequence must be in the intersection of the lan­guages accepted by the transducers (viewed as acceptors); in procedural terms, this is often referred to as "having the transducers executed in parallel." Hence, when work­ing with the transducer formalism the linguist has to devise independent transducers whose intersection is the required language. These transducers, like the two-level rules, define a mapping between equal-length symbol sequences, as described earlier; there are no "empty transitions." Under these conditions, since the intersection of a set of regular languages is also a regular language, it follows that these parallel finite-state acceptors define exactly the regular sets of symbol-pair sequences.

As observed earlier, Kaplan's work on regular relations shows that the "parallel transducer" model is at least as powerful as the two-level grammar model defined earlier. The obvious question is whether there is a difference in power; in fact, there is:

Theorem 1

There are regular sets of symbol-pair sequences (i.e. symbol-pair languages character­ized by regular expressions of symbol-pairs) that cannot be generated by any two-level morphological grammar.

This follows directly from Lemma 3 above. Any language L generated by a two-level morphological grammar must have the property that if E2, E3, and E1E2E3E4 G L, then e2e3 6 L. There are regular symbol-pair languages that do not have this property, such as the language defined by the regular expression which contains b:b and a:a b:b but not b:b a:a b:b, even though that sequence is a subsequence of other elements of the language. ■

It was already clear that there are some regular relations that cannot be generated by a two-level grammar, since a regular relation can put into correspondence symbol-sequences of different lengths; what the above result shows is that there are some equal-length regular relations that cannot be generated by any two-level grammar. That is, we have the following proper inclusions:

languages generated by regular two-level morphological grammars C regular sets of symbol-pair sequences C regular relations

There is another, rather trivial, difference between the power of two-level morpholog­ical rules and arbitrary regular expressions. According to the definitions given here, the empty sequence of symbol-pairs is in every language generated by a two-level morphological grammar, since it conforms to the definition regardless of the content of the rules. The definitions could be altered to exclude the empty sequence from ev­ery language, but it is hard to see how the rule mechanism could be used to allow the empty sequence in some languages but not others.

Karttunen et al. (1987) allow a further rule operator /<= in their rules. A rule of the form means, informally, that the pair a:b must not occur if the contexts LC and RC are present. It would be straightforward to extend the definition of a two-level morpho­logical grammar to cover this symbol. What would be needed would be to allow a grammar to be a triple (CR, SC, CE) in which CR and SC are as before, but CE is a set of context exclusion rules of this new sort. The applicability of such rules could be defined thus:

A symbol-pair sequence E is contextually excluded by a rule (a:b, C) if there is a partition (Si, a:b, S2) of E and some context-pair (LC, RC) G C such that LC matches Si at the right end and RC matches S2 at the left end.

The definition of generation of a sequence E by a grammar (CR, SC, CE) would have to be amended to include the additional stipulation (see Section 6 above):

4. no rule in CE contextually excludes E.

Inclusion of this type of context exclusion rule does not affect the proof of Lemmas 1 and 2, or their corollaries, and the Concatenation Property (Lemma 3) can still be proven since the following lemma can be proven by a very simple modification to the proof for Lemma 2 :

Let R = (a:a', C) be a two-level morphological rule. Let Ei, E2, e3 be symbol-pair se­quences such that EiE2E3 is not contextually excluded by R. Then E2 is not contextually excluded by R.

Hence, the use of context exclusion rules does not affect any of the results about generative power given earlier, or the lack of closure demonstrated in Section 11 below.

David Weir (personal communication) has pointed out that the use of context ex­clusion rules makes surface coercion rules technically redundant, since any stipulation of surface coercion can be restated as a set of context exclusion rules.

The two-level formalism, as defined above (and as originally defined by Koskenniemi) has no word-boundary symbol to mark the end of a sequence of symbols. Although the linguist is free to introduce any symbols that seem empirically useful, none of these symbols has any specially defined status beyond what the linguist chooses to state in the actual two-level morphological grammar, and the apparatus does not stipulate that a particular symbol occurs only at the start or end of a complete string. The

PC-KIMMO implementation of two-level morphology (Antworth 1990) has a special boundary marker "#" with the following properties:

1. # can appear in both the lexical and the surface component of a symbol-pair.

2. # can be paired only with itself, not with any other symbol.

3. The pair # : # occurs at the extreme ends of every string.

4. The pair # : # never occurs at any internal position of a string.

5. Any two-level rule may refer to the pair # : #, thereby making phenomena it describes relative to the end of the sequence.

6. The symbol-pair # : # is not regarded as part of the sequence generated; that is, the rules characterize an extended string with # : # at each end, but the generated sequence is defined to be the sequence without boundary markers.

It would be possible, with careful design of the two-level rules, to enforce points 1-5 without resorting to special treatment for # : #, but point 6 steps outside the two-level mechanism; a grammar that merely enforced 1-5 using its rules would, according to the basic definitions of generation, have in its language strings that contained # : # (at the ends). Karttunen et al. (1987) also allow a lexical boundary marker #, such that the symbol-pair # : 0 meets points 3, 4, and 5 above (which can be achieved by writing suitable two-level rules without any need for special treatment). However, they do not need to rely on the additional stipulation given in 6, since the use of a surface null will give the desired effect when considering the entire mapping from lexical forms to surface form, with explicit null symbols being removed in the manner outlined in Section 2 above (see Ritchie 1989 for a formal definition of this phase).

We can show that the inclusion of a specially treated boundary symbol slightly alters the generative power of the formalism, since this allows any set that can be included as a context set to constitute the entire language, as follows.

Given two sets A and A', a symbol a € A, a symbol ß e A', and a two-level morphological grammar G based on alphabets A and A', then a symbol-pair sequence E e (A x A')* is said to be generated by G with boundary a:ß iff

2. a:ß E a:ß is generated by G.

Notice that under this definition, a sequence E may well be "generated by G with boundary a:ß" even though E itself is not generated by G.

Theorem 2

Let A and A' be sets of symbols, let a be some symbol not in A, ß some symbol not in A' and let L be some set of symbol-pair sequences from A x A'. Then there is a two-level morphological grammar G based on A U { a } and A' U { ß } such that

L = {£ I G generates E with boundary a : ß}

Graeme Ritchie

Corollary

Consider the two-level grammar (written in the usual textual notation):

where C is some expression denoting the set L (in the usual notation based on regular sets, C would be a regular expression). More formally, this grammar G (= (CR, SC)) could be written set-theoretically as where L\ is the set consisting of all possible concatenations of elements of L with a:ß, and L2 is the set consisting of all possible concatenations of a:ß with elements of L.

It is straightforward to verify that this grammar generates all and only strings of the form a:ßY,a:ß for E <E L. ■

This theorem may sound rather odd, since it implies that any set whatsoever can be generated in this manner (in contrast to the earlier result that there are some regular languages of symbol-pairs that cannot be generated by two-level rules). However, it is important to note that this theorem not only uses a special boundary symbol, it relies on the generalization of the usual regular-set based two-level rules to rules that allow any set as a context set. Essentially it says that, with the augmentation of a special boundary symbol, the generative power is limited only by the limits placed on the class of sets that can appear as contexts. A more useful and natural specialization of this theorem is as follows:

For any regular set L of symbol-pair sequences, there is a regular two-level morpho­logical grammar that generates L with boundary a:ß, for some symbols a, ß that are not in the alphabets used for L.

We define a two-level morphological grammar as being in normal form if there is no symbol-pair that is the subject of more than one context restriction rule or more than one surface coercion rule. More formally, a grammar (CR, SC) is said to be in normal form iff whenever (a:b, C) e CR and (a:b, C) G CR, C = C, and whenever (a:b, C) e SC and (a:b, C) e SC, C = C.

Theorem 3

For any two-level morphological grammar G, there is a corresponding two-level mor­phological grammar G' in normal form that generates the same symbol-pair language.

Suppose G = (CR, SC) is not in normal form. Create the grammar G' = (CR', SC) as follows. If there is exactly one rule (a:b, C) in CR for the symbol-pair a:b, include that rule in CR'. For any set of two or more rules in CR with the same symbol-pair a : b

include in CR' a rule: (a : b, C\ U C2... C„).

Create SC from SC in exactly the same way. It should be clear from this that (a) any symbol-pair a:b occurs in the rules in G iff it occurs in the rules in G'; (b) any pair of context sets (LC, RC) appears in a rule in CR iff it appears in a rule in CR' with the same symbol-pair; (c) any pair of context sets (LC, RC) appears in a rule in SC iff it appears in a rule in SC with the same symbol-pair.

It is straightforward to prove the equivalence for G and G' of the three defining criteria for language membership. ■

Most classes of formal languages (e.g. regular languages, context-free languages) are closed under at least some of the simple set-theoretic operations such as union and intersection. Certain results can be proved for two-level morphological languages con­cerning their closure (or lack of it).

Theorem 4

The set of languages generated by two-level morphological grammars is not closed under union.

The proof follows from the "concatenation property" of Lemma 3 above, which allows a simple counterexample to be constructed. Consider the grammar G\ which would conventionally be written as (if we follow the normal practice of omitting textually any context set containing only the empty sequence). This generates the language (b : b)*; that is,

The grammar G2 given by the rules (again, omitting mention of contexts that are trivially satisfiable) generates the lan­guage (a:a b:b)*:

e, a:a b:b, a:a b:b a:a b:b, ... The union of these languages contains the sequences

Graeme Ritchie but does not contain the sequence

and hence, by Lemma 3, cannot be a language generated by a two-level language. ■ The following result and construction are due to David Weir (personal communi­cation).

Theorem 5

The set of languages generated by two-level morphological grammars is closed under intersection.

Let L] and L2 be two-level morphological languages, generated by the grammars Gi=(GRi,SCi) and G2=(CR2, SC2) respectively. It is possible to construct a two-level grammar for the intersection L = Li n L2. Without loss of generality, assume (merely to simplify the construction):

1. For every rule in CRi or CR2 of the form (P,C) let the set C contain just one pair of the form (LC,RC). This assumes that the set of context-pairs in a rule is never an infinite set—see Section 4 above. There can be several such rules for any given symbol-pair P (i.e. we have to allow the grammar to be not in the "normal form" defined in Section 10 above). Note that we do not need to make this assumption for the rules in SCi and SC2.

2. For every pair a : b that is a feasible pair in both Gi and G2 there is at least one rule in CRi and CR2 for the symbol pair a : b. In the absence of any other rule include (a : b, ({e}, {e})).

Suppose that $ is the set of symbol-pairs that are feasible in both Gi and G2, so that <I>* is the set of sequences of such symbol-pairs.

To create the context restriction rules for the intersection grammar G, take where CR' is defined to be:

where a' is a new symbol that is not in a feasible pair in either Gj or G2.

The set SC' defined below has the effect of making the context a' : a' unobtainable. Hence, the above rule makes each a : b that is not feasible in both grammars behave as if it were not feasible in the new grammar. There will be no symbol-pair that is the subject of a rule in CR' and in CR", since the first of these sets covers the jointly feasible pairs and the latter handles others.

I would like to thank Alan Black, Ron Kaplan, and David Weir for useful discussions of this material.