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Chapter 12
Closer to the Truth: A New Model Theory
for HPSG
Frank Richter
Eberhard Karls Universit¨at T ¨ubingen
T ¨ubingen, Germany
[email protected]
The paper proceeds in three steps. Section 12.3 reviews problems with models of typical grammars (irrespective HPSG is a model theoretic grammar framework in which of the choice of meta-theory) and suggests universal re- a grammar is formulated as a pair consisting of (a) a sig- strictions on the form of HPSG grammars to amend them.
nature which generates a space of possible structures and Section 12.4 presupposes these amendments and investi- (b) a set of grammar principles which single out the well- gates the models which the existing three meta-theories formed structures among them. There are three proposals postulate. In response to the shortcomings we find, Sec- of how to precisely define the denotation of grammars tion 12.5 proposes a new definition of the meaning of within this general setting. Each proposal is accompa- HPSG grammars, together with a meta-theory of the re- nied by its own meta-theory of the ontological nature of lationship between the set of structures denoted by an the structures in the denotation of the grammar and their HPSG grammar and empirical linguistic phenomena. In relationship to empirically observable phenomena. I will the final section I conclude with a few remarks on the re- show that all three model theories face serious, if not fatal, lationship of the new proposal to its predecessors.
problems: One of them makes very idiosyncratic funda- For space reasons, I will concentrate on a rather infor- mental assumptions about the nature of linguistic theories mal discussion of the problems and their solutions. The which many linguists might not share; the other two fail presentation of the mathematical details is left for a dif- to capture the concepts they were designed to make math- ferent occasion.
ematically precise. I will propose an alternative modeltheory which takes into account the shape of actual gram-mars and fixes the shortcomings of its predecessors.
Instead of taking a realistic grammar of a natural lan- guage as my object of study, I approach the questionsof Section 12.2 with a very simple toy grammar which HPSG is an attractive candidate for studying a model the- is built in such a way that it reflects crucial properties oretic linguistic framework. It has a history of over 20 which all actual HPSG grammars in the literature share.
years, many HPSG grammars of different languages have This simplification helps to keep our modeling structures been written, and there are mathematically precise pro- at a manageable (i.e., readable) size. Crucially, for our posals about the denotation of HPSG grammars. Thus it toy grammar below it will be obvious which structures is possible to take actual grammar specifications written form its intended denotation, and we can easily investi- by linguists and investigate the classes of structures the gate whether the logical formalism supports the apparent grammars denote according to the different model theo- expectations of the linguist.
Here I want to take advantage of this fortunate situation An Example
to address the following questions: An HPSG grammar consists of (a) a signature, Σ, declar- 1. Do the models of HPSG grammars meet the appar- ing a sort hierarchy, attribute appropriateness conditions, ent intentions of the linguists who write them? And and a set of relations and their arity, and (b) a set of log- if they do not, how can we repair the problem(s) as ical statements, θ usually called the principles of gram- conservatively as possible? mar. The grammar hΣ1 θ 1i in (7) and (8) is a particularly simple example which, however, is structured like a typi- 2. Are the structures in the denotation of the grammars cal linguistic grammar.
actually compatible with the meta-theories of the A most general sort, top, is the supersort of all other meaning of grammars formulated within the HPSG sort symbols in the sort hierarchy. The attributes PHON Closer to the Truth: A New Model Theory for HPSG (for phonology) and CAT (syntactic category) are appro- c. HEAD FEATURE PRINCIPLE: priate to all signs, with values list and cat, respectively.
phrase → Attribute appropriateness is inherited by more specific sorts, in this case word and phrase, with the possibility d. CONSTITUENT ORDER PRINCIPLE: of subsorts adding further appropriate attributes. Here the sort phrase also bears the attributes H DTR (head phrase → daughter) and NH DTR (non-head daughter) for the syn- tactic tree structure. Another important feature of the  H DTR PHON 2 ∧ append( 1 2 3 ) present signature is the attribute SUBCAT, appropriate to cat. SUBCAT will be used for the selection of syntactic ar- e. APPEND PRINCIPLE: guments. Finally, the signature introduces a relation sym- bol for a ternary relation, (7) The signature Σ 1 elist∧ 2 list∧ 2 = 3 ∃ 4 ∃ 5 ∃ 6 ∧ append( 5 2 6 ) Models only contain objects labeled with maximally specific sorts (sorts without any proper subsorts in the sort hierarchy). For each appropriate attribute, there is one outgoing arc which points to an object labeled with an appropriate maximally specific sort. Informally, HPSG grammars denote a class of structures comprising all structures licensed by the signature such that all nodes in these structures also obey the well-formedness require- ments imposed by the theory. In other words, the denota- tion of the grammar comprises at least one copy of each possible well-formed structure. Such ‘complete' modelsare called exhaustive models.
The signature Σ1 together with the theory θ1 predicts Which structures do linguists expect to find in the de- exactly three well-formed signs: The words Uther and notation of grammar hΣ1 θ1i? Fig. 12.1 shows the most walks and the phrase Uther walks.
likely candidate (omitting the relation). The configura- Uther and walks are not only words in our grammar, tion with the phrasal root node 16 represents the sentence they may also occur as complete independent utterances, Uther walks; the configurations with root nodes 30 and e.g. in exclamations and elliptical statements. θ1 incor- 19 represent the words Uther and walks.
porates important HPSG principles: A WORD PRINCI- Upon reflection it is not difficult to see that these are by PLE specifies the well-formed words, a (trivial) IMMEDI- far not the only configurations licensed by our grammar.
ATE DOMINANCE (ID) PRINCIPLE specifies admissible Three kinds of problems can be readily distinguished, phrase structures, a HEAD FEATURE PRINCIPLE makes which I will call the intensionality of lists, twin structures, category information travel up syntactic head projections, and stranded structures.
and a CONSTITUENT ORDER PRINCIPLE regulates word The intensionality of lists is a side effect of the partic- order. The last principle fixes the intended meaning of the ular feature logical encoding of lists standardly adopted relation symbol append.
in HPSG. Consider the structure for the word walks un-der node 19 above. It contains three distinct elist objects (8) The theory θ1: (22, 24, 28) at the end of the PHON and SUBCAT lists of a. WORD PRINCIPLE: the verb and at the end of the SUBCAT list of its selected word  → argument. Nothing in the grammar prevents any two or even all three elist objects from being the same object.
This way we get five possible configurations for the word SUBCAT elist walks which the linguist presumably never intended to distinguish. We should clearly treat this ambiguity as an accident of encoding and get rid of it.
Twin structures are structures with more than one root SUBCAT elist node. For example, nothing would prevent the HEAD arc originating at the subcategorized object 23 in the word CAT SUBCAT elist walks from pointing to the object 35 of the word Uther in- phrase → H-DTR CAT SUBCAT stead of to the object 25. The noun object 35 would then belong to the word walks and to the word Uther. No re-strictions of the grammar would be violated, but what em- Fig. 12.2 shows such a monster in the denotation of Figure 12.1: The intended hΣ1 θ Figure 12.2: A stranded monster structure in a hΣ1 θ h5 4 4i h5 5 5i h5 11 11i h5 12 12i append = h11 2 2i h11 3 3i h11 4 4i h11 5 5i h11 11 11i h11 12 12i h12 2 2i h12 3 3i  h12 4 4i h12 5 5i h12 11 11i h12 12 12i  The monster in Fig. 12.2 is a nominal cat object whose SUBCAT list contains the phonetic string Uther and se-lects a verb and a noun. Although no such category existsin a word in the denotation of our grammar, it exists as astranded structure because the constraints that prevent itsexistence in words all operate at the sign level. It is im-mediately clear that our grammar denotes infinitely manystranded monster structures. Even worse, the architec-ture of signs in HPSG and the standard grammar princi-ples guarantee the existence of infinite classes of strandedmonster structures in realistic grammars.
Contrary to first appearances, there is no simple rem- edy for this problem. Consider a brute force restriction pirical phenomenon should correspond to linguistic struc- which states that only configurations with root nodes of ture belonging to two (or even more) independent utter- sort word and phrase may populate the linguistically rele- ances? It seems obvious to me that this kind of configura- vant models, configurations which are empirically acces- tion is not intended by linguists, and it should not occur in sible through their phonology. However, there are phrases the intended models. In this paper I will not elaborate on which require a licensing environment. In HPSG this the causes of the problem and on the full range of possible environment may in fact contribute crucial structural re- solutions. It will disappear as a side effect of the solution strictions, and its absence leads to absurd phrasal struc- to the third problem of our grammar, stranded structures.
tures. Slashed constituents – phrases which contain an Stranded structures constitute the most serious one of extraction site for a constituent without their correspond- the three types of problems with the grammar hΣ ing filler – are a straightforward example. Their seman- Stranded structures are typically structures which are tics will partly depend on the extracted constituent as ‘smaller' than utterances. As an immediate consequence, recorded in the SLASH set.
According to HPSG sig- they tend to be inaccessible to empirical observation. A natures, configurations in SLASH are smaller than signs trivial example is a configuration which looks just like (they are of sort local).
Moreover, there are hardly the configuration under the cat object 34 of Uther in any well-formedness restrictions on these local config- Fig. 12.1, the only difference being that there is no arc urations as long as the extracted constituent is not real- pointing to the cat object: It is stranded and inacces- ized as a sign in the syntactic tree. Therefore the con- sible to empirical observation, since it is not connected figurations under local objects in the SLASH set of a to a phonological value. While some of the stranded slashed constituent without its complete licensing envi- structures in the denotation of grammars are isomorphic ronment are usually not configurations which may actu- to structures which occur in observable linguistic signs ally occur in signs according to the grammar principles.
(such as the one just described), stranded monster struc- A slashed constituent without its embedding matrix envi- tures are of a shape which prevents them from being ronment might thus have an arbitrary and even impossi- possible substructures of well-formed linguistic signs.
ble semantics, due to the unrestricted local configurationin SLASH and its contribution to the meaning of the con- Closer to the Truth: A New Model Theory for HPSG stituent. This means that monster structures are back, and words and phrases. Syntactic daughters are always em- this time they even have a phonology and make empiri- bedded signs. The specification in the signature of the cally false predictions.
EMBEDDED value u sign for each object ensures that ev- The grammars in the HPSG literature are not precise ery object in an interpretation is tied to an unembedded enough for their models to match the intentions of lin- sign. The dots under list stand for all declarations under guists. Independent of the choice of model theory they list in (7), including append.
denote structures that their authors do not intend to pre-dict. As the considerations about slashed constituents (10) Normal form extension Σ2 of signature Σ1: show, this is not a problem of the model theories. It is preferable to solve it by amending the grammars.
Normal Form Grammars
What we saw in the previous section was a weakness of the linguistic theory rather than of the logical formal- ism. Stranded structures are often inaccessible to empiri- cal observation and should not be predicted. In grammars with interesting coverage stranded structures also materi- alize as phrasal stranded monster structures. These have a e phraseu phrase phonology, which means that they should be observable, but their internal structure prevents them from occurring as part of an actual utterance.
Appropriate extensions of the linguistic theory elimi- (11) shows the logical statements which must be added nate the spurious structures and can simply be added to most HPSG grammars. The extensions consist of general 1 in (8) to obtain the corresponding normal form grammar hΣ θ assumptions about the signature and of a number of logi- 2, 2i. The new theory, θ2, incorporates all principles from θ cal statements to be included among the grammar princi- 1 in (8), adding four new restrictions on admissible models. For each of the new principles the corresponding formulation in (9) is indicated. The rela- The first move is to single out utterances from other tion component is defined with respect to all attributes types of signs as the only ones that are immediately em- A in the signature. (11i) states that each pair of nodes x pirically accessible. Every kind of linguistic structure is and y in a configuration is in the component relation iff a ultimately part of an utterance. Since no linguistic struc- sequence of attributes leads from y to x.
ture can simultaneously belong to two utterances, twinstructures are ruled out. A minor technical amendment (11) Normal form extension θ2 of theory θ1:1 concerns lists: For their encoding we fix a unique struc-ture that excludes spurious ambiguities that stem from f. (3c) U-SIGN COMPONENT CONDITION: multiple elist objects. In sum, I add to each HPSG gram- 1 top→ ∃ 2 component 1 2 u sign

g. (3d) UNIQUE U-SIGN CONDITION: (9) a. a sort hierarchy of signs which distinguishes unembedded signs from embedded signs, 1 u-sign ∧ 2 u-sign
→ 1 = 2 b. an attribute, appropriate to each sort, which h. (3e) UNIQUE EMPTY LIST CONDITION: articulates the insight that each entity in the linguistic universe has the property of belonging 1 elist ∧ 2 elist
→ 1 = 2 to an unembedded sign, i. COMPONENT PRINCIPLE: c. a principle which requires that each entity be a component of an unembedded sign, ∀ 1 ∀ 2component( d. a principle which requires the uniqueness of unembedded sign entities in connected configu- rations of entities, and, finally, 3 ∧ component( 1 3 )
e. a principle which formulates the weak exten- sionality of elist entities.
The effect of normalizing the grammar hΣ1 θ inspected in Fig. 12.3. For readability I systematically A grammar which incorporates these restrictions will omit the attribute EMBEDDED, which points from each be called a normal form grammar.
node to the unique u sign node to which the node be- the normal form grammar derived from the grammar longs. For example, each node in the configuration with 1i is shown in (10).
The hierarchy of signs dis- tinguishes between unembedded signs (u sign) and em- 1 The logical expressions are RSRL descriptions (Richter, 2004). ‘∀' bedded signs (e sign), a distinction which is inherited by is not the first order universal quantifier.
the u phrase 10 – representing the sentence Uther walks (Pollard and Sag 1994), (2) Pollard's theory of mathemat- – has an outgoing EMBEDDED arc pointing to 10. The ical idealizations of utterance tokens (Pollard 1999), and reader may want to verify that there are no other possi- (3) King's theory of exhaustive models containing sets of ble configurations in the denotation of the grammar. It possible utterance tokens (King 1999). In order to make should also be noted that the independent words Uther sure that all three logical formalisms can easily be com- (under u word node 15) and walks (under u word node pared and are comprehensive enough for a full formaliza- 21) are no longer isomorphic to the occurrences of these tion of HPSG grammars of the kind introduced by Pol- words in the sentence, because they are now marked as lard and Sag (1994), I use them in their variants defined in (Richter, 2004), which expresses them in terms of Re-lational Speciate Re-entrant Language (RSRL).
Figure 12.3: An exhaustive hΣ2 θ 2i model, systemat- ically omitting the attribute EMBEDDED for readability (see the explanation in the text) The formalization of the model theory of (1) and (2) failsto produce models that agree with their respective meta-theories of the structures in their grammar models. Inessence, the problem is that both (1) and (2) intend tocapture the idea that for each isomorphism class of well-formed utterances in a language, we find exactly onestructure in the denotation of the grammar which mod-els the members of the isomorphism class. For example,take a realization of the utterance I am sitting in a 370year old house in Engadin. The intention of the modeltheory of (1) is to have exactly one abstract feature struc-ture in the denotation a grammar of English which mod-els – or stands for the utterance type of – the utterancetoken. Similarly, the intention of the model theory of (2)is to have exactly one mathematical idealization of theisomorphism class of tokens of the given sentence in thedenotation of the grammar. However, this intention is notborne out in either formalism. Their models are definedin such a way that we necessarily find a large number ofmodeling structures for the given sentence in the denota-tion of a correct grammar of English. Subsection 12.4.2sketches the properties of the formalisms which are re-sponsible for this result.
The problem with (3) is not of a technical nature, it comes from the meta-theory itself. King postulates thatthe intended model of a grammar is an exhaustive modellike the one shown in Fig. 12.3 for the grammar hΣ2 θ According to King, the exhaustive model of a language h1 13 1i h1 5 11i h13 1 1i h13 13 13i h13 5 5i  that the linguist aims for does not contain utterance types h13 9 9i h13 11 11i h5 13 5i h9 13 9i or mathematical idealizations of utterance tokens. Instead append = h11 13 11i h16 17 16i h17 16 16i h17 17 17i it contains the utterance tokens of the language them- h22 23 22i h23 22 22i h23 23 23i h23 26 26i selves. Since we cannot know how many tokens of a given utterance there have been and will be in the world, we never know how many isomorphic copies of each ut- terance token the intended model contains. The definitionof exhaustive models permits an arbitrary number of iso- Problems in Previous Model
morphic copies of each possible configuration, all that isrequired is the presence of at least one representative of each. From the definition we only know that the classof exhaustive models of a grammar comprises, among On the basis of the notion of normal form HPSG gram- many others, the particular exhaustive model which, for mars I can now investigate the previous mathematical each utterance, contains the right number of tokens (if characterizations of the meaning of HPSG grammars.
the grammar is correct). However, since there will be These are (1) Pollard and Sag's original theory of linguis- grammatical utterances of a language which have never tic utterance types modeled by abstract feature structures occurred and will never occur, this is not yet the full Closer to the Truth: A New Model Theory for HPSG story. As exhaustive models (by definition) contain at Uther = hβU, U, U, Ui with least one copy of each potential grammatical utterance PHON PHON REST PHON FIRST in the language, the intended exhaustive model must also CAT CAT SUBCAT CAT HEAD comprise possible (as opposed to actual) utterance tokens, at least for those well-formed utterances of a language hε εi hPHON PHONi hCAT CATi which never occur. This means that the configurations in hPHON FIRST PHON FIRSTi exhaustive models are potential utterance tokens. These hPHON REST PHON RESTi potential utterance tokens are a dubious concept if tokens hPHON REST CAT SUBCATi are supposed to be actual occurrences of a linguistic form.
hCAT SUBCAT PHON RESTi In light of this problem, King's model theory has been un- hCAT SUBCAT CAT SUBCATi acceptable to some linguists.
hCAT HEAD CAT HEADi hε u wordi h PHON nelisti hPHON REST elisti hCAT SUBCAT elisti In this section I substantiate my claim that the model the- hPHON FIRST utheri hCAT cati  ories based on abstract feature structures by Pollard and hCAT HEAD nouni Sag (1994) and on mathematical idealizations of linguis- tic utterance tokens by Pollard (1999) do not achieve what append PHON PHON REST PHONi their meta-theories call for. Henceforth I refer to these append PHON REST PHON PHONi two theories as AFS and MI, respectively.
append PHON CAT SUBCAT PHONi Let us first consider AFS. The underlying idea is that append CAT SUBCAT PHON PHONi the denotation of a grammar is a set of relational ab- stract feature structures as determined by an admission relation. Each abstract feature structure in the set of rela- tional abstract feature structures admitted by a grammar π1 ∈ βU π2 ∈ βU & is a unique representative of exactly one utterance type of the natural language which the grammar is supposed to 2is a prefix of π1 capture. This means that there is a one-to-one correspon- Note that the set theoretical definition of abstract dence between the utterance types of the natural language feature structures guarantees that every abstract feature and the abstract feature structures which the grammar ad- structure isomorphic to another one is identical with it.
mits. A grammar can then be falsified by showing eitherthat there is no feature structure admitted by the grammarwhich corresponds to a particular utterance type of the Figure 12.4: The utterance type Uther and its reducts, language or that the grammar admits an abstract feature without relations and the EMBEDDED attribute structure which does not correspond to any grammaticalutterance type in the language.
Relational abstract feature structures consist of four sets: A basis set, β, which provides the basic syntacticmaterial; a re-entrancy relation, ρ, which is an equiva-lence relation that can be understood as an abstract rep-resentation of the nodes in connected configurations; alabel function, λ, which assigns species to the abstractnodes; and a relation extension, symbolized below as ξ,which represents the tuples of abstract nodes which are inthe relations of a grammar.
How these four components of a relational abstract fea- ture structure conspire to produce a representation of theutterance type Uther from Fig. 12.3 can be seen in (12).2The symbol ε stands for the empty path, i.e., an emptysequence of attributes. The basis set, βU, contains all at-tribute paths which can be created by following sequences Fig. 12.4 repeats the Uther configuration from of arcs from 15. The re-entrancy relation, ρU, enumer- Fig. 12.3 and adds a few more configurations. They are ates all possibilities of getting to the same node by a pair all rooted at a distinguished node (marked by a circle).
of attribute paths; and the label function, λU, assigns the The significance of the new configurations is the fact correct species to each attribute path.
that the set of abstract feature structures admitted by ourgrammar does not only contain the abstract feature struc- 2For expository purposes I pretend that the attribute EMBEDDED is ture corresponding to the Uther configuration under F7 not in the grammar. See footnote 3 for further remarks on this simplifi-cation.
(beside the two corresponding to walks and Uther walks).
Since the abstract feature structure for Uther is in the set, of the grammar. The abstract feature structures admitted it also contains abstract feature structures corresponding by the grammar predict six different types for this single to the configurations under A0, B3, C6, D13 and E14.
expression. The six types are distinct, and they are un- The reason for this is to be found in the definition of avoidable by construction if the grammar predicts the re- relational abstract feature structures and the ensuing ad- lational abstract feature structure which is an abstraction mission relation based on the traditional satisfaction rela- of a Uther configuration. The fundamental problem of the tion for feature structures, and it is an artifact of the con- construction is that the well-formedness of AUther is only struction. Intuitively, this is what happens: Abstract fea- guaranteed by the well-formedness of all of its reducts.
ture structures lack an internal recursive structure. Since Hence we do not get a one-to-one correspondence be- the admission relation must ensure that the entire abstract tween the types predicted by the grammar and the empiri- feature structure including all of its abstract nodes satis- cally observable expressions. Rather, it is the case that the fies the set of principles of a grammar, an auxiliary notion abstract feature structures admitted by a grammar neces- of reducts provides the necessary recursion. The idea is sarily introduce a version of stranded structures, although that a relational abstract feature structure is admitted by there are no stranded monster structures among them as a theory if and only if the feature structure itself and all long as the grammar is a normal form grammar.3 its reducts satisfy the theory. But that means that not only I conclude that AFS fails to behave in the intended way.
the relational abstract feature structure but also all of its Even if one is willing to accept types of linguistic expres- reducts are in the set of abstract feature structures admit- sions as an appropriate target for linguistic theory, rela- ted by the theory.
tional abstract feature structures are not adequate to make The definition of reducts is straightforward. Any at- this approach to the theory of grammatical meaning tech- tribute path in the basis set may be followed to get to an nically precise.
abstract node in the feature structure. At the end of each Let us now turn to the second theory, MI. Pollard path we find a new abstract root node of a reduct. This can (1999) postulates that a formal grammar as a scientific best be seen by considering the corresponding pictures of theory should predict the grammatical utterance tokens of configurations in Fig. 12.4 again. The configuration un- a natural language by specifying a set of structures which der A0 corresponds to the PHON reduct of the Uther con- contains an idealized mathematical structure for each ut- figuration; the configuration under B3 corresponds to the terance token (and for nothing else). For two utterance CAT reduct of the Uther configuration; C6 to the PHON tokens of the same expression there should only be one REST and CAT SUBCAT reduct; and analogously for the mathematical structure in the set. Moreover, the idealized two remaining atomic configurations. (13) contains an mathematical structure should be structurally isomorphic example of the reducts depicted in Fig. 12.4, an abstract to the utterance tokens it represents. This last condition is feature structure corresponding to the configuration with in fact much stronger than what (Pollard and Sag, 1994) root node E14. The reducts can be obtained either by ab- asks from its linguistic types. Pollard and Sag's linguistic straction from the configurations in Fig. 12.4 or directly types merely stand in a relationship of conventional cor- from AUther by a reduct formation operation. In contrast respondence to utterance tokens. The conventional corre- to the depictions of the corresponding graphical config- spondence must be intuited by linguists without any fur- uration in Fig. 12.4, the PHON FIRST reduct of Uther in ther guidance with respect to the correctness of these in- (13) contains the relation(s).
tuitions from the meta-theory of linguistic meaning.
The most significant technical difference compared to (13) The PHON FIRST reduct of AUther: AFS resides in how Pollard sets out to construct the mathematical idealizations of utterance tokens. Pollard's ρPF = {hε εi}, construction eschews relational abstract feature structures λPF = {hε utheri}, and and consequently does not need the specialized feature structure satisfaction and admission relations of strictly The scientific purpose of relational abstract feature feature structure based grammar formalisms.
structures in linguistic theory is their use as conveniently Pollard starts from the conventional grammar models of structured mathematical entities which correspond to King (1999). From these standard models he proceeds to types of linguistic entities. The relational abstract feature define singly generated models and then canonical rep- structures admitted by a grammar are meant to constitute resentatives of singly generated models as mathematical the predictions of the grammar (Pollard and Sag, 1994, p.
idealizations of utterance tokens.
A singly generated model is a connected configuration In the context of our example, we are talking about one under an entity which is actually a model of a grammar.
empirical prediction of the grammar hΣ2 θ Nothing substantial changes when we include the structure gen- tion that the described language contains the utterance erated by the attribute EMBEDDED in the relational abstract feature Uther. The exhaustive models mirror this prediction by structures. All four component sets of AUther as well as those of its containing (potentially multiple but isomorphic) Uther five reducts become infinite, but the six feature structures remain dis-tinct mathematical entities seemingly representing six different linguis- configurations. There is nothing else in the exhaustive models which has to do with this particular prediction Closer to the Truth: A New Model Theory for HPSG In other words, a singly generated model has a topmost six entities there are six distinct canonical representatives entity such that all other entities in the model are com- for it, although I assume that they would constitute one ponents of it. However, this is not yet the whole pic- single prediction in Pollard's sense. The intended pre- ture. Pollard defines the structures of interest as mod- diction seems to be that utterance tokens isomorphic to els together with their distinguished topmost entity. They the Uther configuration are grammatical. In fact, for each are pairs, hu hU S A R n with 15 ≤ n ≤ 20, all hU S A R u, u, uii, usually simply written as n, n, ni in (14) are iso- morphic, but this is not relevant in the construction. MI The subscripts indicate that all entities in the universe U are components of u. We could say that I is distinguishes between the corresponding entities in the a connected configuration under u which happens to be a universes because they are made of different equivalence model of a given grammar. Pollard then uses the distin- classes of terms. Intuitively, the problem is that the enti- guished entity in the configuration to define the canonical ties are in different locations relative to their root entity, representative for each hu I which entails that they are in a different equivalence class ui of the grammar. In essence, the entities in the canonical representatives are defined as of terms defined on the root entity.6 equivalence classes of terms relative to the distinguished I conclude that Pollard's construction fails to behave in root entity. Not all details are relevant here,5 the only im- the intended way. Pollard suggests that an HPSG gram- portant thing to note is that the standard model-theoretic mar should be interpreted as specifying a set of canon- technique of using terms of the logical language in the ical representatives such that no two members of the construction of a canonical model guarantees the unique- set are isomorphic, and utterance tokens of the language ness of each hu hU S A R which are judged grammatical are isomorphic to one of u, u, uii by the extensionality of the set-theoretic entities which serve as the elements of the canonical representatives. Even if one is prepared to the universe Uu. As a result, Pollard manages to fix the share Pollard's view of the goal of linguistics as a scien- canonical structure which stands for all isomorphically tific theory, the particular construction proposed in (Pol- configured structures or utterance tokens. In order to have lard, 1999) is not suited to realize this conception without a name for them, I will henceforth call them canonical serious problems. For normal form grammars it intro- representatives. The collection of all canonical represen- duces exactly the multiplicity of canonical representatives tatives of a grammar is the prediction of a grammar.
which it was designed to eliminate.
As in the investigation of AFS, I will focus on one pre- To sum up the preceding discussion, AFS and MI diction of hΣ2 θ clearly fall short of the goals their proponents set for 2i, the prediction that the utterance Uther will be judged grammatical. Although the structures of themselves. Neither Pollard and Sag's set of structures MI are defined quite differently from the set of relational corresponding to linguistic utterance types nor Pollard's abstract feature structures admitted by it, we will see im- set of canonical representatives isomorphic to grammati- mediately that AFS and MI share closely related prob- cal utterance tokens meets the intentions of their respec- lematic aspects.
tive authors.
Assume that we apply Pollard's method of construct- ing the canonical universes of Σ2 interpretations as equiv-alence classes of Σ Minimal Exhaustive Models
(14) shows schematically which canonical representatives Pollard's constructionyields for the Uther configuration when it is applied to I will now present an extension of King's theory of ex- our exhaustive model. The subscripts indicate which en- haustive models which avoids his problematic ontological tity of the exhaustive model of Fig. 12.3 is turned into the commitment to possible utterance tokens, while retaining root entity of each of the six canonical representatives. By all other aspects of his model theory. At the same time, construction, each of the canonical representatives in (14) I also avoid the commitments to the ontological reality is a different set-theoretic entity. In brackets I mention the of utterance types or to the mathematical nature of the species of each root entity.
grammar models, which are characteristic of the meta-theories (1) and (2). My starting point are the structural (14) a. hu15 hU assumptions of normal form HPSG grammars, which I take to be independently motivated by the arguments in Section 12.3. For normal form grammars I define unique models which contain exactly one structure which is iso- d. hu18 hU morphic to each utterance of a language considered well- e. hu19 hU formed by an ideal speaker of the language. This is, of f. hu20 hU course, what (1) and (2) essentially wanted to do, except It is immediately obvious that we observe here the 6 It should be pointed out that the six interpretations in (14) are only same effect which we saw before with Pollard and Sag's isomorphic because we assume normal form grammars with an attribute utterance types. Since the Uther configuration contains EMBEDDED. However, without the EMBEDDED attribute we would runinto the problems discussed in Section 12.3. In particular we would have stranded monster structures, and they would occur as canonical The notation is explained in some detail in Section 12.5.
representatives which should correspond to possible utterance tokens, They can be found in (Pollard, 1999, pp. 294–295) and even more contrary to fact.
explicitly in (Richter, 2004, pp. 208–210).
that I define minimal exhaustive models in such a way that language is isomorphic to a maximal connected configu- I am not forced to make any commitments to the ontologi- ration in a minimal exhaustive model. The definitions of cal nature of the structures in them. Given the philosoph- maximal connected configurations and minimal exhaus- ical intricacies of such commitments, I take this to be a tive models will be supplied directly below. Note that this highly desirable property of my proposal.
condition endorses all arguments which King adduced to The goal is to characterize the meaning of grammars motivate exhaustive models, except for the ontological in terms of a set of structures, M , which should have claim that the intended model is a system of possible (ac- at least the following three properties: Each structure in tual and non-actual) tokens.
M should have empirical consequences, i.e., there must Connected configurations in interpretations have been be empirical facts which can falsify the predictions em- a leading intuitive concept since the first examples above.
bodied by the structure; there should not be isomorphic Their definition is straightforward. It presupposes the fa- copies of any empirically significant structure in the set miliar RSRL signatures with a sort hierarchy hG ⊑i, a of structures M assigned to each grammar; and finally, in distinguished set of maximally specific sorts S , a set of accordance with one of Pollard's criteria, actual utterance attributes A, an appropriateness function F , and a set of tokens which are judged grammatical must be isomorphic relation symbols R whose arity is determined by a func- to precisely one element in M .
tion AR . Interpretations consist of a universe of objects At first this small collection of desirable properties of U, a sort assignment function S which associates a sym- M might seem arbitrary, even if every one of them can be bol from S with each object in U, an attribute interpreta- individually justified. However, there is a way of integrat- tion function A which treats each attribute symbol as the ing them with King's well-motivated theory of exhaustive name of a partial function from U to U, and a relation interpretation function R which interprets each relation King's theory of grammatical truth conceives of lan- symbol as a set of tuples of the appropriate arity. Cou is guage as a system of possible linguistic tokens. It claims the set of those objects in U which can be reached from u that the system of possible tokens can be described as an by following a (possibly empty) sequence of attributes.
exhaustive model of a grammar. The controversial aspectof this theory concerns the idea that language is a system of possible (i.e., actual and non-actual) tokens. Assume hG ⊑ S A F R AR i, Σ interpretation that we give up this aspect of King's theory. Instead we take an agnostic view toward language and say that we i is a connected configuration in I iff do not really know what it consists of. In our grammars 1. U′ ⊆ U, we only make predictions about the discernible shapes of 2. for some u′ ∈ U′, Cou′ = U′, the empirical manifestations of language. We can oper- 3. S′ = S ∩ (U′ × S ), ationalize this conception as follows: We want to write 4. A′ = A ∩ (A × {U′ × U′}), grammars such that whenever we encounter an actual ut- terance token, it will be judged grammatical if and only 5. R′ = R ∩ R × Pow if there is an isomorphically structured connected con-figuration in an exhaustive model of the grammar. The Certain connected configurations in interpretations are connected configurations of interest will turn out to be of special interest to us. These are connected configura- the familiar connected configurations under unembedded tions which are not properly contained within other con- signs. The choice of exhaustive model will not matter, nected configurations in their interpretation. I will call since we are only concerned with the shape of the con- them maximal: figurations, and we know that all shapes are present inany exhaustive model (by definition). However, since we Definition 12.5.2. For each signature Σ, for each Σ in-
are no longer after a system of possible tokens with an terpretation I = hU S A Ri, unknown number of isomorphic copies of configurations, hU′ S′ A′ R′i is a maximal connected configuration in we can be more precise about our choice of exhaustive model. It suffices to choose one which contains just one hU′ S′ A′ R′i is a connected configuration in I, copy of each relevant connected configuration.
and for some u′ ∈ U′: The theory of meaning we obtain from these considera- Cou′ = U′ 6⊂ Cou′′ , and for every u′′ ∈ U, Cou′ tions is a weakened form of King's theory. King says that There are three maximal connected configurations in a grammar is true of a natural language only if the lan- the interpretation of Fig. 12.1. Their topmost elements guage can be construed as a system of possible tokens, are the phrase entity 16, which is the topmost entity in the and the system of possible tokens forms an exhaustive connected configuration with the phonology Uther walks; model of the grammar. The theory proposed here as an al- the word entity 30, which is the topmost entity in the con- ternative refrains from making such strong claims about nected configuration with the phonology Uther; and the the nature of language. It says that a grammar is true word entity 19, which is the topmost entity in the con- of a natural language only if each actual utterance token nected configuration with the phonology walks.
which is judged grammatical by an ideal speaker of the Closer to the Truth: A New Model Theory for HPSG We can prove important properties of maximal con- u sign entity in each u-sign configuration contains all nected configurations in models of normal form gram- other elements of the configuration as its components, it mars: No two of them overlap. Each of them contains is quite natural to define the entities in the u-sign con- exactly one u sign entity, which guarantees that they are figurations as equivalence classes of paths which lead to empirical structures. Each entity in a model actually be- them from their individual u sign. This of course is essen- longs to a maximal connected configuration, which en- tially Pollard's construction of canonical representatives, sures the empiricity of all entities. Every u sign entity is except that I avoid the multiplicity of representatives for contained in a maximal connected configuration, which one and the same prediction because my mathematical guarantees that maximal connected configurations indeed idealizations do not consist of pairs of entities and con- capture all empirically relevant predictions without miss- figurations. Instead, I exploit the special properties of the ing any. From now on I refer to maximal connected con- models of normal form grammars and am thus able to figurations in models of normal form grammars as u-sign make do with bare u-sign configurations.
configurations. The u-sign configurations in models of But although the construction of minimal exhaustive our grammars constitute the empirical predictions of the models from mathematical entities is simple, I am not aware of any convincing argument for them. In my opin- I define minimal exhaustive grammar models as ex- ion, DEFINITION 12.5.3 completes the explanation of the haustive models which contain exactly one copy of each meaning of normal form HPSG grammars.
possible u-sign configuration.
Definition 12.5.3. For each signature Σ, for each Σ-
theory θ, for each exhaustive hΣ θi model I, I is a minimal exhaustive hΣ θi model iff King, Paul J. (1999).
Towards Truth in Head-driven for each maximal connected configuration I1 in I, for Phrase Structure Grammar.
In Valia Kordoni, ed., each maximal connected configuration I2 in I: T¨ubingen Studies in Head-Driven Phrase Structure if I1 and I2 are isomorphic then I1 = I2. Grammar, Arbeitspapiere des SFB 340, Nr. 132, Vol-ume 2, pp. 301–352. Eberhard Karls Universit¨at T¨ubin- The exhaustive hΣ2 θ 2i model of Fig. 12.3 is an exam- ple of a minimal exhaustive grammar model. It containsexactly one copy of each u-sign configuration predicted Pollard, Carl and Ivan A. Sag (1994).
by the grammar hΣ2 θ Phrase Structure Grammar.
University of Chicago According to the properties of u-sign configurations, a minimal exhaustive model of a normal form grammaris partitioned into separate u-sign configurations. Each Pollard, Carl J. (1999). Strong generative capacity in pair of u-sign configurations in it is structurally distinct HPSG. In Gert Webelhuth, Jean-Pierre Koenig, and and thus constitutes a different prediction of the grammar.
Andreas Kathol, eds., Lexical and Constructional As- Since all connected configurations in these models are u- pects of Linguistic Explanation, pp. 281–297. CSLI sign configurations, they do not contain anything which is empirically vacuous.
Richter, Frank (2004).
A Mathematical Formalism With my construction I have not made any ontological for Linguistic Theories with an Application in Head- commitments. I have claimed that the internal structure Driven Phrase Structure Grammar. Phil. dissertation of actual utterance tokens can be discovered, and that this (2000), Eberhard Karls Universit¨at T¨ubingen.
structure is mirrored precisely in u-sign configurations inminimal exhaustive grammar models. This did not pre-suppose saying anything about the ontology of linguisticobjects. It was not even necessary to say what kinds ofentities populate the minimal exhaustive models.
Should there be any concern about the undetermined na-ture of the entities in minimal exhaustive models, or apreference for mathematical models, it is possible to pickout one mathematical model and fix it as the intendedminimal exhaustive model of a given normal form gram-mar. The architecture of minimal exhaustive models ofnormal form grammars suggests strongly how to do this.
Since the minimal exhaustive models are populated by acollection of u-sign configurations, and since the unique

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