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<Paper uid="J97-4003">
  <Title>A Computational Treatment of Lexical Rules in HPSG as Covariation in Lexical Entries</Title>
  <Section position="4" start_page="544" end_page="559" type="intro">
    <SectionTitle>
6 The Partial-VP Topicalization Lexical Rule proposed by Hinrichs and Nakazawa (1994, 10) is a
</SectionTitle>
    <Paragraph position="0"> linguistic example. The in-specification of this lexical rule makes use of an append relation to constrain the valence attribute of the auxiliaries serving as its input. In the lexicon, however, the complements of an auxiliary are uninstantiated because it raises the arguments of its verbal complement.</Paragraph>
    <Paragraph position="1">  Computational Linguistics Volume 23, Number 4 simple-word ---* LE1 v .. * V LEn derived-word ~ (\[IN LRl-in\] A LRl-out) V ... V (\[IN LRm-in\] A LRm-oUt) Figure 1 The extended lexicon under the DLR approach.</Paragraph>
    <Paragraph position="2"> partially be dealt with, for example, by using a depth bound on lexical rule application to ensure that a finite number of lexical entries is obtained. 7  as relations between word objects. Lexical rules under this approach are part of the theory, just like any other constraint of the grammar, and they relate the word objects licensed by the base lexical entries to another set of well-formed word objects. Thus, under the DLR approach, no new lexical entries are created, but the theory itself is extended in order to include lexical rules. One possibility for extending the theory is to introduce two subtypes of word, i.e., simple-word and derived-word, and define an additional feature IN with appropriate value word for objects of type derived-word. The principles encoding the extended lexicon in such an approach are shown in Figure 1. Each basic lexical entry is a disjunct LE in an implicative constraint on simple-word. This disjunction thus constitutes the base lexicon. The disjuncts in the constraint on derived-word, on the other hand, encode the lexical rules. The in-specification of a lexical rule specifies the IN feature, the out-specification, the derived word itself. Note that the value of the IN feature is of type word and thus also has to satisfy either a base lexical entry or an out-specification of a lexical rule. While this introduces the recursion necessary to permit successive lexical rule application, it also grounds the recursion in a word described by a base lexical entry. Contrary to the MLR setup, the DLR formalization therefore requires all words feeding lexical rules to be grammatical with respect to the theory.</Paragraph>
    <Paragraph position="3"> Since lexical rules are expressed in the theory just like any other part of the theory, they are represented in the same way, as unary immediate dominance schemata. 8 This conception of lexical rules thus can be understood as underlying the computational approach that treats lexical rules as unary phrase structure rules as, for example, adopted in the LKB system (Copestake 1992). Both the input and output of a lexical rule, i.e., the mother and the daughter of a phrase structure rule, are available during a generation or parsing process. As a result, in addition to the information present in the lexical entry, syntactic information can be accessed to execute the constraints on the input of a lexical rule. The computational treatment of lexical rules that we propose in this paper is essentially a domain-specific refinement of such an approach to lexical rules. 9  immediate dominance schemata and lexical rules, however, is that immediate dominance schemata are fully specified in the linguistic theory and can thus be directly interpreted as a relation on objects. Lexical rules, on the other hand, are usually not 7 This approach is, for example, taken in the ALE system. See Section 7 for more discussion of different computational approaches.</Paragraph>
    <Paragraph position="4"> 8 Elaborating this analogy, the IN feature of derived words can be understood as the DTRS feature of a phrase.</Paragraph>
    <Paragraph position="5"> 9 See Section 7 for a more detailed discussion of the relation between our approach and this perspective on lexical rules.</Paragraph>
    <Paragraph position="6">  A passivization lexical rule.</Paragraph>
    <Paragraph position="7"> written as fully specified relations between words, rather, only what is supposed to be changed is specified.</Paragraph>
    <Paragraph position="8"> Consider, for example, the lexical rule in Figure 2, which encodes a passive lexicai rule like the one presented by Pollard and Sag (1987, 215) in terms of the setup of Pollard and Sag (1994, ch. 9). This lexical rule could be used in a grammar of English to relate past participle forms of verbs to their passive form2 deg The rule takes the index of the least oblique complement of the input and assigns it to the subject of the output. The index that the subject bore in the input is assigned to an optional prepositional complement in the output.</Paragraph>
    <Paragraph position="9"> Only the verb form and some indices are specified to be changed, and thus other input properties, like the phonology, the semantics, or the nonlocal specifications, are preserved in the output. This is so since the lexical rule in Figure 2 &amp;quot;(like all lexical rules in HPSG) preserves all properties of the input not mentioned in the rule.&amp;quot; (Pollard and Sag \[1994, 314\], following Flickinger \[1987\]). This idea of preserving properties can be considered an instance of the well-known frame problem in AI (McCarthy and Hayes 1969), and we will therefore refer to the specifications left implicit by the linguist as the frame specification, or simply frame, of a lexical rule. Not having to represent the frame explicitly not only enables the linguist to express only the relevant things, but also allows a more compact representation of lexical rules where explicit framing would require the rules to be split up (Meurers 1994).</Paragraph>
    <Paragraph position="10"> One thus needs to distinguish the lexical rule specification provided by the linguist from the fully explicit lexical rule relations integrated into the theory. The formalization of DLRs provided by Meurers (1995) defines a formal lexical rule specification language and provides a semantics for that language in two steps: A rewrite system enriches the lexical rule specification into a fully explicit description of the kind shown in Figure 1. This description can then be given the standard set-theoretical interpretation of King (1989, 1994). 11 10 Note that the passivization lexical rule in Figure 2 is only intended to illustrate the mechanism. We do not make the linguistic claim that passives should be analyzed using such a lexical rule. For space reasons, the SYNSEM feature is abbreviated by its first letter. The traditional (First I Rest) list notation is used, and the operator * stands for the append relation in the usual way. 1l Manandhar (1995) proposes to unify these two steps by including an update operator in the  Computational Linguistics Volume 23, Number 4 The computational treatment we discuss in the rest of the paper follows this setup in that it automatically computes, for each lexical rule specification, the frames necessary to preserve the properties not changed by it. 12 We will show that the detection and specification of frames and the use of program transformation to advance their integration into the lexicon encoding is one of the key ingredients of the covariation approach to HPSG lexical rules.</Paragraph>
    <Paragraph position="11">  3. Lexical Covariation: Encoding Lexical Rules and their Interaction as Definite Relations  Having situated the computational approach presented in this paper as a computational treatment of DLRs that emphasizes their domain-specific properties, we now turn to the compiler that realizes this approach. We describe four compilation steps that translate a set of lexical rules, as specified by the linguist, and their interaction into definite relations to constrain lexical entries. To give the reader a global idea of our approach, we focus on those aspects of the compiler that are crucial to the presented conception of lexical rules. The different steps of the compiler are discussed with emphasis on understandability and not on formal details. 13 Figure 3 shows the overall setup of the compiler. The first compilation step, discussed in Section 3.1, translates lexical rules into a definite clause representation and derives, for each lexical rule, a frame predicate that ensures the transfer of properties that remain unchanged. In the second compilation step (Section 3.2), we determine the possible interaction of the lexical rules. This results in a finite-state automaton representing global lexical rule interaction, i.e., the interaction of lexical rules irrespective of the lexical entries in the lexicon. In the subsequent step of word class specialization (Section 3.3) this finite-state automaton is fine-tuned for each of the natural classes of lexical entries in the lexicon. In the fourth compilation step (Section 3.4) these automata are translated into definite relations and the lexical entries are adapted to call the definite relation corresponding to the automaton fine-tuned for the natural class to which they belong.</Paragraph>
    <Section position="1" start_page="547" end_page="551" type="sub_section">
      <SectionTitle>
3.1 Lexical Rules as Definite Relations and the Automatic Specification of Frames
</SectionTitle>
      <Paragraph position="0"> We start by translating each lexical rule into a definite clause predicate, called the lexical rdle predicate. The first argument of a lexical rule predicate corresponds to the in-specification of the lexical rule and the second argument to its out-specification.</Paragraph>
      <Paragraph position="1"> Assume the signature in Figure 4 on which we base the example throughout the paper and suppose the lexical rule specification shown in Figure 5.14 This lexical rule applies to base lexical entries that unify 15 with the in-specification, i.e., lexical entries specifying B and Y as -. The derived lexical entry licenses word objects with + as the value of x and Y, and b as that of A.</Paragraph>
      <Paragraph position="2"> The translation of the lexical rule into a predicate is trivial. The result is displayed description language. 12 In order to focus on the computational aspects of the covariation approach, in this paper we will not go into a discussion of the full lexical rule specification language introduced in Meurers (1995). The reader interested in that language and its precise interpretation can find the relevant details in that paper. 13 A more detailed presentation can be found in Minnen (in preparation).</Paragraph>
      <Paragraph position="3"> 14 We use rather abstract lexical rules in the examples to be able to focus on the relevant aspects. 15 Hinrichs and Nakazawa (1996) show that the question of whether the application criterion of lexical rules should be a subsumption or a unification test is an important question deserving of more attention. We here assume unification as the application criterion, which formally corresponds to the conjunction of descriptions and their conversion to normal form (G6tz 1994). Computationally, a subsumption test could equally well be used in our compiler.</Paragraph>
      <Paragraph position="4">  Meurers and Minnen Covariation Approach to HPSG Lexical Rules input: output: Figure 3 The compiler setup.</Paragraph>
      <Paragraph position="5">  interaction into definite relations in Figure 6. Though this predicate represents what was explicitly specified in the lexical rule, it does not accomplish exactly what is intended. As discussed in Section 2.2.3, features specified in a lexical entry unifying with the in-specification of the lexical rule that are not specified differently in the out-specification of the lexical rule are intended to receive the same value on the derived word as on the input: The compiler implements this by enriching the lexical rule with type specifications and path equalities between the in- and the out-specification to arrive at an explicit representation of its frame.</Paragraph>
      <Paragraph position="6"> The detection of which additional specifications are intended by the linguist crucially depends on the interpretation of the signature assumed in HPSG, discussed in Section 2.1. This interpretation makes it possible to determine which kind of word objects (by ontological status fully specified) may undergo the rule. A type can always be replaced by a disjunction of its most specific subtypes and the appropriate features  A sample lexical entry.</Paragraph>
      <Paragraph position="7"> of each type are known. So, on the basis of the signature, we can determine which &amp;quot;appropriate&amp;quot; paths the linguist left unspecified in the out-specification of the lexical rule. For those appropriate paths not specified in the out-specification, one can then add path equalities between the in- and the out-specifications of the lexical rule to ensure framing of those path values.</Paragraph>
      <Paragraph position="8"> Frame specification becomes slightly more difficult when one considers type specifications of those paths in words serving as input to a lexical rule that occur in the out-specification of the lexical rule but are not assigned a type value. For example, the lexical rule 1 of Figure 6 applies to word objects with tl as their c value and to those having t2 as their c value. With respect to frame specification this means that there can be lexical entries, such as the one in Figure 7, for which we need to make sure that tl as the value of c gets transferred. 16 One would think that the type information tl, which is more specific than that 16 A linguistic example based on the signature given by Pollard and Sag (1994) would be a lexical rule deriving predicative signs from nonpredicative ones, i.e., changing the PRD value of substantive signs from - to +, much like the lexical rule for NPs given by Pollard and Sag (1994, p. 360, fn. 20). In such a Predicative Lexical Rule (which we only note as an example and not as a linguistic proposal) the subtype of the head object undergoing the rule as well as the value of the features only appropriate for the subtypes of substantive either is lost or must be specified by a separate rule for each of the subtypes.  Meurers and Minnen Covariation Approach to HPSG Lexical Rules b lex lel, \[:, _,\]ollx framel Figure 8 Lexical rule predicate representing lexical rule 1.</Paragraph>
      <Paragraph position="9"> \[B o lIB \]c w r frame_l( , ). frame_l( |W tl \[ ~1\] Ch\[W ~\] L t2LZ Figure 9 Definition of the frame predicate for lexical rule 1.</Paragraph>
      <Paragraph position="10"> given in the output of the lexical rule, can be specified on the out-specification of the lexical rule if the specification of c is transferred as a whole (via structure sharing of the value of c). This is not possible, though, since the values of x and Y are specified in the out-specification of the lexical rule. The problem seems to be that there is no notion of sharing just the type of an object. However, introducing such type sharing would not actually solve the problem, since one also needs to account for additional appropriate features. The subtypes of t have different appropriate features, the values of which have to be preserved. In particular, in case the lexical entry has t2 as the value of c, we need to ensure that the value of the feature z is transferred properly. To ensure that no information is lost as a result of applying a lexical rule, it seems to be necessary to split up the lexical rule to make each instance deal with a specific case. In the above example, this would result in two lexical rules: one for words with tl as their c value and one for those with t2 as their c value. In the latter case, we can also take care of transferring the value of z. However, as discussed by Meurers (1994), creating several instances of lexical rules can be avoided. Instead, the disjunctive possibilities introduced by the frame specification are attached as a constraint to a lexical rule. This is accomplished by having each lexical rule predicate call a so-called frame predicate, which can have multiple defining clauses. So for the lexical rule 1, the frame specification is taken care of by extending the predicate in Figure 6 with a call to a frame predicate, as shown in Figure 8.17 On the basis of the lexical rule specification and the signature, the compiler deduces the frame predicates without requiring additional specifications by the linguist. The frame predicate for lexical rule 1 is defined by the two clauses displayed in Figure 9. The first case applies to lexical entries in which c is specified as tl. We have to ensure that the value of the feature w is transferred. In the second case, when feature c has t2 as its value, we additionally have to ensure that z gets transferred. Note that neither clause of the frame predicate needs to specify the features A, X, and Y since these features are changed by lex_rule_l. Furthermore, filling in features of the structure below z is unnecessary as the value of z is structure shared as a whole. Finally, if a lexical entry specifies c as t, bothframe_l clauses apply. TM 17 We use indexing of predicate names to be able to indicate later on which lexical rule a frame predicate belongs to. 18 Since in computational systems, in contrast to the general theoretical case, we only need to ensure transfer for the properties actually specified in the lexical entries of a given grammar, some of the distinctions made in the signature can possibly be ignored. One could therefore improve the calculation of frame predicates by taking the base lexical entries into account at this stage of the  Computational Linguistics Volume 23, Number 4 i V --.V n Figure 10 Finite-state automaton representing free application.</Paragraph>
      <Paragraph position="11"> Summing up, we distinguish the lexical rule predicates encoding the specification of the linguist from the frame predicates taking care of the frame specification. Based on the signature, the frame predicates are automatically derived from the lexical rule predicates and they can have a possibly large number of defining clauses. In Section 4 we will show that the encoding can be advanced in a way that eliminates the nondeterminism introduced by the multiply defined frame predicates.</Paragraph>
    </Section>
    <Section position="2" start_page="551" end_page="554" type="sub_section">
      <SectionTitle>
3.2 Determining Global Lexical Rule Interaction
</SectionTitle>
      <Paragraph position="0"> In the second compilation step, we use the definite clause representation of a set of lexical rules, i.e., the lexical rule and the frame predicates, to compute a finite-state automaton representing how the lexical rules interact (irrespective of the lexical entries).</Paragraph>
      <Paragraph position="1"> In general, any lexical rule can apply to the output of another lexical rule, which is sometimes referred to as free application. As shown in Figure 10, this can be represented as a finite-state automaton that consists of a single state with a cycle from/into this state for all lexical rules. 19 When looking at a specific set of lexical rules though, one can be more specific as to which sequences of lexical rule applications are possible. One can represent this information about the interaction of lexical rules as a more complex finite-state automaton, which can be used to avoid trying lexical rule applications at run-time that are bound to fail. To derive a finite-state automaton representing global lexical rule interaction, we first determine which lexical rules can possibly follow which other lexical rules in a grammar. The set of follow relationships is obtained by testing which in-specifications unify with which out-specifications. 2deg To illustrate the steps in determining global lexical rule interaction, let us add three more lexical rules to the one discussed in Section 3.1. Figure 11 shows the full set of four lexical rules.</Paragraph>
      <Paragraph position="2"> Figure 12 shows the definite clause representations of lexical rules 2, 3, and 4 and the frame predicates derived for them. The definite clauses representing lexical rule 1 and its frame were already given in Figures 8 and 9. The follow relation obtained for the set of four lexical rules is shown in Figure 13, where follow(LR,ListOfLRs) specifies compilation process.</Paragraph>
      <Paragraph position="3"> 19 We use the following conventions with respect to finite-state automata to represent lexical rule interaction: The state annotated with an angle bracket represents the initial state. All states (including the initial state) are final states. The labels of the transitions from one state to another are (disjunctions of) the lexical rule predicate indices, i.e., the lexical rule names constitute the alphabet of the finite-state automaton.</Paragraph>
      <Paragraph position="4"> 20 For the computation of the follow relationships, the specifications of the frame predicates are taken into account. In case the frame relation called by a lexical rule has several defining clauses, the generalization of the frame possibilities is used.</Paragraph>
      <Paragraph position="5">  A set of four lexical rules.</Paragraph>
      <Paragraph position="7"> The definite clause encoding of lexical rules 2, 3, and 4.</Paragraph>
      <Paragraph position="9"> follow(I, \[2, 3, 4\]). follow(2, \[1, 3, 4\]). follow(3, \[3, 4\]). follow(4, \[\]).</Paragraph>
      <Paragraph position="10"> Figure 13 The follow relation for the four lexical rules of the example. that only the lexical rules in ListOfLRs can possibly be applied to a word resulting from the application of lexical rule LR.</Paragraph>
      <Paragraph position="11"> Once the follow relation has been obtained, it can be used to construct an automaton that represents which lexical rule can be applied after which sequence of lexical rules. Special care has to be taken in case the same lexical rule can apply several times in a sequence. To obtain afinite automaton, such a repetition is encoded as a transition cycling back to a state in the lexical rule sequence preceding it.</Paragraph>
      <Paragraph position="12"> In order to be able (in the following steps) to remove a transition representing a certain lexical rule application in one sequence without eliminating the lexical rule application from other sequences, every transition, except those introducing cycles, is taken to lead to a new state. The finite-state automaton in Figure 14 is constructed on the basis of the follow relation of Figure 13.</Paragraph>
      <Paragraph position="13">  Computational Linguistics Volume 23, Number 4</Paragraph>
      <Paragraph position="15"> Finite-state automaton representing global lexical rule interaction.</Paragraph>
      <Paragraph position="16"> The finite-state automaton representing global lexical rule interaction can be used as the backbone of a definite clause encoding of lexical rules and their interaction (see Section 3.4). Compared to free application, the finite-state automaton in Figure 14 limits the choice of lexical rules that can apply at a certain point. However, there still are several places where the choices can be further reduced. One possible reduction of the above automaton consists of taking into account the propagation of specifications along each possible path through the automaton. This corresponds to actually unifying the out-specification of a lexical rule with the in-specification of the following lexical rule along each path in the automaton, instead of merely testing for unifiability, which we did to obtain the follow relation. 21 As a result of unifying the out-specification of a lexical rule in a path of the finite-state automaton with the in-specification of the following lexical rule, the out-specification of the second rule can become more specific. This is because of the structure sharing between the second lexical rule's inand out-specifications, which stem from the lexical rule and its frame specification.</Paragraph>
      <Paragraph position="17"> This makes it possible to eliminate some of the transitions that seem to be possible when judging on the basis of the follow relation alone. 22 For example, solely on the basis of the follow relation, we are not able to discover the fact that upon the successive application of lexical rules 1 and 2, neither lexical rule 1 nor 2 can be applied again. Taking into account the propagation of specifications, the result of the successive application of lexical rule 1 and lexical rule 2 in any order (leading to state q7 or q9) bears the value + on features w and Y. This excludes lexical 21 The reason for first determining the automaton on the basis of the follow relation alone, instead of taking propagation of specifications into account right from the start, is that the follow relations allow a very simple construction of a finite-state automaton representing lexical rule interaction. Using unification right away would significantly complicate the algorithm, in particular for automata containing cycles.</Paragraph>
      <Paragraph position="18"> 22 Note that in the case of transitions belonging to a cycle, only those transitions can be removed that are useless at the first visit and after any traversal of the cycle.</Paragraph>
      <Paragraph position="19">  A lexical entry.</Paragraph>
      <Paragraph position="20"> rules 1 and 2 as possible followers of that sequence since their in-specifications do not unify with those values. As a result, the arcs 1(q7, q2) and 2(q9, q3), which are marked with grey dots in Figure 14, can be removed.</Paragraph>
      <Paragraph position="21"> Two problems remain: First, because of the procedural interpretation of lexical rules, duplicate lexical entries can possibly be derived. And second, relative to a specific lexical entry, many sequences of lexical rules that are bound to fail are tried anyway. We tackle these problems by means of word class specialization, i.e., we prune the automaton with respect to the propagation of specifications belonging to the base lexical entries.</Paragraph>
    </Section>
    <Section position="3" start_page="554" end_page="556" type="sub_section">
      <SectionTitle>
3.3 Word Class Specialization of Lexical Rule Interaction
</SectionTitle>
      <Paragraph position="0"> In the third compilation step, the finite-state automaton representing global lexical rule interaction is fine-tuned for each base lexical entry in the lexicon. The result is a pruned finite-state automaton. The pruning is done by performing the lexical rule applications corresponding to the transitions in the automaton representing global lexical rule interaction. To ensure termination in case of direct or indirect cycles, we use a subsumption check. If the application of a particular lexical rule with respect to a lexical entry fails, we know that the corresponding transition can be pruned for that entry. In case of indirect or direct cycles in the automaton, however, we cannot derive all possible lexical entries, as there may be infinitely many. Even though one can prune certain transitions even in such cyclic cases, it is possible that certain inapplicable transitions remain in the pruned automaton. However, this is not problematic since the lexical rule application corresponding to such a transition will simply fail at run-time.</Paragraph>
      <Paragraph position="1"> Consider the base lexical entry in Figure 15. With respect to this base lexical entry, we fine-tune the finite-state automaton representing global lexical rule interaction by pruning transitions. In the automaton of Figure 14, we can prune the transitions {3(q2, q8), 4(q2, q6), 3(q3, q11), 4(q3, ql0), 3(ql, q4), 4(ql, q5)}, because the lexical rules 3 and 4 can not be applied to a (derived) lexical entry that does not have both w and x of value +. As a consequence, the states q8, q15, q11, q18, q4, and q12 are no longer reachable and the following transitions can be eliminated as well: {3(q8,q8), 4(q8, q15), 3(q11, q11), 4(q11, q18), 3(q4, q4), 4(q4, q12)}. We can also eliminate the transitions {4(q7,q13),4(q9, q17)}, because the lexical rule 4 requires the value of z to be empty list. Note that the lexical rules 3 and 4 remain applicable in q14 and q16.</Paragraph>
      <Paragraph position="2"> Furthermore, due to the procedural interpretation of lexical rules in a computational system (in contrast to the original declarative intention), there can be sequences of lexical rule applications that produce identical entries. 23 To avoid having arcs in the pruned automaton leading to such identical entries, we use a tabulation method 23 Note that the order in which two lexical rules are applied is immaterial as long as both rules modify the value of different features of a lexical entry.</Paragraph>
      <Paragraph position="3">  during word class specialization that keeps track of the feature structures obtained for each node. If we find a feature structure for a node qn that is identical to the feature structure corresponding to another node qm, the arc leading to qn or the arc leading to qm is discarded. 24 In the example, q7 and q9 are such identical nodes. So we can discard either 2(q2, q7) or 1(q3, q9) and eliminate the arcs from states that then become unreachable. Choosing to discard 1(q3, q9), the pruned automaton for the example lexical entry looks as displayed in Figure 16. 25 Note that word class specialization of lexical rule interaction does not influence the representation of the lexical rules themselves. Pruning the finite-state automaton representing global lexical rule interaction only involves restricting lexical rule interaction in relation to the lexical entries in the lexicon.</Paragraph>
      <Paragraph position="4"> The fine-tuning of the automaton representing lexical rule interaction results in a finite-state automaton for each lexical entry in the lexicon. However, identical automata are obtained for certain groups of lexical entries and, as shown in the next section, each automaton is translated into definite relations only once. We therefore automatically group the lexical entries into the natural classes for which the linguist intended a certain sequence of lexical rule applications to be possible. 26 No additional hand-specification is required. Moreover, the alternative computational treatment to expand out the full lexicon at compile-time is just as costly and, furthermore, impossible in case of an infinite lexicon.</Paragraph>
      <Paragraph position="5"> An interesting aspect of the idea of representing lexical rule interaction for particular word classes is that this allows a natural encoding of exceptions to lexical rules. More specifically, the linguist specifies exceptions as a special property of either a lexical rule or a lexical entry. During word class specialization, the compiler then deals with such specifications by pruning the corresponding transitions in the finite-state automaton representing global lexical rule interaction for the particular lexical entry under consideration. This results in an encoding of exceptions to a lexical rule in the interaction predicate called by the irregular lexical entries. An advantage of the setup presented is that entries that behave according to subregularities will automatically be grouped together again and call the same interaction predicate. The final representa24 In general, there is not always enough information available to determine whether two sequences of lexical rule applications produce identical entries. This is because in order to be able to treat recursive lexical rules producing infinite lexica, we perform word class specialization of the interaction predicate instead of expanding out the lexicon.</Paragraph>
      <Paragraph position="6"> 25 Note that an automaton can be made even more deterministic by unfurling instances of cycles prior to pruning. In our example, unfurling the direct cycle by replacing 3(q14, q14) with {3(q14, q14~), 3(q14 ~, q14~), 4(q14 ~, q19~)} would allow pruning of the cyclic transition 3(q14 ~, q14 ~) and the transition 4(q14, q19). Note, however, that unfurling of the first n instances of a cycle does not always allow pruning of transitions, i.e., reduce nondeterminism.</Paragraph>
      <Paragraph position="7"> 26 The pruned finite-state automaton constitutes valuable feedback, as it represents the interaction of the set of lexical rules possible for a word class in a succinct and perspicuous manner.</Paragraph>
      <Paragraph position="9"> An extended lexical entry.</Paragraph>
      <Paragraph position="10"> tion of the lexical rules and the lexical entries remains, without a special specification of exceptions. 27</Paragraph>
    </Section>
    <Section position="4" start_page="556" end_page="559" type="sub_section">
      <SectionTitle>
3.4 Lexical Rule Interaction as Definite Relations
</SectionTitle>
      <Paragraph position="0"> In the fourth compilation step, the finite-state automata produced in the last step are encoded in definite clauses, called interaction predicates. The lexical entries belonging to a particular natural class all call the interaction predicate encoding the automaton representing lexical rule interaction for that class. Figure 17 shows the extended version of the lexical entry of Figure 15. The base lexical entry is fed into the first argument of the call to the interaction predicate q_l. For each solution to a call to q_l the value of ~ is a derived lexical entry.</Paragraph>
      <Paragraph position="1"> Encoding a finite-state automaton as definite relations is rather straightforward.</Paragraph>
      <Paragraph position="2"> In fact, one can view the representations as notational variants of one another. Each transition in the automaton is translated into a definite relation in which the corresponding lexical rule predicate is called, and each final state is encoded by a unit clause. Using an accumulator passing technique (O'Keefe 1990), we ensure that upon execution of a call to the interaction predicate q_l a new lexical entry is derived as the result of successive application of a number of lexical rules. Because of the word class specialization step discussed in Section 3.3, the execution avoids trying out many lexical rule applications that are guaranteed to fail.</Paragraph>
      <Paragraph position="3"> We illustrate the encoding with the finite-state automaton of Figure 16. As the lexical rules themselves are already translated into a definite clause representation in the first compilation step, the interaction predicates only need to ensure that the right combination of lexical rule predicates is called. The interaction predicate encoding the finite-state automaton of Figure 16 is shown in Figure 18. 28 We now have a first complete encoding of the lexical rules and their interaction represented as covariation in lexical entries. The encoding consists of three types of definite  clause predicates: 1. Lexical rule predicates representing the lexical rules; 2. Frame predicates specifying the frame for the lexical rule predicates; and 3. Interaction predicates encoding lexical rule interaction for the natural  classes of lexical entries in the lexicon.</Paragraph>
      <Paragraph position="4"> The way these predicates interconnect is represented in Figure 19.</Paragraph>
      <Paragraph position="5"> 27 Briscoe and Copestake (1996) argue that semi-productivity of lexical rules, which can be understood as a generalization of exceptions to lexical rules, can be integrated with our approach by assigning probabilities to the automaton associated with a particular lexical entry.</Paragraph>
      <Paragraph position="6"> 28 In order to distinguish the different interaction predicates for the different classes of lexical entries, the compiler indexes the names of the interaction predicates. Since for expository reasons we will only discuss one kind of lexical entry in this paper, we will not show those indices in the examples given.</Paragraph>
      <Paragraph position="8"> The definite relations representing the pruned finite state automaton of Figure 16.</Paragraph>
      <Paragraph position="9">  The automata resulting from word class specialization group the lexical entries into natural classes. In case the automata corresponding to two lexical entries are identical, the entries belong to the same natural class. However, each lexical rule application, i.e., each transition in an automaton, calls a frame predicate that can have a large number of defining clauses. Intuitively understood, each defining clause of a frame predicate corresponds to a subclass of the class of lexical entries to which a lexical rule can be applied. During word class specialization, though, when the finite-state automaton representing global lexical rule application is pruned with respect to a particular base lexical entry, we know which subclass we are dealing with. For each interaction definition we can therefore check which of the flame clauses are applicable and discard the non-applicable ones. We thereby eliminate the redundant nondeterminism resulting from multiply defined frame predicates.</Paragraph>
      <Paragraph position="10"> The elimination of redundant nondeterminism is based on Unfold/Fold transformation techniques (Tamaki and Sato 1984). 29 The unfolding transformation is also referred to as partial execution, for example, by Pereira and Shieber (1987). Intuitively understood, unfolding comprises the evaluation of a particular literal in the body of a clause at compile-time. As a result, the literal can be removed from the body of 29 This improvement of the covariation encoding can also be viewed as an instance of the program transformation technique referred to as deletion of clauses with a finitely failed body (Pettorossi and Proietti 1994).</Paragraph>
      <Paragraph position="11">  Schematic representation of the partial unfolding transformation.</Paragraph>
      <Paragraph position="12"> the clause. Whereas unfolding can be viewed as a symbolic way of going forward in computation, folding constitutes a symbolic step backwards in computation.</Paragraph>
      <Paragraph position="13"> Given a lexical entry as in Figure 15, we can discard all frame clauses that presuppose tl as the value of c, as discussed in the previous section. To eliminate the frame predicates completely, we can successively unfold the frame predicates and the lexical rule predicates with respect to the interaction predicates. 3deg The successive unfolding steps are schematically represented in Figure 20.</Paragraph>
      <Paragraph position="14"> Such a transformation, however, would result in the loss of a representation of the lexical rule predicates that is independent of a particular word class, but an independent representation of lexical rules constitutes an advantage in space in case lexical rules can be applied across word classes. Our compiler therefore performs what can be viewed as &amp;quot;partial&amp;quot; unfolding: it unfolds the frame predicates directly with respect to the interaction predicates, as shown in Figure 21.</Paragraph>
      <Paragraph position="15"> One can also view this transformation as successive unfolding of the frame predicates and the lexical rule predicates with respect to the interaction predicates followed by a folding transformation that isolates the original lexical rule predicates. The definite clause encoding of the interaction predicates resulting from unfolding the frame predicates for the lexical entry of Figure 15 with respect to the interaction predicate of Figure 18 is given in Figure 22. The lexical rule predicates called by these interaction predicates are defined as in Figures 8 and 12, except that the frame predicates are no longer called.</Paragraph>
      <Paragraph position="16"> 30 Note that it is only possible to eliminate the frame predicates, since they are never called independently of the covariation encoding.</Paragraph>
      <Paragraph position="17">  Computational Linguistics Volume 23, Number 4</Paragraph>
      <Paragraph position="19"> Unfolding the frame predicates for the example entry with respect to the interaction predicate.</Paragraph>
    </Section>
  </Section>
class="xml-element"></Paper>