Categorial Type Logics

perspective, labels represent an extra `informational resource' which is present in deduction and can play a variety of roles. One obtains a scala of labeling regimes depending on the degree of autonomy between the formula component and the label component of the basic declarative units. At the conservative end of the spectrum, the label simply keeps a passive record of inferential steps. Curry-Howard labeling is a case in point. In this simple situation, the labels provide no feedback to in uence the deductive process. A more complex situation arises in cases in which the operations on labels are only partially de ned: under these conditions, the labeling system acts as a lter on deduction, since an inference step requiring a label operation that cannot be carried out cannot go through. In more complex cases, there is a genuine two-way communication between labels and formulas. Initial investigations of such possibilities can be found in [Gabbay 94],[Blackburn & de Rijke 94]. Within the categorial context, labeled deduction has appeared in a variety of di erent research directions. One focus of interest has been the use of labels to codify resourcemanagement in di erent systems of proof, especially in adaptations of Girard's `proof-nets' to various forms of categorial inference. We discuss this use in x7. Another focus of work on labeled deduction has been the investigation of how various logical systems are related to particular model structures. Already in 1986, [Buszkowski 86] employed -term labels in his study of completeness proofs for various members of the Lambek family of substructural logics. An LDS presentation ofNL in [Venema 94] forms the basis for completeness proofs for tree-based models. We have seen in x2 that the unlabeled system NL is incomplete for this type of semantics: labeling, in this case, makes an irreducible contribution to the deductive strength of the logic. In a similar vein, [Kurtonina 95] shows how the simple and multimodal Lambek systems of x2 and x4.1 can be reconstructed in such a way that they share a common system of type-inference for formulas and distinct systems of resource-management involving only the associated labels, and provides completeness proofs for the appropriate ternary-relation frames. This direction of work illustrates a basic slogan of labeled deduction: `bringing semantics into syntax'. The labels make it possible to discriminate among cases which the formula-language alone cannot tell apart. The work just described re ects traditional logical questions of proofand model-theory. But the study of labeled deduction for type-logical grammatical inference has intrinsic motivation as well: natural language (as stressed in [Oehrle 88]) is characteristically multidimensional, in a way that pairs phonological, syntactic, and interpretive information. Labeling can be used to explicitly manipulate these grammatical dimensions and their complex interactions. We mention some relevant studies. The system of `Natural Logic' of [Sanchez 95] presents a combined calculus of categorial deduction and natural language inference: the inferential contribution of subexpressions is accounted for in terms of a labeling system decorating Lambek deductions with monotonicity markings. The work on ellipsis and cross-over phenomena of [Gabbay & Kempson 92] similarly exploits the structure of a labeled natural deduction system in order capture the context-dependency of natural language interpretation. This form of labeled deduction, and the systems proposed in [Oehrle 95] in studies of quanti cation and binding, essentially interleave phonological, syntactic, semantic and pragmatic information, and assign to the labels an active role in controlling and constraining the course of proof. Finally, the general framework of labeled deduction provides a unifying point of view in which to study di erent proposals concerning the multidimensional nature of linguistic structure, ranging from the sequential architecture of the Government & Binding school to the family of parallel architectures found in the `sign'-based system of HPSG, the correspondences between f-structure, c-structure, and -structure of LFG, and 61 various members of the family of Tree Adjoining Grammars. These systems can be simulated (at least partially) in the framework of labeled type-logical deduction, in a way that may reveal underlying points of similarity and sharpen understanding of points of essential di erence. See [Joshi & Kulick 95] for an exploration of this perspective. 7 Categorial parsing as deduction In this section we turn to the computational study of categorial type logics, and discuss some aspects of their algorithmic proof theory, under the slogan `Parsing as Deduction' | a slogan which in the type-logical framework assumes a very literal interpretation. In x7.1 we return to a problem that was already signalled in x3: the many-to-one correspondence between Gentzen proofs and -term meaning recipes. We discuss the procedural control strategies that have been proposed to remove this source of `spurious ambiguity' from Gentzen proof search. In x7.2, we present an attractive alternative for Gentzen proof search, inspired by what Girard [Girard 95a] has called the Natural Deduction for resource logics | the `proof nets' of Linear Logic. In the spirit of x6.2, we show how labeling can be used to implement an appropriate level of control over the linguistic resources. 7.1 Proof normalization in Gentzen calculus From the literature on automated deduction, it is well known that Cut-free Gentzen proof search is still suboptimal from the e ciency perspective: there may be di erent (Cut-free!) derivations leading to one and the same proof term. Restricting ourselves to the implicational fragment, the spurious non-determinism in the search space has two causes ([Wallen 90]): (i) permutability of [L] and [R] inferences, and (ii) permutability of [L] inferences among themselves, i.e. non-determinism in the choice of the active formula in the antecedent. A so-called goal-directed (or: uniform) search regime performs the non-branching [R] inferences before the [L] inferences (re (i)), whereas head-driven search commits the choice of the antecedent active formula in terms of the goal formula (re (ii)). Such optimized search regimes have been proposed in the context of Linear Logic programming in [Hodas & Miller 94, Andreoli 92]. In the categorial setting, goal-directed head-driven proof search for the implicational fragment L was introduced in [Konig 91] and worked out in [Hepple 90] who provided a proof of the safeness (no proof terms are lost) and non-redundancy (each proof term has a unique derivation) of the method. We present the search regime in the format of [Hendriks 93] with Curry-Howard semantic term labelling. Definition 7.1. Goal-directed head-driven search for product-free L ([Hendriks 93]). [Ax/?L] x : p? ) x : p ; u : B?; 0 ) t : p ; u : B; 0 ) t : p? [?R] [/R] ; x : B ) t : A? ) x:t : A=B? ) u : B? ; x : A?; 0 ) t : C ; s : A=B?; ; 0 ) t[su=x] : C [/L] [nR] x : B; ) t : A? ) x:t : BnA? ) u : B? ; x : A?; 0 ) t : C ; ; s : BnA?; 0 ) t[su=x] : C [nL] 62 The above L* calculus eliminates the spurious non-determinism of the original presentation L by annotating sequents with a procedural control operator `*'. Goal sequents ) t : A in L are replaced by L* goal sequents ) t : A?. With respect to the rst cause of spurious ambiguity (permutability of [L] and [R] inferences), the control part of the [R] inferences forces one to remove all connectives from the succedent until one reaches an atomic succedent. At that point, the `*' control is transmitted from succedent to antecedent: the [*R] selects an active antecedent formula the head of which ultimately, by force of the control version of the Axiom sequent [*L], will have to match the (now atomic) goal type. The [L] implication inferences initiate a `*' control derivation on the minor premise, and transmit the `*' active declaration from conclusion to major (right) premise. The e ect of the ow of control information is to commit the search to the target type selected in the [*R] step. This removes the second source of spurious ambiguity: permutability of [L] inferences. Prop 7.2 sums up the situation with respect to proofs and readings in L and L*. Syntactically, derivability in L and L* coincide. Semantically, the set of L* proof terms forms a subset of the L terms. But, modulo logical equivalence, no readings are lost in moving from L to L*. Moreover, the L* system has the desired one-to-one correspondence between readings and proofs. Proposition 7.2. Proofs and readings ([Hendriks 93]). 1. L? ` ) A? i L ` ) A 2. L? ` ) t : A? implies L ` ) t : A 3. L ` ) t : A implies 9t0; t0 = t and L? ` ) t0 : A? 4. if 1 is an L* proof of ) t : A and 2 is an L* proof of ) t0 : A and t = t0, then 1 = 2 Example 7.3. Without the constraint on uniform head-driven search, there are two L sequent derivations for the Composition law, depending on the choice of a=b or b=c as active antecedent type. They produce the same proof term. Of these two, only the rst survives in the L* regime. (c)? ) c ?L c) (c)? ?R (b)? ) b ?L (b=c)?; c) b =L b=c; c) (b)? ?R (a)? ) a ?L (a=b)?; b=c; c) a =L a=b; b=c; c ) (a)? ?R a=b; b=c) (a=c)? =R (c)? ) c ?L c) (c)? ?R fail a=b; (b)? ) a a=b; (b=c)?; c) a =L a=b; b=c; c) (a)? ?R a=b; b=c) (a=c)? =R Uniform proof search: modal control. The control operators 3;2# make it possible to enforce the Konig-Hepple-Hendriks uniform head-driven search regime via a modal translation, as shown in [Moortgat 95]. This illustrates a second type of control that can be logically implemented in terms of the unary vocabulary: a procedural/dynamic form of control rather than the structural/static control of x4.2.3. The 3;2# annotation is a variation on the \lock-and-key" method of [Lincoln e.a. 95]: one forces a particular execution strategy for successful proof search by decorating formulae with the 2# (`lock') and 3 63 (`key') control operators. For the selection of the active formula, one uses the distributivity principesK1;K2, in combination with the base residuation logic for 3;2#. To establish the equivalence with L* search, one can use the sugared presentation of L where Associativity is compiled away so that binary punctuation ( ; ) can be omitted (but not the unary ( ) !). This gives the following compiled format for K1;K2: ; (A) ; 0 ) B ( ; A; 0) ) B K 0 (43) The mappings (44) ( )1; ( )0 : F(=; n) 7! F(=; n;3;2#), for antecedent and succedent formula occurrences, respectively, are de ned as follows. (p)1 = p (p)0 = 2#p (A=B)1 = (A)1=(B)0 (A=B)0 = (A)0=2#(B)1 (BnA)1 = (B)0n(A)1 (BnA)0 = 2#(B)1n(A)0 (44) We have the following proposition ([Moortgat 95]). L? ` ) A? i L3K0 ` 2#( )1 ) (A)0 (45) Example 7.4. Uniform head-driven search: modal control. We illustrate how a wrong identi cation of the antecedent head formula leads to failure. Below the modal translation of the crucial upper part of the failing L* derivation in Ex 7.3. c) c (c)1 ) c ( )1 (2#(c)1) ) c 2#L 2#(c)1 ) 2#c 2#R 2#(c)1 ) (c)0 ( )0 fails 2#(a=b)1; (b)1 ) a y 2#(a=b)1; (b)1=(c)0;2#(c)1 ) a =L 2#(a=b)1; (b=c)1;2#(c)1 ) a ( )1 2#(a=b)1; (2#(b=c)1) ;2#(c)1 ) a 2#L (2#(a=b)1;2#(b=c)1;2#(c)1) ) a K 0 2#(a=b)1;2#(b=c)1;2#(c)1 ) 2#a 2#R Consider rst the interaction of [/R] rules and selection of the active antecedent type. Antecedent types all have 2# as main connective. The 2# acts as a lock: a 2#A formula can only become active when it is unlocked by the key 3 (or ( ) in structural terms). The key becomes available only when the head of the goal formula is reached: through residuation, [2#R] transmits 3 to the antecedent, where it selects a formula via [K 0]. There is only one key 3 by residuation on the 2# of the goal formula. As soon as it is used to unlock an antecedent formula, that formula has to remain active and connect to the Axiom sequent. In the derivation above, the key to unlock 2#(a=b)1 has been spent on the wrong formula. As a result, the implication in (a=b)1 cannot become active. The normalization strategy for the implicational fragment can be extended to cover the connective as well, as shown in Andreoli's `focusing proofs' approach for LP (the 64 multiplicative fragment of Linear Logic). But in the case, a residue of `spurious' nondeterminism remains. Summarizing the above, we see that the Gentzen format has certain limitations as a vehicle for e cient categorial computation. To overcome these limitations, the above proposals increase the Gentzen bookkeeping by adding procedural control features. The alternative is to switch to a data structure for categorial derivations which e ectively removes the sources of computational ine ciency. We turn to such an alternative below. 7.2 Proof nets and labeled deduction In Sections 4.1 and 4.2 we have described the shift from the study of individual categorial systems to the mixed architecture of multimodal type logics. In this section, we consider the mixed architecture from a computational point of view. The central objective in this area is to develop a general algorithmic proof theory for the multimodal grammar logics. This is an active area of research (see a.o. [Moortgat 92, Morrill 95c, Hepple 94, Oehrle 95]) with an evolving methodology centering around the combination of the techniques of proof nets and labeled deduction, i.e. resolution theorem proving over labeled literals (rather than Gentzen style proof search over structured formula databases). Here are some desiderata that guide current work. In practice, one nds the expected tension between generality/expressivity and e ciency. {soundness and completeness of the labeling regime w.r.t. the interpretation; {expressivity: support for the full language 3;2#; =; ; n; {modular treatment of `logic' (residuation) and `structure' (resource management); {reversibility: neutrality w.r.t. parsing and generation; {e cient compilation techniques. In Linear Logic, [Girard 87] has advocated `proof nets' as the appropriate proof-theoretic framework for resource logics. The proof net approach has been studied in the context of categorial type logics in [Roorda 91]. In this section, we rst discuss semantic -term labelling for categorial proof nets. On the basis of the semantic labelling one can characterize the appropriate well-formedness conditions for proof nets and the correspondence between nets and LP sequent derivations. In order to capture the syntactic ne-structure of systems more discriminating than LP, and multimodal architectures with interacting unary and binary multiplicatives, we complement the semantic labeling with structure labeling. Both types of labeling `bring semantics into the syntax', as the slogan has it: for the structure labeling, this is the semantics for the form dimension of the linguistic resources, as discussed in x2. semantic labeling. Building a proof net corresponding to a sequent ) A is a three stage process. The rst stage is deterministic and consists in unfolding the formula decomposition tree for the Ai antecedent terminal formulae and for the goal formula A. The unfolding keeps track of the antecedent/succedent occurrence of subformulae: we distinguish ( )1 (antecedent) from ( )0 (succedent) unfolding, corresponding to the sequent rules of use and proof for the connectives. We call the result of the unfolding a proof frame. The second stage, corresponding to the Axiom case in the Gentzen presentation, consists in linking the literals with opposite signature. We call an arbitrary linking connecting the leaves of the proof frame a proof structure. Not every proof structure corresponds to a sequent derivation. The nal stage is to perform a wellformedness check on the proof structure graph in 65 order to identify it as a proof net, i.e. a structure which e ectively corresponds to a sequent derivation. Definition 7.5. Formula decomposition. Antecedent unfolding ( )1, succedent unfolding ( )0. 9-type (3L, L, =R,nR) and 8-type (2#L,=L,nL, R) decomposition. (A)1 (B)0 (A=B)1 8 (B)1 (A)0 (A=B)0 9 (B)0 (A)1 (BnA)1 8 (A)0 (B)1 (BnA)0 9 (A)1 (B)1 (A B)1 9 (B)0 (A)0 (A B)0 8 (A)1 (3A)1 9 (A)0 (3A)0 8 (A)1 (2#A)1 8 (A)0 (2#A)0 9 For the checking of the well-formedness conditions, there are various alternatives, such as Girard's original `long trip' condition, or the coloring algorithm of [Roorda 91]. Here we develop a labeling approach, because it is naturally adaptable to the linguistic application of the type logics in parsing and generation. The purpose of the labeling regime in this case is to push all relevant information about the structure of the proof frame up to the atomic leaf literals, so that the wellformedness check can be formulated locally in terms of resolution on the axiom links. Definition 7.6. Labelled formula decomposition. Positive (antecedent) unfolding ( )1, negative (succedent) unfolding ( )0. We use x; y; z (t; u; v) for object-level variables (terms), M;N for meta-level variables. Newly introduced object-level variables and metavariables in the rules below are chosen fresh. Axiom links t : (A)1 M : (A)0 M : (A)0 t : (A)1 with M := t t(M) : (A)1 M : (B)0 t : (A=B)1 x : (B)1 N : (A)0 x:N : (A=B)0 M : (B)0 t(M) : (A)1 t : (BnA)1 N : (A)0 x : (B)1 x:N : (BnA)0 (t)0 : (A)1 (t)1 : (B)1 t : (A B)1 N : (B)0 M : (A)0 hM;Ni : (A B)0 [t : (A)1 t : (3A)1 M : (A)0 \M : (3A)0 _t : (A)1 t : (2#A)1 M : (A)0 ^M : (2#A)0 For the binary vocabulary, the wellformedness conditions of Def 7.7 identify LP proof nets among the proof structures. Roorda also shows that one can narrow this class to the L proof nets by imposing an extra planarity constraint forbidding `crossing' axiom linkings 66 (interpreting the formula decomposition steps of Def 7.6 in an order-sensitive way). Evaluating this proposal, [Hendriks 93] observes that don't care non-determinism is removed at the declarative level, not at the algorithmic/procedural level: the conditions act as lters, in the `generate-and-test' sense, rejecting structures that have already been computed. In [Hendriks 93], the set of proof nets is de ned in an alternative, purely inductive way, which does away with proof net conditions for rejecting structures that have already been computed. Definition 7.7. Proof net conditions ([Roorda 91]). Let t be the term assigned to the ( )0 goal formula. (P1) there is precisely one terminal ( )0 formula (P2) all axiom substitutions can be performed (P3) if t contains a subterm x:u then x occurs in u and x does not occur outside u (P4) every variable assigned to a terminal ( )1 formula occurs in t (P5) every subterm of t counts for one in t (P6) t has no closed subterms structure labeling. The semantic labeling of Def 7.6 checks for LP derivability, and relies on a geometric criterion (planarity) for the L re nement. It is not clear how the geometric approach would generalize to other systems in the categorial landscape, and to the multimodal systems. Below, we give a system of structure labeling, that functions with respect to the `structural' interpretation of the type language. The labeling regime of Def 7.8 is related to proposals in [Hepple 94, Morrill 95c, Oehrle 95], but makes adjustments for the full multimodal architecture. Definition 7.8. Structure labels: syntax. The labeling system uses atomic formula labels x and structure labels ; ( ), for the 8 formula decomposition nodes. For the 9 nodes, we use di erence structures: expressions that must be rewritten to structure/formula labels under the residuation reductions of Def 7.9. ; ! x (atoms) (constructor 3) t (destructor 3) u (goal 2#) ( ) (constructor ) /( ) (left-destructor ) ( ). (right-destructor ) xn (goal n) =x (goal =) Definition 7.9. Labelled formula decomposition and residuation term reductions (boxed). Positive (antecedent) unfolding ( )1, negative (succedent) unfolding ( )0. We use x; y; z (t; u; v) for object-level formula (structure) labels, ; for meta-level structure label variables. Newly introduced formula labels and metavariables in the rules below are chosen fresh. (t ) : (A)1 : (B)0 t : (A=B)1 x : (B)1 : (A)0 =x : (A=B)0 (t x )=x t 67 : (B)0 ( t) : (A)1 t : (BnA)1 : (A)0 x : (B)1 xn : (BnA)0 xn(x t) t /(t) : (A)1 (t). : (B)1 t : (A B)1 : (B)0 : (A)0 ( ) : (A B)0 (/(t) (t).) t tt : (A)1 t : (3A)1 : (A)0 : (3A)0 tt t t : (A)1 t : (2#A)1 : (A)0 u : (2#A)0 u t t The basic residuation reductions in Def 7.9 are dictated by the identities for complex formulae 3A;2#A;A B;A=B;BnA. Structural postulates A! B translate to reductions (B) (A), where ( ) is the structure label translation of a formula. The reduction for the distributivity postulate K is given as an illustration in (46). Notice that both residuation reductions and structural postulate reductions are asymmetric, capturing the asymmetry of the derivability relation. 3(A B)! 3A 3B ( ) ; ( t u) (t u) (46) The parsing problem now assumes the following form. To determine whether a string x1 : : : xn can be assigned the goal type B on the basis of a multiset of lexical assumptions = x1 : A1; : : : ; xn : An, one takes the formula decomposition of ( )1; (B)0, and resolves the literals t : (p)1, : (p)0, with matching := t under the residuation and/or structural postulate rewritings. The string x1 : : : xn has to be the yield of the structure label assigned to the goal type B. Example 7.10. Compare the unfoldings (y) for the theorem A! 2#3A and (z) for the non-theorem 2#3A! A. Matching 1,2 gives the instantiation := x. For the goal type (2#3A)0 we get u = u x x. In the case of (z), matching 3,4 yields := t x which does not reduce to x. (y) x : (A)1 1 : (A)0 2 : (3A)0 u : (2#3A)0 (z) t x : (A)1 3 x : (3A)1 x : (2#3A)1 : (A)0 4 efficient compilation techniques. The labeling regime of Def 7.9 covers the multimodal architecture in a general fashion. In designing e cient compilation techniques one can exploit the properties of speci c multimodal grammars. Such techniques have been developed in [Morrill 95b, Morrill 95c] for the grammar fragment covered by [Morrill 94a] and some extensions. With a restriction on the use of the product connective, the compiler can translate lexical type assignments into a higher-order clausal fragment of linear logic. The clauses are processed by a linear-logical clausal engine and the normalised structure labels and semantic labels yielding the string are enumerated. The structural, sublinear properties, are represented by the term structure of the linear clauses. Satisfaction of constraints 68 under associativity can be met not by enumerating and testing uni ers, but by propagatingsuccessive constraints through string position or di erence list representations, as used inthe compilation of context-free grammar. One thus obtains a linear logical programmingparadigm for categorial grammar which is the counterpart of the logic programing paradigmfor phrase structure grammar. But whereas the latter lacks resource-consciousness (in aderivation a rule may be used once, more than one, or not at all), sensitivity for the lexicalresources processed is a built-in feature of the categorial paradigm.8 Conclusions, directions for further researchIf one compares the present state of the eld with the situation as described seven years agoin the overview chapter of [Moortgat 88], one can describe the changes as dramatic. On thelevel of `empirical coverage', one has witnessed a strong increase in linguistic sophistication.On the level of `logical foundations', Lambek's architecture for a grammar logic has been sig-ni cantly generalized without sacri cing the attractive features of the original design. Belowwe summarize some key themes of the type-logical research that may facilitate comparisonwith related grammatical frameworks.Design of a speci c grammar logic, i.e. a logic with a consequence relation attunedto the resource-sensitivity of grammatical inference | to be contrasted with `gen-eral purpose' speci cation languages for grammar development, where such resourcesensitivity has to be stipulated, e.g. the language of feature logic used in HPSG.A uni ed deductive perspective on the composition of form and meaning in naturallanguage | to be contrasted with rule-based implementations of the compositionalityprinciple.Radical lexicalism. Properties of the macro-grammatical organisation are fully pro-jected from lexical type declarations.Integration of the grammar logic in a wider landscape of reasoning systems, so thatthe transition between the formal systems characterizing `knowledge of language' andthe systems of inference underlying more general cognitive/reasoning capacities canbe seen as gradual.The change of emphasis from individual type logics to mixed architectures suggestsnew lines of research. The following themes, among others, would seem relevant for futureexploration.Descriptive studies. Although, according to [Carpenter 96], the current descriptivecoverage of type logical grammar rivals that of competing grammar formalisms, onecan expect a wide range of further descriptive studies exploiting the expressivity ofinteractive modes of structural composition. Contrastive studies could deepen ourunderstanding of the `logical' perspective on parametric variation, characterized interms of language speci c structural rule packages.Formal learnability theory. From a cognitive point of view, the radical lexicalismof the type-logical approach makes the acquisition problem acute. This requires atheory explaining how multimodal type assignments (with their resource management69 properties) could be inferred from exposition to raw data. Learnability theory for theextended type logics can build on results that have been obtained for individual systems(e.g. [Buszkowski & Penn 90, Kanazawa 92], but will have to remove the unrealisticinput conditions for the learning algorithm assumed in these studies.Computational complexity. Even at the level of individual systems, our knowledge ispartial, see [van Benthem 91,95, Aarts 94, Aarts & Trautwein 95] for discussion andresults. The context-free recognizing power result for L of [Pentus 93], for example,has not yielded a polynomial complexity result for this system. In the multimodalsetting, one would like to have a systematic theory linking complexity to the algebraicproperties of the characterizing rule packages.Connections between categorial type-logics and Linear Logic. In this chapter, we havepresented an interpretation of the categorial formalism in terms of structural com-position of grammatical resources. Recent studies of applications of Linear Logic inlinguistic analysis suggest an interesting alternative interpretation in terms of temporalcomposition of grammatical processes. See [Lecomte & Retore 95] for an illustration.An integration of these two complementary perspectives would o er a uni ed frame-work for the study of the static aspects of linguistic structure and the dynamics ofnatural language communication.AcknowledgementsThis chapter is based in part on work supported by the National Science Foundation underGrant No. SBR-9510706, and on research conducted in the context of the Esprit BRAproject 6852 `Dynamic interpretation of natural language'. I thank the editors, GosseBouma and Martin Emms for comments on preliminary versions. The gestation periodwas somewhat longer than anticipated: it started with [Moortgat & Oehrle 93], and pro-duced [Moortgat 95, Kurtonina & Moortgat 95] as preparations for the nal delivery. I amdeeply indebted to Natasha Kurtonina, Dick Oehrle and Glyn Morrill: it was a pleasure towork together and exchange ideas with them over these years. The chapter has greatly ben-e ted from their e orts. 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