Dirk Roorda, Resource logics: proof-theoretical investigations, Ph.D. thesis, University of Amsterdam, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Further, our results from section 2 provide a new proof of the context freeness of categorial grammars based on Nonassociative Lambek Calculus and ( rst proven in [4] for the product free system, then in [11] for the full system, and recently proven in [10] by a modi cation of results of Roorda [25] and Pentus [22] Nonassociative Lambek Calculus with permutation (proven in [12, 10] and Generalized Lambek Calculus (proven in [13, 10] The polynomial time decidability of Nonassociative Lambek Calculus with modalities and or permutation and Generalized Lambek Calculus seems to be new (it is ....

D. Roorda, Resource Logics: Proof Theoretical Investigations, Ph.D. Thesis, University of Amsterdam, 1991. 14

....give an idea and an illustration of it. This paper is thus mostly devoted to the properties on which the method is based. We call connection graphs the special kind of proof nets we explore, just in order to make explicit some difference with the usual method of proof nets, as it can be found in [Roorda, 1991; 1992] and [Moortgat 1992] but the two concepts are very similar. In many respects, connection graphs are a mere conservative extension of the earlier method of syntactic connection, discovered by Ajduckiewicz [1935] The method amounts to link the nodes of an ordered sequence of trees in such a ....

....graph. We assume here that connection graphs provide a semantics for derivations. It is possible to show that this semantics is isomorphic to the associative directed lambda calculus (see Wansing 1990) 2. 6 Soundness of Connection Graphs with respect to A This paragraph is very similar to Roorda 1991, chap lII, 4. Lemma 1: If we remove a type 1 link from a connection graph G, we keep a connection graph. Proof. we may assume that this link has been added at the last stage of the construction.0 Definition 4: a type 2 link is called separable if it could have been added in the last stage of ....

Dirk Roorda. Resource Logics: Prooftheoretical Investigations, PhD Thesis, Faculteit van Wiskunde en Informatica, Amsterdam.

....have seen that (A B) B # A already. Another case is t # (A B) # (B A) So the integers do not give an exact fit for distributive linear logic. Others have been aware that simple counting mechanisms can provide a useful filter for issues of validity in substructural logics [37, 148, 237, 210]. The logic here is known as abelian logic: It was introduced by Meyer and Slaney, who show that it is the logic of ordered abelian groups [173] EXAMPLE 21 (# UNDER DIVISION) Using number systems as structures gives us rich mathematical tradition upon which we can build. However, the structures ....

D. ROORDA. Resource Logics: Proof-theoretical Investigations. PhD thesis, Amsterdam, 1991.

....the proof published in [12, 14, 16] Section 6 deals with the Craig interpolation property in elementary fragments of the Lambek calculus. We prove that the fragments L( L( and L( have the interpolation property (6.1) The same about other elementary fragments is known due to D. Roorda [18, 19]. In addition, we introduce the notion of generalized interpolation property, which is of interest in fragments without multiplication. It is proved that the fragments L( and L( have the generalized interpolation property (6.3) whereas L( the product free Lambek calculus) does not ....

....type. Next we spread this replacement down along the derivation tree. This is possible due to the fact that in all derivation rules except the cut rule every primitive type occurrence in the consequence has exactly one predecessor in the premises of the rule. 4. Interpolation In 1991 D. Roorda [18] proved (using the method of Maehara and Schutte [20] that the calculus L # has the Craig interpolation property. In the paper [19] he remarked that the proof handles also the case of L. In Section 4.1 we present a proof of the interpolation theorem for L. Essentially this proof copies D. ....

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D. Roorda, Resource logics: proof-theoretical investigations, Ph.D. thesis, University of Amsterdam, 1991.

....In this section we prove the interpolation theorem for the product free fragment of the Lambek calculus and obtain a corollary for interpolation of thin sequents. Interpolation in the product free fragment of the Lambek calculus is more complicated than in the full Lambek calculus. See [12] for the proof of the interpolation theorem in the full Lambek calculus allowing empty antecedents. Namely, in the product free fragment we must allow not only single types, but also finite sequences of types to appear as interpolants. Lemma 7 Let # # Tp( # , # # Tp( # , # # Tp( ....

D. Roorda. Resource Logics: Proof-theoretical Investigations. PhD thesis, Fac. Math. and Comp. Sc., University of Amsterdam, 1991.

....any proofnet in some other proof net that matches some given (phonological or prosodic) bracketing. 1 Introduction Almost a decade ago, Girard invented linear logic together with the notion of proof net [7] Girard s proof nets have been subsequently adapted to the Lambek calculus by Roorda [16] and, since then, many authors have advocated the notion of proof net as the right parsing structure in the framework of categorial grammars [11, 13, 14, 16] Nevertheless, if one wants to take this proposal seriously, one must be able to perform, on the proof nets, all the computations that one ....

.... invented linear logic together with the notion of proof net [7] Girard s proof nets have been subsequently adapted to the Lambek calculus by Roorda [16] and, since then, many authors have advocated the notion of proof net as the right parsing structure in the framework of categorial grammars [11, 13, 14, 16]. Nevertheless, if one wants to take this proposal seriously, one must be able to perform, on the proof nets, all the computations that one usually performs on Gentzen s sequential derivations. From a theoretical standpoint, the above possible objection is actually not a problem. Indeed, by ....

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D. Roorda. Resource Logics: proof-theoretical investigations. PhD thesis, University of Amsterdam, 1991. 16

....computing semantic recipes. The absence of structural rules enables the consideration of non commutative restrictions of linear logic, first introduced by Abrusci and Yetter [2, 99] For instance one can rediscover the Lambek calculus as being exactly intuitionistic multiplicative linear logic [2, 81, 48, 75]. Non commutative linear logic proofs i.e. parse structures can be viewed as linear logic proofs, even when the proof is lifted to the corresponding semantical types; using the embedding of intuitionistic logic into linear logic, semantical terms which by Curry Howard isomorphism are ....

Dirk Roorda. Resource logic: proof theoretical investigations. PhD thesis, FWI, Universiteit van Amsterdam, 1991.

.... Lamping, Pereira, and Saraswat have shown how deduction in linear logic can be used to enforce various constraints during the construction of semantic terms [1, 2, 3] On the syntax side, Hepple, Johnson, Moortgat, Morrill, Oehrle, Roorda, and others have related linear logic to formal grammars [8, 13, 20, 21, 23, 27]. The latter body of work has focused primarily on the connection between linear logic and categorial grammars. This is natural as the Lambek Calculus [17, 18, 29] can be seen as a non commutative linear logic. In this work we show how linear logic can be used to provide a straightforward, ....

Dirk Roorda. Resource Logics: Proof-Theoretical Investigations. PhD thesis, University of Amsterdam, 1991.

....classes of string rewriting derivations. Corresponding structures for categorial grammar must be deeper, since they incorporate also semantics. Here we pursue the idea that proof nets (Girard 1987, Danos and Regnier 1990 1 ) are those structures (see e.g. Moortgat 1990b, 1992; Hendriks and Roorda 1991; Lecomte 1992, 1993; Lecomte and Retor e 1995; Oehrle 1994, 1995; Morrill 1996, 1999; Merenciano and Morrill 1997; de Groote and Retor e 1996) that proof nets are for categorial grammar what parse trees are for CFG. This provides a particularly vivid realisation of the notion of categorial ....

....calculus L of Lambek (1958) provides a logical model of language which presents formulas as categories and proofs as derivations. Proof nets for the calculus, recognizable as a multiplicative fragment of non commutative intuitionistic linear logic (Girard 1989; Abrusci 1990) were developed in (Roorda 1991). The question arises as to how to characterise proof nets for phenomena which go beyond the expressivity of L. A line of approach will be described here. 1.1 Associative Lambek calculus In the (associative) Lambek calculus L the category formulas F are constructed from atomic category formulas ....

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Roorda, Dirk: 1991, Resource Logics: proof-theoretical investigations, Ph.D. dissertation, Universiteit van Amsterdam.

....which would be tested when searching for a proof in the original sequent or Natural Deduction version of the Lambek calculus. Other proof techniques based on proof nets and quantum graphs, which originate from (Linear) logic rather than from parsing theory, have been presented in Roorda s thesis [28]. These techniques provide new perspectives of Lambek theorem proving. It would be interesting to see how structure sharing among derivations and incremental sentence processing could be integrated into the proof net approach. The use of a more general calculus of hypothetical reasoning, ....

Dirk Roorda. Resource Logics: Proof-theoretical Investigations. PhD thesis, University of Amsterdam, Amsterdam, 1991.

....one must use an extended lambda calculus [2, 21] 11 To accomplish this goal we need the BR lemma for product free types, which will be proven below. The proof follows the Pentus proof rather closely, but an essential change must be done in the interpolation lemma, established for LP in Roorda [26]. By ae(p; a) we denote the number of occurrences of the atomic type p in type a, and ae(p; X) ae(p; X a) are defined as ae(X) ae(X a) Let LP XY Z a with Y 6= The type y is called an interpolant of string Y in the latter context, if the following conditions are satisfied: I1) LP ....

....are defined as ae(X) ae(X a) Let LP XY Z a with Y 6= The type y is called an interpolant of string Y in the latter context, if the following conditions are satisfied: I1) LP Y y and LP XyZ a, I2) ae(p; y) min(ae(p; Y ) ae(p; XZ a) for every atomic type p. As shown in [26], interpolants exist for all strings Y 6= in any context LP XY Z a. The Pentus proof of the BR lemma relies on this interpolation property: the type d is chosen as an interpolant of an interval bc in LP Y bcZ a. For the case of L, the Roorda interpolation property does not hold. Consider ....

D. Roorda, Resource Logics: Proof-theoretical Investigations, Ph.D. Thesis, Faculty of Mathematics and Computer Science, University of Amsterdam, 1991. 21

....relation between incremental combinatory processing and the kind of processing phenomena cited in the introduction. 2 Proof nets Lambek categorial derivations are usually presented in the style of natural deduction or sequent calculus. Here we concern ourselves with categorial proof nets (Roorda 1991) as the fundamental structures of proof in categorial logic, in the same sense that linear Morrill Processing and Acceptability proof nets were originally introduced by Girard (1987) as the fundamental structures of proof in linear logic. Cut free) proof nets exhibit no spurious ambiguity and ....

Roorda, Dirk: 1991, Resource Logics: proof-theoretical investigations, Ph.D. dissertation, Universiteit van Amsterdam.

....serially according to the construction of formulas. The latter provides a phase of unfolding in which all the parts of a formula are made available in parallel, and then a non deterministic phase of linking which builds proofs from the axioms, but requires a certain correctness condition. Roorda (1991) expresses this condition by reference to labelling by lambda terms corresponding to proofs under the Curry Howard correspondence. Roorda (1991) and Moortgat (1991) do so by reference to labelling by groupoid terms of the algebras in which we interpret by residuation. We aim to improve the latter ....

....available in parallel, and then a non deterministic phase of linking which builds proofs from the axioms, but requires a certain correctness condition. Roorda (1991) expresses this condition by reference to labelling by lambda terms corresponding to proofs under the Curry Howard correspondence. Roorda (1991) and Moortgat (1991) do so by reference to labelling by groupoid terms of the algebras in which we interpret by residuation. We aim to improve the latter method, which as it stands presents the task of correctness checking in terms of intractable problems such as semigroup unification, i.e. it ....

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Roorda, Dirk: 1991, Resource Logics: proof-theoretical investigations, Ph.D. dissertation, Universiteit van Amsterdam.

....tasks of interpretation and generation. E mail: meren lsi.upc.es. E mail: morrill lsi.upc.es, http: www lsi.upc.es glyn . 3 The work we report was partially supported by project koala: dgicyt pb95 0787. Methods for categorial interpretation based on proof nets ( 5] 1] [18]) and labelling of deductive systems ( 3] have been developed in [12] 14] 15] and [11] The formalism of proof nets provides a representation of the fundamental structure of proofs, in the same way that parse trees do for context free grammar derivations. Using proof nets we avoid spurious ....

.... the long trip condition) But our first concern here is with the generality of our methodology for generation, which does not need to rely on any noncommutativity and which extends to all manner of sublinear calculi through unification under theory as in [14] Although introduced as long ago as [18], whether the prosodic unifiability alone assures the long trip condition has not been shown. Nor does it appear that unification under associativity (nondeterministic) is imperative for L: 15] and [16] propose formulations on the basis of just structural term unification (deterministic) Still, ....

Dirk Roorda. Resource Logics: Proof-theoretical Investigations. PhD thesis, Universiteit van Amsterdam, 1991.

....for context free grammar cannot suffice. Hence we find, for example, recursive emission of minicharts in Konig (1994) In chart methods we seek (simulation of) parallelism. When one identifies the notion of parallelism in categorial grammar it is found not in sequent proof but in proof nets. Roorda (1991) develops the notion of proof net for Lambek calculus in a manner corresponding to its original introduction in Girard (1987) for linear logic. If proof nets represent parallelism in categorial grammar it is perhaps in relation to these that one should develop tabular methods. In fact, Roorda ....

....representing equivalence classes of sequent proofs: not just normal forms subject to a particular ordering of steps, but structures embodying the partial ordering on inference steps in equivalence classes of sequential proofs. We consider the presentation of proof nets for Lambek calculus of Roorda (1991). Proof nets are built not over formulas but over the construction trees of formulas: trees with atoms at their leaves, and in which each mother node indicates the combination of its daughter subformulas by some connective. Thus the parts of a formula are laid out ( unfolded ) for potential ....

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Roorda, Dirk: 1991, Resource Logics: proof-theoretical investigations, Ph.D. dissertation, Universiteit van Amsterdam.

....Salgado, 1 3 E 08034 Barcelona E mail: morrill lsi.upc.es HTTP: www lsi.upc.es glyn 11th October 1997 Abstract Girard (1987) introduced proof nets as a syntax of linear proofs which eliminates inessential rule ordering manifested by sequent calculus. Proof nets adapted to the Lambek calculus (Roorda 1991) fulfill a role in categorial grammar analogous to that of phrase structure trees in CFG so that categorial proof nets have a central part to play in computational syntax and semantics; in particular they allow a reinterpretation of the problem of spurious ambiguity as an opportunity for ....

....Thus we prefer the term derivational equivalence to spurious ambiguity and interpret the phenomenon not as a problem for sequencialisation, but as an opportunity for parallelism. This opportunity is grasped in proof nets. 5. Proof nets for L Proof nets for L were first developed in detail by Roorda (1991), adapting their original introduction for linear logic in Girard (1987) In proof nets, the opposition of types arising from their location in either the antecedent or the succedent of sequents is replaced by assignment of negative (antecedent) or positive (succedent) polarity. A proof net here ....

Roorda, Dirk (1991), Resource Logics: Proof-theoretical Investigations. Ph.D. thesis, Universiteit van Amsterdam.

....correspondance proposed between these two aspects provides us with a method of parsing related to the conception of parsing as deduction , together with a method for avoiding spurious ambiguities. We will show that it is isomorphic to the method of proof nets (Girard 1987, Danos and Regnier 1989, Roorda 1990, 1991), but that it has the advantage over this last method of being more efficient and of providing more clarity on the result of processing. The devices we obtain are more readable, because they are interpretable in terms of dependency structures. Otherwise, the parsing method can be an incremental ....

....readable, because they are interpretable in terms of dependency structures. Otherwise, the parsing method can be an incremental one. 2. The Method of Proof Nets in the Lambek Calculus The problem of spurious ambiguities in Categorial Grammar is very often discussed (see for instance Hendriks and Roorda (1991)) A proof net is a device which contains all the equivalent proofs of the same result. As Roorda (1990) says: A proof net can be viewed as a parallellized sequent proof [ It is a concrete structure, not merely an abstract equivalence class of derivations, and surely not a special derivation ....

Roorda, D.: 1991, Resource Logics: Proof-theoretical Investigations, PhD Thesis, Faculteit van Wiskunde en Informatica, Amsterdam.

....any proofnet into some other proof net that matches some given (phonological or prosodic) bracketing. 1 Introduction Almost a decade ago, Girard invented linear logic together with the notion of proof net [7] Girard s proof nets have been subsequently adapted to the Lambek calculus by Roorda [16] and, since then, many authors have advocated the notion of proof net as the right parsing structure in the framework of categorial grammars [11, 13, 14, 16] Nevertheless, if one wants to take this proposal seriously, one must be able to perform, on the proof nets, all the computations that one ....

.... invented linear logic together with the notion of proof net [7] Girard s proof nets have been subsequently adapted to the Lambek calculus by Roorda [16] and, since then, many authors have advocated the notion of proof net as the right parsing structure in the framework of categorial grammars [11, 13, 14, 16]. Nevertheless, if one wants to take this proposal seriously, one must be able to perform, on the proof nets, all the computations that one usually performs on Gentzen s sequential derivations. From a theoretical standpoint, the above possible objection is actually not a problem. Indeed, by ....

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D. Roorda. Resource Logics: proof-theoretical investigations. PhD thesis, University of Amsterdam, 1991.

....some results on an extension of Gabbay s Labelled Deductive Systems (LDS) Gab90] for the case of a modal substructural logic. A weak implicational logic Concatenation Logic, due to Gabbay [Gab90] is combined with a modal operator as a case study. The logic is considered to be a resource logic [Gir87, Ben91, Roo91, Avr88], and a fragment of the Lambek Calculus [Lam58] This type of logic has recently attracted considerable attention in theoretical computer science [Abr93] and also has applications to natural language processing [Ben91] and AI planning [Hol92, MTV90] The logic is not completely arbitrary. It is ....

D. Roorda. Resource Logics -- Proof Theoretical Investigations. PhD thesis, FWI, Univ. Amsterdam, The Netherland, September 1991.

....parallelised. Thus we prefer the term derivational equivalence to spurious ambiguity and interpret the phenomenon not as a problem for sequentialisation, but as an opportunity for parallelism. This opportunity is grasped in proof nets. 4 Lambek proof nets Proof nets for L were developed by Roorda (1991), adapting their original introduction for linear logic in Girard (1987) In proof nets, the opposition of formulas arising from their location in either the antecedent or the succedent of sequents is replaced by assignment of polarity: input (negative) for antecedent and output (positive) for ....

Roorda D. (1991) Resource Logics: Prooftheoretical Investigations. Ph.D. thesis, Universiteit van Amsterdam.

....for the connectives. 1 XfflY = fx Deltay 2 L j x 2 X y 2 Y g X Y = fx 2 L j 8y 2 Y : x Deltay 2 X g YnX = fx 2 L j 8y 2 Y : y Deltax 2 X g X 1 , X n = fx 1 Delta: Deltax n 2 L j x 1 2 X 1 : xn 2 Xn g 1 The alternative formulations include e.g. sequent (Lambek 1958) proof net (Roorda 1991), and natural deduction systems (Morrill et al. 1990, Barry et al. 1991) Alternative formulations carry different advantages, e.g. natural deduction is well suited for linguistic presentation, whereas proof nets have benefits for automated theorem proving. Discontinuous type constructors The ....

Roorda, D. 1991. Resource Logics: Proof Theoretical Investigations. Ph.D. Dissertation, Amsterdam.

....have been considered as triples Syn : Phon : Sem where Syn is the syntactic part, Phon the phonetic part and Sem the semantic part of the sign. At the end of the eighties Girard introduced Linear Logic [53] and the Lambek calculus was recognised as a fragment of non commutative linear logic [2,103], thus putting a strong connection between syntax (categorial grammar) and logic (proof theory) Indeed Linear Logic provides a neat logical system to describe resource consumption, and, through the unary connectives called exponentials or modalities , neatly relates the Lambek calculus to ....

....theorem [74 76] is then highly important: it restricts the search space to a finite number of formulae, so this property makes the parsing obviously decidable and guides each step. Another important proof theoretical property of the Lambek calculus is interpolation, established by Roorda in [103], which states that whenever an implication holds there always exists an intermediate formula, whose only atomic formulae are common to the antecedent and consequent. This enables, as shown in this volume, a transformation from one bracketing of a sentence to another, e.g. from the syntagmatic one ....

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Dirk Roorda. Resource logic: proof theoretical investigations. PhD thesis, FWI, Universiteit van Amsterdam, 1991.

....solves an open question raised by Retor# [7] Indeed, in the literature, proof nets for the Lambek calculus are de ned in terms of conditions that ensure commutative correctness, together with an additional condition that ensures non commutativity. The latter is, most often, a planarity condition [7, 9]. In contrast, when using our criterion, commutative correctness and non commutativity are not checked independently. In [9, Chap. III, #6, pp. 3840] Roorda de nes a way of decorating proofnets that is almost identical to ours. He then observes that the existence of such a decoration is ....

D. Roorda. Resource Logics: proof-theoretical investigations. PhD thesis, University of Amsterdam, 1991.

....logics have a type theoretical side, via (adaptations of) the Curry Howard interpretation, cf. Wansing [24] van Benthem [2] How) can we assign terms to proofs in our calculi 3. Besides linear logic itself, Girard also invented a new proof method for it, viz. via proofnets. In his dissertation [20], Roorda extended this method to the Lambek calculus. Can we also find proof nets for the extended logic discussed here ....

Roorda, D., Resource Logics: Proof-theoretical Investigations, PhD Dissertation, University of Amsterdam, 1991.

....tries to capture this behavior using finite dynamic structures. Girard compares denotational semantics to the part of mechanics called Statics, and desires what would be the Dynamics of computation. Proof theoretical results for LL and other resource logics can be found in Dirk Roorda s thesis [145]. 1.5 Complexity of LL Fortunately, we have a quite good understanding of the computational complexity of various fragments of LL, mainly due to Patrick Lincoln [108] and Max Kanovich. Unfortunately, this complexity is rather high. ffl Propositional (quantifier free) LL is undecidable, unlike ....

D. Roorda. Resource Logics: Proof-theoretical Investigations. Dissertation, University of Amsterdam, Sept. 1991.

....discussed in section 4.3, this is handled in PPTS by use of the stretching operation. 5. 2 PPTS and Proof Nets There has been increasing interest in the last few years regarding the use of a linear logic proof net system for natural language uses, particularly for categorial grammar systems (e.g. [Roorda, 1991], Morrill, 1995] As just discussed, we take the view that natural deduction (or at least something very much like natural deduction) is more appropriate for a linguistic application than a proofnet representation. However, in some ways PPTS still has very much the flavor of proof nets ....

.... what would be involved in trying to recast PPTS in such a framework A number of questions immediately arise. First, the basic PPTs combine in a non commutative way. Therefore, some sort of non commutative proof net framework is required, such as those developed by [Abrusci, 1991] or [Roorda, 1991]. However, there is also the use of the limited permutation within a basic PPT. So, there needs to be some intermingling of commutativity and non commutativity in the proof net. This is akin to the use of some type of structural modality for permutation in linear logic or associative Lambek ....

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Dirk Roorda. Resource Logics: proof-theoretical investigations, Ph.D. dissertation, Universiteit van Amsterdam. 1991.

....serially according to the construction of formulas. The latter provides a phase of unfolding in which all the parts of a formula are made available in parallel, and then a non deterministic phase of linking which builds proofs from the axioms, but requires a certain correctness condition. Roorda (1991) expresses this condition by reference to labelling by lambda terms corresponding to proofs under the Curry Howard correspondence. Roorda (1991) and Moortgat (1990, 1992) do so by reference to labelling by groupoid terms of the algebras in which we interpret by residuation. We aim to improve the ....

....available in parallel, and then a non deterministic phase of linking which builds proofs from the axioms, but requires a certain correctness condition. Roorda (1991) expresses this condition by reference to labelling by lambda terms corresponding to proofs under the Curry Howard correspondence. Roorda (1991) and Moortgat (1990, 1992) do so by reference to labelling by groupoid terms of the algebras in which we interpret by residuation. We aim to improve the latter method, which as it stands presents the task of correctness checking in terms of intractable problems such as semigroup unification, i.e. ....

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Roorda, Dirk: 1991, Resource Logics: proof-theoretical investigations, Ph.D. dissertation, Universiteit van Amsterdam.

....to summarize in a few words the logical principles underlying categorial grammars [15, 17, 24] these could well be: Parsing as Deduction and Grammar Theory as Proof Theory. Indeed, during the last decade, proof theoretical investigations of categorial grammars have been extremely fruitful, e.g. [21, 24]. On the syntactic side, Roorda advocates the notion of proof net as an appropriate parsing structure [21] Proof nets are a new proof theoretic tool introduced by Girard in the framework of linear logic [6] They allow several proofs of the sequent calculus to be represented by the same structure ....

....be: Parsing as Deduction and Grammar Theory as Proof Theory. Indeed, during the last decade, proof theoretical investigations of categorial grammars have been extremely fruitful, e.g. 21, 24] On the syntactic side, Roorda advocates the notion of proof net as an appropriate parsing structure [21]. Proof nets are a new proof theoretic tool introduced by Girard in the framework of linear logic [6] They allow several proofs of the sequent calculus to be represented by the same structure when they do not dioeer in an essential way. In this sense, they correspond to unambiguous ....

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D. Roorda. Resource Logics: proof-theoretical investigations. PhD thesis, University of Amsterdam, 1991.

....a syntactic analysis of permutabilities of rules and apply it to cut elimination; our own analysis does not go quite as far, but we have a simpler proof of cut elimination. We conjecture that their presentation could be streamlined using our presentation of the linear sequent calculus. Roorda [Roo91] gives a different proof of cut elimination by generalizing the cut rule to multiple occurrences of modal formulas. The main challenge is to isolate the non linear reasoning and the associated structural rules. Our solution is close to Andreoli s Sigma 2 [And92] and Girard s LU [Gir93] in that ....

Dirk Roorda. Resource Logics: ProofTheoretical Investigations. PhD thesis, University of Amsterdam, September 1991. I \Psi; A \Gamma! A; \Theta \Psi; \Gamma 1 \Gamma! A; \Delta 1 ; \Theta \Psi; \Gamma 2 \Gamma! B; \Delta 2 ;

....used as a general framework for categorial deduction, via the use of such translations. 2 Approaches include sequent proof normalisation methods for L (Konig, 1989; Hepple, 1990; Hendriks, 1992) chart parsing methods for L (Konig, 1990; Hepple, 1992) and proof net methods for a range of systems (Roorda, 1991; Moortgat, 1992) 2 Implicational Linear Logic Linear logic is an example of a resource sensitive logic, requiring that in any deduction, every assumption ( resource ) is used precisely once. We consider only the implicational fragment of (intuitionistic) linear logic. 3 The set of ....

Roorda, D. 1991. Resource Logics: Proof Theoretical Investigations. Ph.D. Dissertation, Amsterdam.

....solves an open question raised by Retor# [8] Indeed, in the literature, proof nets for the Lambek calculus are de ned in terms of conditions that ensure commutative correctness, together with an additional condition that ensures non commutativity. The latter is, most often, a planarity condition [8, 10]. In contrast, when using our criterion, commutative correctness and non commutativity are not checked independently. In his thesis [10, Chap. III, #6, pp. 3840] Roorda de nes a way of decorating proof nets that is almost identical to ours. He then observes that the existence of such a decoration ....

D. Roorda. Resource Logics: proof-theoretical investigations. PhD thesis, University of Amsterdam, 1991.

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Dirk Roorda, Resource logics: proof-theoretical investigations, Ph.D. thesis, University of Amsterdam, 1991.

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Dirk Roorda, Resource logics: proof-theoretical investigations, Ph.D. thesis, University of Amsterdam, 1991.

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Dirk Roorda, Resource logics: proof-theoretical investigations, Ph.D. thesis, University of Amsterdam, 1991.

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Roorda, D., \Resource Logics: Proof-theoretical Investigations," Ph.D. thesis, Universiteit van Amsterdam (1991). 22

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Dirk Roorda. Resource Logics: proof-theoretical investigations. PhD thesis, University of Amsterdam, 1991.

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Roorda, D., "Resource Logics: Proof-theoretical Investigations," Ph.D. thesis, Universiteit van Amsterdam (1991). 22

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Dirk Roorda, Resource logics: proof-theoretical investigations, Ph.D. thesis, University of Amsterdam, 1991.

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Dirk Roorda, Resource logics: proof-theoretical investigations, Ph.D. thesis, University of Amsterdam, 1991.

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D. Roorda. Resource Logics: Proof-theoretical Investigations. Ph.D. Dissertation, Department of Mathematics and Computer Science of the University of Amsterdam, September 1991.

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D. Roorda. Resource Logics: Proof-theoretical Investigations. PhD thesis, Fac. Math. and Comp. Sc., University of Amsterdam, 1991.

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D. Roorda. Resource logics: proof-theoretical investigations. Dissertation, University of Amsterdam, September 1991.

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Roorda, D. 1991. Resource Logics: Proof-Theoretical Investigations, Ph.D. Dissertation, Amsterdam.

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D. Roorda. Resource Logics: Proof-theoretical Investigations. Ph.D. Thesis, University of Amsterdam. 1991.

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Roorda, D. 1991. Resource Logics: Proof Theoretical Investigations. Ph.D. Dissertation, Amsterdam.

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