## Symmetric categorial grammar (2009)

Venue: | Journal of Philosophical Logic |

Citations: | 6 - 0 self |

### BibTeX

@ARTICLE{Moortgat09symmetriccategorial,

author = {Michael Moortgat},

title = {Symmetric categorial grammar},

journal = {Journal of Philosophical Logic},

year = {2009},

pages = {681--710}

}

### OpenURL

### Abstract

is lost or not), is a phenomenon which a linguistic semantics ought to explain, rather than ignore. (van Benthem 1986, p 213) The Lambek-Grishin calculus is a symmetric version of categorial grammar obtained by augmenting the standard inventory of type-forming operations (product and residual left and right division) with a dual family: coproduct, left and right difference. Interaction between these two families is provided by distributivity laws. These distributivity laws have pleasant invariance properties: stability of interpretations for the Curry-Howard derivational semantics, and structure-preservation at the syntactic end. The move to symmetry thus offers novel ways of reconciling the demands of natural language form and meaning. 1 1

### Citations

488 | The mathematics of sentence structure
- Lambek
- 1958
(Show Context)
Citation Context ...ment clause, or giving it wide scope (‘there is a disaster x such that John is afraid x is unavoidable’). But this time, the wide scope reading cannot be obtained in NL, or in the more flexible L of (=-=Lambek 1958-=-), from the s/(np\s) type assignment to ‘a disaster’. In (c) and (d), we see examples of what in generative grammar would be called movement. The relative pronoun ‘that’ in (c) binds the subject of th... |

178 | The duality of computation
- Curien, Herbelin
- 2000
(Show Context)
Citation Context ...ations and to the embedding of LG into single-conclusion LP. In (Bernardi and Moortgat 2009), the term language for LG derivations is a customized version of the λµ˜µ calculus of (Herbelin 2005) and (=-=Curien and Herbelin 2000-=-). The λµ˜µ term calculus is isomorphic to LK, Gentzen’s sequent calculus for classical logic. It is embedded into the λ calculus (isomorphic to natural deduction proofs in Intuitionistic Logic) via a... |

172 | Principles and implementation of deductive parsing - Shieber, Schabes, et al. - 1995 |

115 |
On the calculus of syntactic types
- Lambek
- 1961
(Show Context)
Citation Context ...ferent stages of the project. For comments on an earlier draft I thank Arno Bastenhof, Sylvain Pogodalla, Sylvain Salvati, and two anonymous reviewers. Remaining errors are my own. 1�� system NL of (=-=Lambek 1961-=-), these readings correlate with two derivations for the sentence, given types s/(np\s) and (s/(np\s))\s for subject and verb phrase. The (b) example, likewise, has two readings, interpreting ‘a disas... |

85 |
Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic
- Abrusci
- 1991
(Show Context)
Citation Context ...hat term for systems where ⊗ and ⊕ are related by de Morgan dualities, via an involutive negation (de Groote and Lamarche 2002), or cancelling pre- and post-negations, as in classical bilinear logic (=-=Abrusci 1991-=-; Lambek 1993). (Casadio 2001) has argued for the use of such classical systems for linguistic analysis. Compact bilinear logic is obtained by identifying the ⊗ and ⊕ operations; Lambek’s pregroup gra... |

55 | Towards Abstract Categorial Grammars - Groote - 2001 |

42 | Multimodal Linguistic Inference
- Moortgat
- 1995
(Show Context)
Citation Context ...is question by adding structural modalities that provide facilities for reconfiguring the grammatical material in a controlled way; examples of such work are (Morrill, Leslie, Hepple, and Barry 1990; =-=Moortgat 1996-=-). In the Discontinuous Lambek Calculi of Morrill and co-workers the categorial ontology is enriched with expressions consisting of detached parts (‘split’ strings) that wrap themselves around the phr... |

39 | Substructural logics on display
- Goré
- 1998
(Show Context)
Citation Context ...ws into well-behaved structural rules that do not interfere with the cut elimination process. A comprehensive presentation of the landscape of substructural logics as display calculi can be found in (=-=Goré 1997-=-). Goré discusses symmetric Lambek calculus and (a selection of) Grishin’s distributivity laws. Below we amend Goré’s presentation in two respects. First, in anticipation of the Curry-Howard interpret... |

34 |
Mathematical linguistics and proof theory
- Buszkowski
(Show Context)
Citation Context ...on the semantics discussed here. Reading R⊗xyz as x = y · z where · is a multiplicative operation, not necessarily associative, one obtains the powerset residuated groupoid interpretation of NL, see (=-=Buszkowski 1997-=-). A second groupoid operation, say +, can similarly interpret the operations of the ⊕ family, with R⊕xyz now seen as x = y + z. Next to the ‘intensional’ multiplicative operations considered in this ... |

34 |
Call-by-value, call-by-name, and strong normalization for the classical sequent calculus
- Lengrand
(Show Context)
Citation Context ... Moortgat 2009). Depending on the targeted fragment of Multiplicative Intuitionistic Linear Logic, the CPS translation can be defined in a number of ways. The version in (19), (20) below is based on (=-=Lengrand 2003-=-), extended with clauses for the co-implications. This translation stays within the implication fragment of LP/MILL. The target language for the translation in (Curien and Herbelin 2000) has conjuncti... |

33 | The semantics of variety in categorial grammar - Benthem - 1983 |

25 |
C’est maintenant qu’on calcule, au cœur de la dualité, 2005. Habilitation thesis
- Herbelin
(Show Context)
Citation Context ...retation of LG derivations and to the embedding of LG into single-conclusion LP. In (Bernardi and Moortgat 2009), the term language for LG derivations is a customized version of the λµ˜µ calculus of (=-=Herbelin 2005-=-) and (Curien and Herbelin 2000). The λµ˜µ term calculus is isomorphic to LK, Gentzen’s sequent calculus for classical logic. It is embedded into the λ calculus (isomorphic to natural deduction proofs... |

25 | Categorial deductions and structural operations - Morrill, Leslie, et al. - 1990 |

23 | Proof theory for full intuitionistic linear logic, bilinear logic
- Cockett, Seely
- 1997
(Show Context)
Citation Context ...eraction principles, which play a key role in what follows, have been intensively studied from a category theoretic perspective; see the work on ‘weak’ or linear distributive categories, for example (=-=Cockett and Seely 1996-=-; de Paiva and Ritter 2006) At the Rome workshops organized in the late 1990s by Abrusci and Casadio, Lambek introduced Grishin’s work to a linguistic audience; an English translation by his student Č... |

20 | Donkey anaphora as in-scope binding
- Barker, Shan
(Show Context)
Citation Context ...inuations back into the syntactic source calculus; their semantic effects now arise through the CPS translation. Barker and Shan, in a series of papers, (Barker 2004; Barker and Shan 2006; Shan 2005; =-=Barker and Shan 2008-=-) among others, have worked out continuation-based analyses for a variety of semantic phenomena. An explicit comparison with LG is worked out in (Bastenhof 2009), where it is shown that the Barker-Sha... |

19 |
On a generalization of the Ajdukiewicz-Lambek system
- Grishin
- 1983
(Show Context)
Citation Context ...a 2009). In this paper, I report on a line of work that extends categorial grammar along a different axis. The point of departure is the generalization of the Ajdukiewicz-Lambek calculus proposed in (=-=Grishin 1983-=-), where in addition to the familiar product operation ⊗ and its residual implications one finds a dual coproduct ⊕ with residual co-implications. The extended vocabulary leads to derivability judgeme... |

18 | Relational semantics for the LambekGrishin calculus
- Kurtonina, Moortgat
(Show Context)
Citation Context ...· B · X ⊢ B focusr � A ⊢ Y X ⊢ B ⇌ (16) 4 From LG to LP: CPS translation Let us turn now to the Curry-Howard interpretation of LG derivations and to the embedding of LG into single-conclusion LP. In (=-=Bernardi and Moortgat 2009-=-), the term language for LG derivations is a customized version of the λµ˜µ calculus of (Herbelin 2005) and (Curien and Herbelin 2000). The λµ˜µ term calculus is isomorphic to LK, Gentzen’s sequent ca... |

18 | Type raising, continuations, and classical logic - Groote - 2001 |

17 | Linguistic side effects
- Shan
- 2005
(Show Context)
Citation Context ... pulls continuations back into the syntactic source calculus; their semantic effects now arise through the CPS translation. Barker and Shan, in a series of papers, (Barker 2004; Barker and Shan 2006; =-=Shan 2005-=-; Barker and Shan 2008) among others, have worked out continuation-based analyses for a variety of semantic phenomena. An explicit comparison with LG is worked out in (Bastenhof 2009), where it is sho... |

14 |
Continuations and the nature of quantification. Natural language semantics 10
- Barker
- 2002
(Show Context)
Citation Context ...⌈v⌉) = (∃ λy.(∀ λx.((teases y) x))) ε( ̂ θ12⌈v⌉) = λu9.(everybody (teases (someone u9))) (24) The use of continuations in natural language semantics was advocated originally in (de Groote 2001b) and (=-=Barker 2002-=-). One could say that the symmetric LG calculus pulls continuations back into the syntactic source calculus; their semantic effects now arise through the CPS translation. Barker and Shan, in a series ... |

14 | Types as graphs: Continuations in type logical grammar
- Barker, Shan
- 2006
(Show Context)
Citation Context ... symmetric LG calculus pulls continuations back into the syntactic source calculus; their semantic effects now arise through the CPS translation. Barker and Shan, in a series of papers, (Barker 2004; =-=Barker and Shan 2006-=-; Shan 2005; Barker and Shan 2008) among others, have worked out continuation-based analyses for a variety of semantic phenomena. An explicit comparison with LG is worked out in (Bastenhof 2009), wher... |

14 | On the expressive power of abstract categorial grammars: Representing context-free formalisms - Groote, Pogodalla - 2004 |

13 | Proof nets for the multimodal Lambek calculus
- Moot, Puite
(Show Context)
Citation Context ...sing approach for context-free rewriting systems and their extensions in (Shieber, Schabes, and Pereira 1995). Proof nets provide a declarative alternative. Moot (2007) shows that the nets for NL of (=-=Moot and Puite 2002-=-) can be straightforwardly generalized to the symmetric calculus LG; he proves soundness and completeness of these nets with respect to a Display Logic presentation of LG without the focusing rules. T... |

11 | Generalized Kripke frames
- Gehrke
- 2006
(Show Context)
Citation Context ...ne can add an ‘extensional’ vocabulary for the lattice operations. For the relational interpretation, these require more structured interpretants, as can be found in the Generalized Kripke Frames of (=-=Gehrke 2006-=-) or in the Generalized Galois Logics of (Bimbó and Dunn 2008). The symmetric situation is discussed in (Bimbó and Dunn 2009). Interaction: distributivity principles The symmetric vocabulary creates t... |

10 |
Algebraic methods in philosophical logic
- Dunn, Hardegree
- 2001
(Show Context)
Citation Context ...duces the inverses of the rules in (10) as an orthogonal choice for the interaction between the product and the coproduct families. Characteristic principles now are (12). These principles appear in (=-=Dunn and Hardegree 2001-=-) under the name of hemi-distributivity laws; their semantic content within the setting of generalized Galois logics is investivated in (Bimbó and Dunn 2009). (C ⊕ B) ⊗ A → C ⊕ (B ⊗ A) A ⊗ (B ⊕ C) → (... |

10 |
From categorial to bilinear logic
- Lambek
- 1993
(Show Context)
Citation Context ...n the West. It came to the attention of a wider public during the Tübingen conference on substructural logics organized by Doˇsen and Schröder-Heister in 1990. In the proceedings of that conference, (=-=Lambek 1993-=-) gives the first thorough presentation of this work, and expresses his scepticism as to the linguistic applicability. Grishin’s interaction principles, which play a key role in what follows, have bee... |

9 | Classical Non-Associative Lambek Calculus - Groote, Lamarche |

8 |
From Word to Sentence: a computational algebraic approach to grammar, Polimetrica
- Lambek
- 2008
(Show Context)
Citation Context ...argued for the use of such classical systems for linguistic analysis. Compact bilinear logic is obtained by identifying the ⊗ and ⊕ operations; Lambek’s pregroup grammars are based on this idea; see (=-=Lambek 2008-=-) for a recent exposition. 2 Symmetry, distributivity Recall the essentials of Lambek’s typelogical programme. A categorial grammar consists of a universal and a language-specific part. The universal ... |

5 |
Non-Commutative Linear Logic in Linguistics
- Casadio
- 2001
(Show Context)
Citation Context ...and ⊕ are related by de Morgan dualities, via an involutive negation (de Groote and Lamarche 2002), or cancelling pre- and post-negations, as in classical bilinear logic (Abrusci 1991; Lambek 1993). (=-=Casadio 2001-=-) has argued for the use of such classical systems for linguistic analysis. Compact bilinear logic is obtained by identifying the ⊗ and ⊕ operations; Lambek’s pregroup grammars are based on this idea;... |

5 | Residuated Lattices: An Algebraic Glimpse at - Galatos, Jipsen, et al. - 2007 |

5 | The conjoinability relation in Lambek calculus and linear logic
- Pentus
- 1993
(Show Context)
Citation Context ...erations do not imply a derivational relation; they would allow, for example, combinations p ⊓ q, p ⊔ q of distinct atomic types. calculus interpretation NL free quasigroup (Foret 2003) L free group (=-=Pentus 1993-=-) LP free Abelian group (Pentus 1993) LG free Abelian group (Moortgat and Pentus 2007) Table 1: Models for type equivalence 17The similarity relation for various typelogical systems has been characte... |

4 | Continuations in natural language syntax and semantics. Master Thesis MPhil Linguistics
- Bastenhof
- 2009
(Show Context)
Citation Context ...) → (A ⊗ B) ⊕ C A ⊗ (C ⊕ B) → C ⊕ (A ⊗ B) (B ⊕ C) ⊗ A → (B ⊗ A) ⊕ C (12) The addition to LG ∅ of both (10) and the inverses no longer satisfies the structure-preservation requirement. It is shown in (=-=Bastenhof 2009-=-) that this combination introduces associative and commutative perturbations in the individual ⊗ and ⊕ families. In the remainder of this paper, we will use the extension of the minimal symmetric syst... |

4 | Lambek grammars, tree adjoining grammars and hyperedge replacement grammars - Moot - 2008 |

4 | Dutch Grammar and Processing: A Case Study in TLG - Morrill, Valentín, et al. - 2009 |

3 |
Explanation in natural language of lambda-mu-comu terms
- Coen
- 2006
(Show Context)
Citation Context ...ural deduction. This many-to-one relation between sequent versus natural deduction derivations is often taken to be a flaw of the sequent calculus (‘spurious ambiguity’, ‘bureaucracy of syntax’). In (=-=Coen 2006-=-), the flexibility of the λµ˜µ calculus and its underlying sequent proof format in choosing the order of activation of formulas is exploited as a feature, rather than being perceived as a bug. Coen pr... |

3 | A Parigot-style linear λ-calculus for full intuitionistic linear logic. Theory and Applications of Categories - Paiva, Ritter |

3 |
Proof nets for display logic
- Moot
- 2007
(Show Context)
Citation Context ... The discussion of type similarity shows that LG achieves the expressivity of LP in a structurepreserving way. With respect to recognizing capacity a similar situation obtains. First, it is shown in (=-=Moot 2007-=-) that the mildly context-sensitive languages recognized by lexicalized tree adjoining grammars are also recognized by LG. The result is established on the basis of a streamlined version of LTAG, for ... |

2 |
Symmetric generalized galois logics
- Bimbo, Dunn
(Show Context)
Citation Context ...more structured interpretants, as can be found in the Generalized Kripke Frames of (Gehrke 2006) or in the Generalized Galois Logics of (Bimbó and Dunn 2008). The symmetric situation is discussed in (=-=Bimbó and Dunn 2009-=-). Interaction: distributivity principles The symmetric vocabulary creates the opportunity for interactions between the product and the coproduct families. We are interested in interactions that respe... |

2 | Chinese numbers, mix, scrambling, and range concatenation grammars
- Boullier
- 1999
(Show Context)
Citation Context ... multiplicity of k alphabet symbols in any order, and counting dependencies an 1 . . . an k for any number k of alphabet symbols. Patterns of this type are recognized by Range Concatenation Grammars (=-=Boullier 1999-=-) and Global Index Grammars (Castaño 2004); a comparison with these formalisms then might be useful to fix the upper bound of the recognizing capacity of LG, which is as yet unknown. Also with respect... |

2 |
Global index grammars and descriptive power
- Castaño
(Show Context)
Citation Context ... order, and counting dependencies an 1 . . . an k for any number k of alphabet symbols. Patterns of this type are recognized by Range Concatenation Grammars (Boullier 1999) and Global Index Grammars (=-=Castaño 2004-=-); a comparison with these formalisms then might be useful to fix the upper bound of the recognizing capacity of LG, which is as yet unknown. Also with respect to computational complexity, there are s... |

2 | The Generative Capacity of the Lambek–Grishin Calculus: A New Lower Bound
- Melissen
- 2009
(Show Context)
Citation Context ...tion by s ⦸ (a ⦸ (s ⊘ c)). MIX is a simple example of a pattern not recognized by LTAG. Melissen (2009) generalizes the type of construction exemplified in (27) and obtains the result below. theorem (=-=Melissen 2009-=-). All languages which are the intersection of a context-free language and the permutation closure of a context-free language are recognizable in LG. In this class, we find generalized forms of MIX, w... |

2 |
Type similarity for the Lambek-Grishin calculus
- Moortgat, Pentus
- 2007
(Show Context)
Citation Context ...mple, combinations p ⊓ q, p ⊔ q of distinct atomic types. calculus interpretation NL free quasigroup (Foret 2003) L free group (Pentus 1993) LP free Abelian group (Pentus 1993) LG free Abelian group (=-=Moortgat and Pentus 2007-=-) Table 1: Models for type equivalence 17The similarity relation for various typelogical systems has been characterized in terms of an algebraic interpretation of the types �·�, in the sense that A ∼... |

2 |
Non-associative Categorial Grammars and Abstract Categorial Grammars
- RETORÉ, SALVATI
- 2007
(Show Context)
Citation Context ...Grammar (ACG) (de Groote 2001a; de Groote and Pogodalla 2004) — after all, the target of the CPS translation is the language where ACG higher order linear signatures are defined. For comparison with (=-=Retoré and Salvati 2007-=-) we also give an ACG-style interpretation | · | extracting the generated string from the abstract CPS proof term, obtaining the mappings in (22). In the case of λLG, Σ0, as the starting point for the... |

1 | Continuations in natural language - unknown authors - 2004 |

1 | Generalized Galois Logics. Relational Semantics of Nonclassical Logical Calculi - Bimbó, Dunn - 2008 |

1 | On the computation of joins for non-associative Lambek categorial grammars - unknown authors - 2003 |

1 | Benthem (eds) Categorial grammar - Marciszewski, van - 1988 |