arXiv subjects

Arno Bastenhof

Publications and source records attributed to Arno Bastenhof.

3 recordsLinked to original sources

arXiv2011

Focalization and phase models for classical extensions of non-associative Lambek calculus

Lambek's non-associative syntactic calculus (NL) excels in its resource consciousness: the usual structural rules for weakening, contraction, exchange and even associativity are all dropped. Recently, there have been proposals for conservative extensions dispensing with NL's intuitionistic bias towards sequents with single conclusions: De Groote and Lamarche's classical non-associative Lambek calculus (CNL) and the Lambek-Grishin calculus (LG) of Moortgat and associates. We demonstrate Andreoli's focalization property for said proposals: a normalization result for Cut-free sequent derivations identifying to a large extent those differing only by trivial rule permutations. In doing so, we proceed from a `uniform' sequent presentation, deriving CNL from LG through the addition of structural rules. The normalization proof proceeds by the construction of syntactic phase models wherein every `truth' has a focused proof, similar to work of Okada and of Herbelin and Lee.

cs.LO↗

arXiv2011

Polarized Montagovian Semantics for the Lambek-Grishin calculus

Grishin proposed enriching the Lambek calculus with multiplicative disjunction (par) and coresiduals. Applications to linguistics were discussed by Moortgat, who spoke of the Lambek-Grishin calculus (LG). In this paper, we adapt Girard's polarity-sensitive double negation embedding for classical logic to extract a compositional Montagovian semantics from a display calculus for focused proof search in LG. We seize the opportunity to illustrate our approach alongside an analysis of extraction, providing linguistic motivation for linear distributivity of tensor over par, thus answering a question of Kurtonina&Moortgat. We conclude by comparing our proposal to the continuation semantics of Bernardi&Moortgat, corresponding to call-by- name and call-by-value evaluation strategies.

cs.CL↗

arXiv2010

Tableaux for the Lambek-Grishin calculus

Categorial type logics, pioneered by Lambek, seek a proof-theoretic understanding of natural language syntax by identifying categories with formulas and derivations with proofs. We typically observe an intuitionistic bias: a structural configuration of hypotheses (a constituent) derives a single conclusion (the category assigned to it). Acting upon suggestions of Grishin to dualize the logical vocabulary, Moortgat proposed the Lambek-Grishin calculus (LG) with the aim of restoring symmetry between hypotheses and conclusions. We develop a theory of labeled modal tableaux for LG, inspired by the interpretation of its connectives as binary modal operators in the relational semantics of Kurtonina and Moortgat. As a linguistic application of our method, we show that grammars based on LG are context-free through use of an interpolation lemma. This result complements that of Melissen, who proved that LG augmented by mixed associativity and -commutativity was exceeds LTAG in expressive power.

cs.CL↗