9 research outputs found
Proof nets for display logic
This paper explores several extensions of proof nets for the Lambek calculus
in order to handle the different connectives of display logic in a natural way.
The new proof net calculus handles some recent additions to the Lambek
vocabulary such as Galois connections and Grishin interactions. It concludes
with an exploration of the generative capacity of the Lambek-Grishin calculus,
presenting an embedding of lexicalized tree adjoining grammars into the
Lambek-Grishin calculus
The Hidden Structural Rules of the Discontinuous Lambek Calculus
The sequent calculus sL for the Lambek calculus L (lambek 58) has no
structural rules. Interestingly, sL is equivalent to a multimodal calculus mL,
which consists of the nonassociative Lambek calculus with the structural rule
of associativity. This paper proves that the sequent calculus or hypersequent
calculus hD of the discontinuous Lambek calculus (Morrill and Valent\'in),
which like sL has no structural rules, is also equivalent to an omega-sorted
multimodal calculus mD. More concretely, we present a faithful embedding
translation between mD and hD in such a way that it can be said that hD absorbs
the structural rules of mD.Comment: Submitted to Lambek Festschrift volum
Comparing and evaluating extended Lambek calculi
Lambeks Syntactic Calculus, commonly referred to as the Lambek calculus, was
innovative in many ways, notably as a precursor of linear logic. But it also
showed that we could treat our grammatical framework as a logic (as opposed to
a logical theory). However, though it was successful in giving at least a basic
treatment of many linguistic phenomena, it was also clear that a slightly more
expressive logical calculus was needed for many other cases. Therefore, many
extensions and variants of the Lambek calculus have been proposed, since the
eighties and up until the present day. As a result, there is now a large class
of calculi, each with its own empirical successes and theoretical results, but
also each with its own logical primitives. This raises the question: how do we
compare and evaluate these different logical formalisms? To answer this
question, I present two unifying frameworks for these extended Lambek calculi.
Both are proof net calculi with graph contraction criteria. The first calculus
is a very general system: you specify the structure of your sequents and it
gives you the connectives and contractions which correspond to it. The calculus
can be extended with structural rules, which translate directly into graph
rewrite rules. The second calculus is first-order (multiplicative
intuitionistic) linear logic, which turns out to have several other,
independently proposed extensions of the Lambek calculus as fragments. I will
illustrate the use of each calculus in building bridges between analyses
proposed in different frameworks, in highlighting differences and in helping to
identify problems.Comment: Empirical advances in categorial grammars, Aug 2015, Barcelona,
Spain. 201