92 research outputs found
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
The Grail theorem prover: Type theory for syntax and semantics
As the name suggests, type-logical grammars are a grammar formalism based on
logic and type theory. From the prespective of grammar design, type-logical
grammars develop the syntactic and semantic aspects of linguistic phenomena
hand-in-hand, letting the desired semantics of an expression inform the
syntactic type and vice versa. Prototypical examples of the successful
application of type-logical grammars to the syntax-semantics interface include
coordination, quantifier scope and extraction.This chapter describes the Grail
theorem prover, a series of tools for designing and testing grammars in various
modern type-logical grammars which functions as a tool . All tools described in
this chapter are freely available
Multiplicative-Additive Focusing for Parsing as Deduction
Spurious ambiguity is the phenomenon whereby distinct derivations in grammar
may assign the same structural reading, resulting in redundancy in the parse
search space and inefficiency in parsing. Understanding the problem depends on
identifying the essential mathematical structure of derivations. This is
trivial in the case of context free grammar, where the parse structures are
ordered trees; in the case of categorial grammar, the parse structures are
proof nets. However, with respect to multiplicatives intrinsic proof nets have
not yet been given for displacement calculus, and proof nets for additives,
which have applications to polymorphism, are involved. Here we approach
multiplicative-additive spurious ambiguity by means of the proof-theoretic
technique of focalisation.Comment: In Proceedings WoF'15, arXiv:1511.0252
On noncommutative extensions of linear logic
Pomset logic introduced by Retor\'e is an extension of linear logic with a
self-dual noncommutative connective. The logic is defined by means of
proof-nets, rather than a sequent calculus. Later a deep inference system BV
was developed with an eye to capturing Pomset logic, but equivalence of system
has not been proven up to now. As for a sequent calculus formulation, it has
not been known for either of these logics, and there are convincing arguments
that such a sequent calculus in the usual sense simply does not exist for them.
In an on-going work on semantics we discovered a system similar to Pomset
logic, where a noncommutative connective is no longer self-dual. Pomset logic
appears as a degeneration, when the class of models is restricted. Motivated by
these semantic considerations, we define in the current work a semicommutative
multiplicative linear logic}, which is multiplicative linear logic extended
with two nonisomorphic noncommutative connectives (not to be confused with very
different Abrusci-Ruet noncommutative logic). We develop a syntax of proof-nets
and show how this logic degenerates to Pomset logic. However, a more
interesting problem than just finding yet another noncommutative logic is to
find a sequent calculus for this logic. We introduce decorated sequents, which
are sequents equipped with an extra structure of a binary relation of
reachability on formulas. We define a decorated sequent calculus for
semicommutative logic and prove that it is cut-free, sound and complete. This
is adapted to "degenerate" variations, including Pomset logic. Thus, in
particular, we give a variant of sequent calculus formulation for Pomset logic,
which is one of the key results of the paper
From Proof Nets to the Free *-Autonomous Category
In the first part of this paper we present a theory of proof nets for full
multiplicative linear logic, including the two units. It naturally extends the
well-known theory of unit-free multiplicative proof nets. A linking is no
longer a set of axiom links but a tree in which the axiom links are subtrees.
These trees will be identified according to an equivalence relation based on a
simple form of graph rewriting. We show the standard results of
sequentialization and strong normalization of cut elimination. In the second
part of the paper we show that the identifications enforced on proofs are such
that the class of two-conclusion proof nets defines the free *-autonomous
category.Comment: LaTeX, 44 pages, final version for LMCS; v2: updated bibliograph
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