367 research outputs found
On Pregroups, Freedom, and (Virtual) Conceptual Necessity
Pregroups were introduced in (Lambek, 1999), and provide a founda-tion for a particularly simple syntactic calculus. Buszkowski (2001) showed that free pregroup grammars generate exactly the -free context-free lan-guages. Here we characterize the class of languages generable by all pre-groups, which will be shown to be the entire class of recursively enumerable languages. To show this result, we rely on the well-known representation of recursively enumerable languages as the homomorphic image of the inter-section of two context-free languages (Ginsburg et al., 1967). We define an operation of cross-product over grammars (so-called because of its behaviour on the types), and show that the cross-product of any two free-pregroup grammars generates exactly the intersection of their respective languages. The representation theorem applies once we show that allowing ‘empty cat-egories ’ (i.e. lexical items without overt phonological content) allows us to mimic the effects of any string homomorphism.
Grammars over the Lambek Calculus with Permutation: Recognizing Power and Connection to Branching Vector Addition Systems with States
In [Van Benthem, 1991] it is proved that all permutation closures of
context-free languages can be generated by grammars over the Lambek calculus
with the permutation rule (LP-grammars); however, to our best knowledge, it is
not established whether converse holds or not. In this paper, we show that
LP-grammars are equivalent to linearly-restricted branching vector addition
systems with states and with additional memory (shortly, lBVASSAM), which are
modified branching vector addition systems with states. Then an example of such
an lBVASSAM is presented, which generates a non-semilinear set of vectors; this
yields that LP-grammars generate more than permutation closures of context-free
languages. Moreover, equivalence of LP-grammars and lBVASSAM allows us to
present a normal form for LP-grammars and, as a consequence, prove that
LP-grammars are equivalent to LP-grammars without product
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
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 Lambek calculus with iteration: two variants
Formulae of the Lambek calculus are constructed using three binary
connectives, multiplication and two divisions. We extend it using a unary
connective, positive Kleene iteration. For this new operation, following its
natural interpretation, we present two lines of calculi. The first one is a
fragment of infinitary action logic and includes an omega-rule for introducing
iteration to the antecedent. We also consider a version with infinite (but
finitely branching) derivations and prove equivalence of these two versions. In
Kleene algebras, this line of calculi corresponds to the *-continuous case. For
the second line, we restrict our infinite derivations to cyclic (regular) ones.
We show that this system is equivalent to a variant of action logic that
corresponds to general residuated Kleene algebras, not necessarily
*-continuous. Finally, we show that, in contrast with the case without division
operations (considered by Kozen), the first system is strictly stronger than
the second one. To prove this, we use a complexity argument. Namely, we show,
using methods of Buszkowski and Palka, that the first system is -hard,
and therefore is not recursively enumerable and cannot be described by a
calculus with finite derivations
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