24 research outputs found
Relational Models for the Lambek Calculus with Intersection and Constants
We consider relational semantics (R-models) for the Lambek calculus extended
with intersection and explicit constants for zero and unit. For its variant
without constants and a restriction which disallows empty antecedents, Andreka
and Mikulas (1994) prove strong completeness. We show that it fails without
this restriction, but, on the other hand, prove weak completeness for
non-standard interpretation of constants. For the standard interpretation, even
weak completeness fails. The weak completeness result extends to an infinitary
setting, for so-called iterative divisions (Kleene star under division). We
also prove strong completeness results for product-free fragments
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
A Polynomial-Time Algorithm for the Lambek Calculus with Brackets of Bounded Order
Lambek calculus is a logical foundation of categorial grammar, a linguistic paradigm of grammar as logic and parsing as deduction. Pentus (2010) gave a polynomial-time algorithm for determining provability of bounded depth formulas in L*, the Lambek calculus with empty antecedents allowed. Pentus\u27 algorithm is based on tabularisation of proof nets. Lambek calculus with brackets is a conservative extension of Lambek calculus with bracket modalities, suitable for the modeling of syntactical domains. In this paper we give an algorithm for provability in Lb*, the Lambek calculus with brackets allowing empty antecedents. Our algorithm runs in polynomial time when both the formula depth and the bracket nesting depth are bounded. It combines a Pentus-style tabularisation of proof nets with an automata-theoretic treatment of bracketing
Unitless Frobenius quantales
It is often stated that Frobenius quantales are necessarily unital. By taking
negation as a primitive operation, we can define Frobenius quantales that may
not have a unit. We develop the elementary theory of these structures and show,
in particular, how to define nuclei whose quotients are Frobenius quantales.
This yields a phase semantics and a representation theorem via phase quantales.
Important examples of these structures arise from Raney's notion of tight
Galois connection: tight endomaps of a complete lattice always form a Girard
quantale which is unital if and only if the lattice is completely distributive.
We give a characterisation and an enumeration of tight endomaps of the diamond
lattices Mn and exemplify the Frobenius structure on these maps. By means of
phase semantics, we exhibit analogous examples built up from trace class
operators on an infinite dimensional Hilbert space. Finally, we argue that
units cannot be properly added to Frobenius quantales: every possible extention
to a unital quantale fails to preserve negations