49 research outputs found
Non-associative, Non-commutative Multi-modal Linear Logic
Adding multi-modalities (called subexponentials) to linear logic enhances its power as a logical framework, which has been extensively used in the specification of e.g. proof systems, programming languages and bigraphs. Initially, subexponentials allowed for classical, linear, affine or relevant behaviors. Recently, this framework was enhanced so to allow for commutativity as well. In this work, we close the cycle by considering associativity. We show that the resulting system (acLLΣ ) admits the (multi)cut rule, and we prove two undecidability results for fragments/variations of acLLΣ
Multi-dimensional Type Theory: Rules, Categories, and Combinators for Syntax and Semantics
We investigate the possibility of modelling the syntax and semantics of
natural language by constraints, or rules, imposed by the multi-dimensional
type theory Nabla. The only multiplicity we explicitly consider is two, namely
one dimension for the syntax and one dimension for the semantics, but the
general perspective is important. For example, issues of pragmatics could be
handled as additional dimensions.
One of the main problems addressed is the rather complicated repertoire of
operations that exists besides the notion of categories in traditional Montague
grammar. For the syntax we use a categorial grammar along the lines of Lambek.
For the semantics we use so-called lexical and logical combinators inspired by
work in natural logic. Nabla provides a concise interpretation and a sequent
calculus as the basis for implementations.Comment: 20 page
Parsing/theorem-proving for logical grammar CatLog3
CatLog3 is a 7000 line Prolog parser/theorem-prover for logical categorial grammar. In such logical categorial grammar syntax is universal and grammar is reduced to logic: an expression is grammatical if and only if an associated logical statement is a theorem of a fixed calculus. Since the syntactic component is invariant, being the logic of the calculus, logical categorial grammar is purely lexicalist and a particular language model is defined by just a lexical dictionary. The foundational logic of continuity was established by Lambek (Am Math Mon 65:154–170, 1958) (the Lambek calculus) while a corresponding extension including also logic of discontinuity was established by Morrill and ValentÃn (Linguist Anal 36(1–4):167–192, 2010) (the displacement calculus). CatLog3 implements a logic including as primitive connectives the continuous (concatenation) and discontinuous (intercalation) connectives of the displacement calculus, additives, 1st order quantifiers, normal modalities, bracket modalities, and universal and existential subexponentials. In this paper we review the rules of inference for these primitive connectives and their linguistic applications, and we survey the principles of Andreoli’s focusing, and of a generalisation of van Benthem’s count-invariance, on the basis of which CatLog3 is implemented.Peer ReviewedPostprint (author's final draft
Kleene Algebras, Regular Languages and Substructural Logics
We introduce the two substructural propositional logics KL, KL+, which use
disjunction, fusion and a unary, (quasi-)exponential connective. For both we
prove strong completeness with respect to the interpretation in Kleene algebras
and a variant thereof. We also prove strong completeness for language models,
where each logic comes with a different interpretation. We show that for both
logics the cut rule is admissible and both have a decidable consequence
relation.Comment: In Proceedings GandALF 2014, arXiv:1408.556
Hybrid Type-Logical Grammars, First-Order Linear Logic and the Descriptive Inadequacy of Lambda Grammars
In this article we show that hybrid type-logical grammars are a fragment of
first-order linear logic. This embedding result has several important
consequences: it not only provides a simple new proof theory for the calculus,
thereby clarifying the proof-theoretic foundations of hybrid type-logical
grammars, but, since the translation is simple and direct, it also provides
several new parsing strategies for hybrid type-logical grammars. Second,
NP-completeness of hybrid type-logical grammars follows immediately. The main
embedding result also sheds new light on problems with lambda grammars/abstract
categorial grammars and shows lambda grammars/abstract categorial grammars
suffer from problems of over-generation and from problems at the
syntax-semantics interface unlike any other categorial grammar
Logical ambiguity
The thesis presents research in the field of model theoretic semantics on the problem of ambiguity,
especially as it arises for sentences that contain junctions (and,or) and quantifiers (every man,
a woman). A number of techniques that have been proposed are surveyed, and I conclude
that these ought to be rejected because they do not make ambiguity 'emergent': they all have
the feature that subtheories would be able to explain all syntactic facts yet would predict no
ambiguity. In other words these accounts have a special purpose mechanism for generating
ambiguities.It is argued that categorial grammars show promise for giving an 'emergent' account. This is
because the only way to take a subtheory of a particular categorial grammar is by changing one
of the small number of clauses by which the categorial grammar axiomatises an infinite set of
syntactic rules, and such a change is likely to have a wider range of effects on the coverage of
the grammar than simply the subtraction of ambiguity.Of categorial grammars proposed to date the most powerful is Lambek Categorial Grammar,
which defines the set of syntactic rules by a notational variant of Gentzen's sequent calculus
for implicational propositional logic, and which defines meaning assignment by using the Curry-
Howard isomorphism between Natural Deduction proofs in implicational propositional logic and
terms of typed lambda calculus. It is shown that no satisfactory account of the junctions and
quantifiers is possible in Lambek categorial grammar.I introduce then a framework that I call Polymorphic Lambek Categorial Grammar, which adds
variables and their universal quantification, to the language of categorisation. The set of syntac¬
tic rules is specified by a notational variant of Gentzen's sequent calculus for quantified proposi¬
tional logic, and which defines meaning assignment by using Girard's Extended Curry-Howard
isomorphism between Natural Deduction proofs in quantified implicational propositional logic
and terms of 2nd order polymorphic lambda calculus. It is shown that this allows an account
of the junctions and quantifiers, and one which is 'emerg