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14
Logical Predicates for Intuitionistic Linear Type Theories
 In Typed Lambda Calculi and Applications (TLCA'99), Lecture Notes in Computer Science 1581
, 1999
"... We develop a notion of Kripkelike parameterized logical predicates for two fragments of intuitionistic linear logic (MILL and DILL) in terms of their categorytheoretic models. Such logical predicates are derived from the categorical glueing construction combined with the free symmetric monoidal co ..."
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Cited by 11 (4 self)
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We develop a notion of Kripkelike parameterized logical predicates for two fragments of intuitionistic linear logic (MILL and DILL) in terms of their categorytheoretic models. Such logical predicates are derived from the categorical glueing construction combined with the free symmetric monoidal cocompletion. As applications, we obtain full completeness results of translations between linear type theories.
The Shuffle Hopf Algebra and Noncommutative Full Completeness
, 1999
"... We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformati ..."
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Cited by 8 (3 self)
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We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffle algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cutfree proofs in CyLL+MIX. This can be viewed as a fully faithful representation of a free *autonomous category, canonically enriched over vector spaces. This paper
Entropic hopf algebras and models of noncommutative linear logic
 THEORY AND APPLICATIONS OF CATEGORIES 10
, 2002
"... We give a definition of categorical model for the multiplicative fragment of noncommutative logic. We call such structures entropic categories. We demonstrate the soundness and completeness of our axiomatization with respect to cutelimination. We then focus on several methods of building entropic ..."
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Cited by 5 (3 self)
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We give a definition of categorical model for the multiplicative fragment of noncommutative logic. We call such structures entropic categories. We demonstrate the soundness and completeness of our axiomatization with respect to cutelimination. We then focus on several methods of building entropic categories. Our first models are constructed via the notion of a partial bimonoid acting on a cocomplete category. We also explore an entropic version of the Chu construction, and apply it in this setting. It has recently been demonstrated that Hopf algebras provide an excellent framework for modeling a number of variants of multiplicative linear logic, such as commutative, braided and cyclic. We extend these ideas to the entropic setting by developing a new type of Hopf algebra, which we call entropic Hopf algebras. We show that the category of modules over an entropic Hopf algebra is an entropic category, (possibly after application of the Chu construction). Several examples are discussed, based first on the notion of a bigroup. Finally the TannakaKrein reconstruction theorem is extended to the entropic setting.
6 Finite Presentations of Pregroups and the Identity Problem
"... We consider finitely generated pregroups, and describe how an appropriately defined rewrite relation over words from a generating alphabet yields a natural partial order for a pregroup structure. We investigate the identity problem for pregroups; that is, the algorithmic determination of whether a w ..."
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Cited by 2 (0 self)
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We consider finitely generated pregroups, and describe how an appropriately defined rewrite relation over words from a generating alphabet yields a natural partial order for a pregroup structure. We investigate the identity problem for pregroups; that is, the algorithmic determination of whether a word rewrites to the identity element. This problem is undecidable in general, however, we give a dynamic programming algorithm and an algorithm of Oerhle (2004) for free pregroups, and extend them to handle more general pregroup structures suggested in Lambek (1999). Finally, we show that the identity problem for a certain class of nonfree pregroups is NPcomplete.
Minimalism and the Logical Structure of the Lexicon
"... This paper explores the linguistic implications of Noncommutative Linear Logic, restricted to its multiplicative fragment NMLL. In the paper particular emphasis is given to the logical representation of lexical information and of the principles of the Xbar theory. ..."
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Cited by 1 (0 self)
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This paper explores the linguistic implications of Noncommutative Linear Logic, restricted to its multiplicative fragment NMLL. In the paper particular emphasis is given to the logical representation of lexical information and of the principles of the Xbar theory.
Finite Presentations of Pregroups and the Identity Problem
, 2005
"... We consider finitely generated pregroups, and describe how an appropriately defined rewrite relation over words from a generating alphabet yields a natural partial order for a pregroup structure. We investigate the identity problem for pregroups; that is, the algorithmic determination of whether a w ..."
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We consider finitely generated pregroups, and describe how an appropriately defined rewrite relation over words from a generating alphabet yields a natural partial order for a pregroup structure. We investigate the identity problem for pregroups; that is, the algorithmic determination of whether a word rewrites to the identity element. This problem is undecidable in general, however, we give a dynamic programming algorithm and an algorithm of Oerhle (2004) for free pregroups, and extend them to handle more general pregroup structures suggested in Lambek (1999). Finally, we show that the identity problem for a certain class of nonfree pregroups is NPcomplete.
Quantifiers and Scope in Pregroup Grammar
"... In this paper we propose a geometrical representation of quantified noun phrases and their scope properties in the framework of Pregroup Grammar ([6][7][8]) and by means of the planar proof nets of noncommutative linear logic ([1][3][4]). 1 ..."
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In this paper we propose a geometrical representation of quantified noun phrases and their scope properties in the framework of Pregroup Grammar ([6][7][8]) and by means of the planar proof nets of noncommutative linear logic ([1][3][4]). 1
Under consideration for publication in Math. Struct. in Comp. Science Path Functors in Cat
, 2013
"... We build an endofunctor in the category of small categories along with the necessary structure on it to turn it into a path object suitable for homotopy theory and modelling identity types in MartinLöf type theory. We construct the free Grothendieck bifibration ..."
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We build an endofunctor in the category of small categories along with the necessary structure on it to turn it into a path object suitable for homotopy theory and modelling identity types in MartinLöf type theory. We construct the free Grothendieck bifibration