Constructing free Boolean categories

François Lamarche 1 Lutz Straßburger 2, 3
1 CALLIGRAMME - Linear logic, proof networks and categorial grammars
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
2 PARSIFAL - Proof search and reasoning with logic specifications
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France
Abstract : By Boolean category we mean something which is to a Boolean algebra what a category is to a poset. We propose an axiomatic system for Boolean categories, which differs in several respects from the one given very recently by Fuehrmann and Pym. In particular everything is done from the start in a *-autonomous category and not a linear distributive one, which simplifies issues like the Mix rule. An important axiom, which is introduced later, is a ''graphical'' condition, which is closely related to denotational semantics and the Geometry of Interaction. Then we show that the proof net model presented in TLCA2005 is the free ''graphical'' Boolean category in our sense. This validates our categorical axiomatization with respect to a real-life example. Another important aspect of this work is that we do not assume a-priori the existence of units in the *-autonomous categories we use. This has some retroactive interest for the semantics of linear logic, and is motivated by the properties of our TLCA proof nets with respect to units.
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François Lamarche, Lutz Straßburger. Constructing free Boolean categories. 20th Annual IEEE Symposium on Logic in Computer Science - LICS 2005, May 2005, Chicago/USA, United States. pp.209- 218, ⟨10.1109/LICS.2005.13⟩. ⟨hal-00012296v2⟩

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