1,491 research outputs found
Dualized Simple Type Theory
We propose a new bi-intuitionistic type theory called Dualized Type Theory
(DTT). It is a simple type theory with perfect intuitionistic duality, and
corresponds to a single-sided polarized sequent calculus. We prove DTT strongly
normalizing, and prove type preservation. DTT is based on a new propositional
bi-intuitionistic logic called Dualized Intuitionistic Logic (DIL) that builds
on Pinto and Uustalu's logic L. DIL is a simplification of L by removing
several admissible inference rules while maintaining consistency and
completeness. Furthermore, DIL is defined using a dualized syntax by labeling
formulas and logical connectives with polarities thus reducing the number of
inference rules needed to define the logic. We give a direct proof of
consistency, but prove completeness by reduction to L.Comment: 47 pages, 10 figure
Involutive Nonassociative Lambek Calculus: Sequent Systems and Complexity
In [5] we study Nonassociative Lambek Calculus (NL) augmented with De Morgan negation, satisfying the double negation and contraposition laws. This logic, introduced by de Grooté and Lamarche [10], is called Classical Non-Associative Lambek Calculus (CNL). Here we study a weaker logic InNL, i.e. NL with two involutive negations. We present a one-sided sequent system for InNL, admitting cut elimination. We also prove that InNL is PTIME.Zadanie „ Wdrożenie platformy Open Journal System dla czasopisma „ Bulletin of the Section of Logic” finansowane w ramach umowy 948/P-DUN/2016 ze środków Ministra Nauki i Szkolnictwa Wyższego przeznaczonych na działalność upowszechniającą naukę
From Proof Nets to the Free *-Autonomous Category
In the first part of this paper we present a theory of proof nets for full
multiplicative linear logic, including the two units. It naturally extends the
well-known theory of unit-free multiplicative proof nets. A linking is no
longer a set of axiom links but a tree in which the axiom links are subtrees.
These trees will be identified according to an equivalence relation based on a
simple form of graph rewriting. We show the standard results of
sequentialization and strong normalization of cut elimination. In the second
part of the paper we show that the identifications enforced on proofs are such
that the class of two-conclusion proof nets defines the free *-autonomous
category.Comment: LaTeX, 44 pages, final version for LMCS; v2: updated bibliograph
One-Sided Sequent Systems for Nonassociative Bilinear Logic: Cut Elimination and Complexity
Bilinear Logic of Lambek amounts to Noncommutative MALL of Abrusci. Lambek proves the cut–elimination theorem for a one-sided (in fact, left-sided) sequent system for this logic. Here we prove an analogous result for the nonassociative version of this logic. Like Lambek, we consider a left-sided system, but the result also holds for its right-sided version, by a natural symmetry. The treatment of nonassociative sequent systems involves some subtleties, not appearing in associative logics. We also prove the PTime complexity of the multiplicative fragment of NBL
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