A new correctness criterion for multiplicative non-commutative proof nets

Abstract. We introduce a new correctness criterion for multiplicative non commutative proof nets which can be considered as the non-commutative counterpart to the Danos-Regnier criterion for proof nets of linear logic. The main intuition relies on the fact that any switching for a proof net (obtained by mutilating one premise of each disjunction link) can be naturally viewed as a series-parallel order variety (a cyclic relation) on the conclusions of the proof net.

[1]  J. Lambek.  The Mathematics of Sentence Structure , 1958 .

[2]  Paul Ruet,et al.  Non-Commutative Logic I: The Multiplicative Fragment , 1999, Ann. Pure Appl. Log..

[3]  V. Michele Abrusci.  Modules in Non-communicative Logic , 1999, TLCA.

[4]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[5]  Paul Ruet,et al.  Non-commutative logic II: sequent calculus and phase semantics , 2000, Mathematical Structures in Computer Science.

[6]  Vincent Danos,et al.  The structure of multiplicatives , 1989, Arch. Math. Log..

[7]  David N. Yetter,et al.  Quantales and (noncommutative) linear logic , 1990, Journal of Symbolic Logic.

[8]  Rolf H. Möhring,et al.  Computationally Tractable Classes of Ordered Sets , 1989 .