eng
episciences.org
Logical Methods in Computer Science
1860-5974
2020-02-28
Volume 16, Issue 1
10.23638/LMCS-16(1:27)2020
5134
journal article
Unique perfect matchings, forbidden transitions and proof nets for linear logic with Mix
Lê Thành Dũng Nguyên
This paper establishes a bridge between linear logic and mainstream graph
theory, building on previous work by Retor\'e (2003). We show that the problem
of correctness for MLL+Mix proof nets is equivalent to the problem of
uniqueness of a perfect matching. By applying matching theory, we obtain new
results for MLL+Mix proof nets: a linear-time correctness criterion, a
quasi-linear sequentialization algorithm, and a characterization of the
sub-polynomial complexity of the correctness problem. We also use graph
algorithms to compute the dependency relation of Bagnol et al. (2015) and the
kingdom ordering of Bellin (1997), and relate them to the notion of blossom
which is central to combinatorial maximum matching algorithms.
In this journal version, we have added an explanation of Retor\'e's
"RB-graphs" in terms of a general construction on graphs with forbidden
transitions. In fact, it is by analyzing RB-graphs that we arrived at this
construction, and thus obtained a polynomial-time algorithm for finding trails
avoiding forbidden transitions; the latter is among the material covered in
another paper by the author focusing on graph theory (arXiv:1901.07028).
https://lmcs.episciences.org/5134/pdf
Computer Science - Logic in Computer Science
03F52, 68R10
F.4.1
G.2.2