Unique perfect matchings, forbidden transitions and proof nets for linear logic with Mix

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...

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Published in:Logical methods in computer science Vol. 16, Issue 1; no. 1
Main Author: Nguyên, Lê Thành Dũng
Format: Journal Article
Language:English
Published: Logical Methods in Computer Science Association 01.02.2020
Logical Methods in Computer Science e.V
Subjects:
03f52, 68r10
Computer Science
computer science - logic in computer science
Discrete Mathematics
f.4.1
g.2.2
Logic in Computer Science
Proof nets
Perfect matching
ISSN:1860-5974, 1860-5974
Online Access:Get full text
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Abstract 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).
AbstractList This paper establishes a bridge between linear logic and mainstream graph theory, building on previous work by Retoré (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 Retoré'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).
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).
Author Nguyên, Lê Thành Dũng
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Issue 1
Keywords Proof nets
Perfect matching
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Snippet 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...
This paper establishes a bridge between linear logic and mainstream graph theory, building on previous work by Retoré (2003). We show that the problem of...
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Computer Science
computer science - logic in computer science
Discrete Mathematics
f.4.1
g.2.2
Logic in Computer Science
Title Unique perfect matchings, forbidden transitions and proof nets for linear logic with Mix
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